͸͡Ίʹ୯ཻࢠྗֶͱ͸ి࣓ੴ΍Ճ଎ۭಎͳͲ͕ൃੜ͢Δి࣓৔தͷ୯ҰཻࢠͷྗֶͰ͋ΔɻՃ଎‫ث‬தΛඈߦ͢Δཻࢠ‫( ܈‬ϏʔϜ) ͷӡಈ͸ɺߏ੒ཻࢠؒͷ૬‫༻࡞ޓ‬ɺϏʔϜ͕ൃ͢Δి࣓৔Λհͨ͠‫؀‬‫ڥ‬ମ (ਅۭύΠϓ) ͱͷ૬‫༻࡞ޓ‬ɺ࢒ཹΨεʹΑΔࢄཚɺߏ੒ཻࢠͷి࣓৔ͷ์ࣹͳͲͷෳ਺ͷཁҼͷ݁Ռͱܾͯ͠·ΔɻҰൠʹՃ଎‫ث‬ͷઃ‫ܭ‬΋͘͠͸ϏʔϜௐ੔Ͱ͸͜ΕΒͷޮՌΛแ‫ׅ‬తʹߟྀͯ͠࠷దͳύϥϝʔλΛબͿඞཁ͕͋Δɻ্ʹ‫ͨ͛ڍ‬શͯͷޮՌΛಉ࣌ʹߟྀͯ͠ɺղੳత΋͘͠͸਺஋‫ʹࢉܭ‬ΑͬͯϏʔϜͷৼΔ෣͍
Λ༧ଌ͢Δ͜ͱ͸౸ఈෆՄೳͰ͋Δɻ͜ͷͨΊɺ௨ৗ͸͜ΕΒͷཁҼΛՄೳͳ‫ݶ‬Γࡉ෼Խͯ͠ɺ‫ݸ‬ผʹ‫ݕ‬౼͢Δɻͦͷࡍɺ୯ཻࢠྗֶ͸ϏʔϜͷෳࡶͳӡಈΛٞ࿦͢Δ্Ͱ‫ج‬൫ͱͳΔͨΊඞཁෆՄܽͳ΋ͷͰ͋ΔɻຊߘͰ͸୯ཻࢠྗֶͷ֓આΛࢼΈ͕ͨɺͦͷྔͷ๲େ͞ͱචऀͷྗྔෆ଍Ͱ
ॏཁͰ͸͋Δ͕औΓѻΘͳ͔߲ͬͨ໨΋਺ଟ͋͘Δ͜ͱ͸அ͓ͬͯ͘ɻ·ͨɺຊߘ͸‫׬‬શʹະࠪಡ
Ͱ͋ΔͨΊ‫ࣈޡ‬୤ࣈ΍චऀͷ‫ͨͬޡ‬ೝࣝʹ‫هͮ͘ج‬ड़͕൱ఆͰ͖ͳ͍ɻΠϯλʔωοτͰ‫͢ࡧݕ‬Ε͹୯ཻࢠྗֶʹؔ͢Δ༏Εͨϊʔτ΍ߨٛͷεϥΠυΛ‫͚ͭݟ‬ΒΕΔɻΑΓચ࿅͞Εͨ‫ࡐڭ‬Λ๬Ήํʹ͸ͦͪΒΛ͓નΊ͍ͨ͠ɻ͓ͦΒ͘೔ຊ‫࠷Ͱޠ‬΋৘ใྔ͕ଟ͍ͷࢿྉ͸ա‫ڈ‬ͷ OHO ͷςΩετ (ΠϯλʔωοτͰӾཡՄೳ) Ͱ͋Ζ͏ɻຊߘͰ͸·ͣୈ̍ষͰՃ଎‫ث‬தͷՙిཻࢠͷӡಈΛࢧ഑͢ΔϋϛϧτχΞϯʹ͍ͭͯઆ໌͢Δɻ
ୈ̎ষͰ͸ϏʔϜਐߦํ޲ʹ௚ަ͢Δԣํ޲ͷӡಈʢϕʔλτϩϯӡಈʣͷఆࣜԽΛߦ͏ɻୈ̏
ষͰ͸ϏʔϜਐߦํ޲ͷӡಈʢγϯΫϩτϩϯӡಈʣΛѻ͏ɻୈ̐ষͰ͸ඇઢ‫ֶྗܗ‬ͷಋೖͱͯ͠γϯϓϨΫςΟοΫม‫׵‬΍਺஋γϛϡϨʔγϣϯख๏ʹ͍ͭͯઆ໌͢Δɻ

2

２－ 1

ୈ 1 ষ ϋϛϧτχΞϯ

1.1

ి࣓৔தͷՙిཻࢠͷϋϛϧτχΞϯ

Α͘஌ΒΕ͍ͯΔΑ͏ʹి৔ E, ࣓৔ B தͷిՙ q ͷՙిཻࢠͷӡಈ͸dP= qE + qv × B
dt

(1.1)

ʹै͏ɻ͜͜Ͱ t, v, P , t ͸ͦΕͧΕ࣌ࠁ‫ͼٴ‬ɺཻࢠͷ଎౓ͱӡಈྔͰ͋Δɻ·ͨɺଠจࣈ͸ϕΫτϧྔΛҙຯ͢Δɻӡಈྔ P ͸ཻࢠͷ੩ࢭ࣭ྔ mɺޫ଎ c ‫ͼٴ‬ϩʔϨϯπҼࢠ β, γ ͱ‫ʹڞ‬

P

= γmv

γ = 1/ 1 − β 2 , β 2 = β · β, β = v/c

(1.2)
(1.3)

ͷؔ܎ʹ͋ΔɻҰํɺղੳྗֶͷ‫͑ڭ‬Δͱ͜ΖʹΑΕ͹ࣜ (1.1) ͸ҰൠԽ࠲ඪΛ q ɺಠཱม਺Λ tͱͨ͠ϥάϥδΞϯ L(q, q̇, t)
(1.4)
L = −mc2 1 − v 2 /c2 − qϕ + qv · A͔Βಋ͔ΕΔɻ͜͜Ͱυοτ͸࣌ؒʹΑΔඍ෼ d/dt Λҙຯ͢Δɻ·ͨεΧϥʔؔ਺ ϕ ‫ͼٴ‬ϕΫτϧؔ਺ A ͸ి࣓৔ͱE = −∇ϕ −

∂A
∂t

(1.5)

B =∇×AΛຬͨ͢ɻࣜ (1.4) ͔ΒϋϛϧτχΞϯ H(q, p, t)
p · q̇ − L = c2 (p − qA)2 + m2 c4 + qϕ
H=

(1.6)

i

͕ಘΒΕΔɻୠ͠ɺਖ਼४ӡಈྔ p = ∂L/∂v = P + qA Ͱ͋Δ͜ͱΛ༻͍ͨɻ͜ͷϋϛϧτχΞϯ
ʹରͯ͠ਖ਼४ํఔࣜ
∂H
∂H
q̇i =
, ṗi = −
(1.7)
∂pi
∂qi
Λ۩ମతʹࣜม‫͢ܗ‬ΔͱɺͦΕ͸౰વͳ͕Βࣜ (1.1) ʹ‫ؼ‬ண͢Δɻ͜͜·Ͱ͸Ճ଎‫ث‬ͷ෼໺ʹ‫ݶ‬Βͣɺղੳྗֶ΍ి࣓‫ֶؾ‬ͷ‫ڭ‬Պॻʹ΋ॻ͔Ε͍ͯΔ͘͝Ұൠతͳ࿩Ͱ͋Δɻ݁‫ہ‬ɺՃ଎‫ث‬தͷ୯ཻࢠͷӡಈΛ༧ଌ͢ΔͨΊʹզʑ͕͢Δ͜ͱ͸ɺࣜ (1.5) Λղ͍ͯి࣓৔΋͘͠͸ϙςϯγϟϧΛ‫ٻ‬Ίɺࣜ (1.1) ΋͘͠͸ࣜ (1.7) Λղ͘͜ͱͰ͋ΔɻຊߘͰ͸ϋϛϧτχΞϯ (1.6) ͱਖ਼४ํఔࣜ (1.7) Λϕʔεʹ͢Δɻॏཁͳ͜ͱ͸୯ཻࢠྗֶͰ͸Ճ଎‫ث‬෼໺ʹಛ༗ͷࣄ৘Λ൓өͨ͠࠲ඪม‫׵‬΍෺ཧྔΛಋೖ͢Δ
͜ͱʹ͋Δɻ΋͏গ͠۩ମతʹड़΂Δͱɺ͋Δ‫ج‬४ͱͳΔཻࢠʢࢀরཻࢠʣͷ‫ي‬ಓʹԊͬͨ࠲ඪ‫ʹܥ‬ҠΓɺͦͷࢀরཻࢠ͔ΒͷζϨͱͯ͠ண໨͍ͯ͠ΔཻࢠͷӡಈΛٞ࿦͢ΔɻͦͷఆࣜԽʹࡍͯ͠ϕʔλؔ਺ɺνϡʔϯɺ෼ࢄؔ਺ɺӡಈྔѹॖ཰ɺΤϛολϯεɺ৭ऩࠩ (ΫϩϚςΟγςΟ)ͳͲͷύϥϝʔλ͕ಋೖ͞ΕΔɻຊߘͰ͸͜ΕΒͷύϥϝʔλͷҙຯ΍ಛੑΛ֓આ͠ɺ࣮ࡍͷઃ‫ܭ‬΍ӡసʹ͓͍ͯ͜ΕΒͷ‫ݴ‬༿͕ͲͷΑ͏ͳҙຯͰ༻͍ΒΕΔ͔Λ‫͍ͯ͘ݟ‬ɻ

3

２－ 2

1.2

࠲ඪ‫ ܥ‬ʙFrenet ඪߏʙ

r0

ey

A

B
C

es

ex

ਤ 1.1: Frenet ඪߏҰൠతͳσΧϧτ࠲ඪ (x, y, z) Ͱ͸ͳ͘ਤ 1.1 ʹࣔͨ͠ Frenet ඪߏ͕޿͘༻͍ΒΕΔɻ͜Ε͸ࡾ࣍‫ۭؒݩ‬Λඈߦ͢Δཻࢠͷ‫ي‬ಓΛ௥੻͢ΔͷʹΑΓศརͳͨΊͰ͋Δɻ‫ج‬४ͱͳΔ఺ A ͱ B Λ݁Ϳ‫׈‬Β͔ͳ‫ۂ‬ઢ C Λߟ͑ɺͦͷ‫ۂ‬ઢ্ͷ௚ަ͢Δ̏ͭͷϕΫτϧɺ୯Ґ઀ઢϕΫτϧ es ɺ୯Ґओ๏ઢϕΫτϧ ex ɺ୯Ґै๏ઢϕΫτϧ ey Λߟ͑Δɻண໨͍ͯ͠ΔཻࢠͷҐஔϕΫτϧ r ͸ r0Λ‫ۂ‬ઢ C ্ͷҐஔʢ‫ج‬४ཻࢠͷҐஔʣϕΫτϧͱͯ͠r = r0 + xex + yey(1.8)

ͱͳΔɻ‫ج‬४ཻࢠͷબͼํ͸ཻࢠͷ‫ۂ‬ઢ͕‫׈‬Β͔Ͱཻࢠͷ࠲ඪ͕ҰҙʹܾఆͰ͖Δ‫ݶ‬Γ͸೚ҙͰ͋ΔɻFrenet ඪߏʹ͓͍ͯ͸‫ج‬ఈϕΫτϧ ex,y,s ʹରͯ͠ҎԼͷ Frenet-Serret ͷެ͕ࣜ੒ཱ͢Δɻ

 


es
0 −κ 0
es
d 
 


(1.9)
 ex  =  κ 0 −τ   ex 
ds
ey
ey
0 τ
0

͜͜Ͱ κ, τ ͸‫ۂ‬ઢ C ͷ‫ͼٴ཰ۂ‬೧཰Ͱ͋Δɻ‫཰ۂ‬͸จࣈ௨Γ‫ۂ‬ઢͷ‫͕ۂ‬Γ۩߹Λఆྔతʹࣔ͢ྔͰ͋Γɺͦͷ‫਺ٯ‬͸‫཰ۂ‬൒‫ ܘ‬ρ Ͱ͋ΔɻҰํɺᎇ཰ͱ͸‫ۂ‬ઢ͕ฏ໘͔ΒͲΕ͚ͩ‫ٸ‬଎ʹ཭Ε͍͔ͯ͘Λද͢ྔͰ͋Δɻ͜ΕΒͷೋͭͷྔΛܾఆ͢Δ͜ͱͰ‫ۂ‬ઢ C ΛҰҙʹܾఆ͢Δ͜ͱ͕Ͱ͖ΔɻҎԼͰ͸؆୯ͷͨΊ೧཰Λθϩͱ͢Δɻ͜Ε͸‫ج‬४‫ۂ‬ઢΛ͋Δฏ໘಺ʹ‫ݶ‬ఆ͢Δ͜ͱΛҙຯ͢Δɻ
·ͨɺ‫ج‬४‫ۂ‬ઢͱͯ͠͸ઃ‫ࢠཻܭ‬ͷ‫ي‬ಓΛબͿ͜ͱ͕ଟ͍ͨΊɺҎԼͰ͸‫ج‬४‫ۂ‬ઢ͓Αͼ‫ج‬४ཻࢠΛઃ‫يܭ‬ಓɺઃ‫ͱࢠཻܭ‬শ͢ɻͯ͞ɺ͜ͷ࠲ඪ‫ܥ‬͸࣌ؒͷ‫ܦ‬աͱͱ΋ʹಈ͘࠲ඪ‫͋Ͱܥ‬ΔɻҰൠͷྗֶͷ‫ڭ‬ՊॻͰ͸ཻࢠͷӡಈΛ‫ه‬ड़͢Δࡍɺಠཱม਺Λ࣌ࠁ t ʹબͿ͜ͱ͕ଟ͍͕ɺՃ଎‫ث‬෼໺Ͱ͸͜ͷઃ‫يܭ‬ಓͷ௕͞ s(఺A ͔ΒଌͬͨಓͷΓ) Λಠཱม਺ʹબͿͷ͕௨ৗͰ͋Δɻಠཱม਺͕ s Ͱ͋Δ͜ͱΛ͋ΒΘʹද‫ݱ‬͢Δͱࣜ (1.8) ͸r(s) = r0 (s) + xex (s) + yey (s)(1.10)
ͱͳΔɻಠཱม਺Λ s ʹબͿͷ͸ϏʔϜϥΠϯ͸ͦͷߏ੒ཁૉ͕৔ॴ s ʹΑͬͯ۠෼͞Ε͓ͯΓɺཧ࿦తऔΓѻ͍͕ศརʹͳΔ͜ͱ͕ଟ͍ͨΊͰ͋Δɻ͜ͷ৔߹ɺϏʔϜਐߦํ޲ͷӡಈʹؔͯ͠͸ཻࢠ͕͋Δ࣌ࠁʹ্ۭؒͷͲ͜ʹҐஔ͢Δ͔Ͱ͸ͳ͘ɺ͋Δ৔ॴ s ʹ౸ୡ͢Δ࣌ࠁΛٞ࿦ͯ͠
͍Δ͜ͱʹͳΔɻ

1.3

Ճ଎‫࢖Ͱث‬ΘΕΔϋϛϧτχΞϯ

લઅͰఆٛͨ͠࠲ඪ‫Ͱܥ‬ϋϛϧτχΞϯ (1.6) Λॻ͖௚͢ɻͦͷͨΊʹҎԼͷ฼ؔ਺ʹΑΔਖ਼४ม‫׵‬Λߟ͑Δɻͳ͓ɺ฼ؔ਺΍ਖ਼४ม‫ݴ͏͍ͱ׵‬༿ʹෆ‫׳‬Εͨಡऀ͸ղੳྗֶͷ‫ڭ‬Պॻ΋͘͠͸

4

２－ 3

ຊߘͷ෇࿥Λࢀর͞Ε͍ͨɻ

F1 (p, x, y, s) = −p · [r0 (s) + xex (s) + yey (s)](1.11)

͜ΕʹΑΓਖ਼४ӡಈྔ͸

∂F1= p · ex ,px = −∂x∂F1= p · ey ,py = −∂y∂F1=
ps = −
∂s



x
1+
ρ



ͱͳΔɻϕΫτϧϙςϯγϟϧ A = (Ax , Ay , Az ) ʹ΋ಉ༷ͷม‫׵‬x
A · esAx = Ax · ex , Ay = Ay · ey , As = 1 +ρ

p · es(1.12)(1.13)

Λࢪ͢ɻ৽ͨͳϋϛϧτχΞϯ H1 ͸

H1 (x, px , y, py , s, ps ; t) = H +

∂F1
∂t


= qϕ + c(1.14)

m2 c2 + (px − qAx )2 + (py − qAy )2 +(ps − qAs )2(1 + x/ρ)2(1.15)

ͱͳΔɻ͜͜Ͱಠཱม਺Λ t ͔Β s ʹมߋ͢Δɻ࣌ࠁ t ͕ϋϛϧτχΞϯ H1 ͷ͕̐ͭΊͷࣗ༝౓
Ͱ͋Δͱ‫ݟ‬၏͢ͱࣜ (1.7) ͔Βɺͦͷ‫ڞ‬໾ͳӡಈྔ͸ −H1 Ͱ͋Δɻैͬͯɺࣜ (1.14) Λ −ps ʹ͍ͭͯղ͘͜ͱͰ৽ͨͳϋϛϧτχΞϯ

H2 (x, px , y, py , t, pt ; s)
= −qAs −



x
1+
ρ

 

pt + qϕc

2− m2 c2 − (px − qAx )2 − (py − qAy )2(1.16)

ΛಘΔɻୠ͠ɺpt = −H1 ͱද‫ͨ͠ه‬ɻզʑ͕͢Δ΂͖͜ͱ͸ɺՃ଎‫ث‬தͷ৔ॴ s ͕༩͑ΒΕͨ࣌ʹͦͷ৔ॴʹରԠ͢Δి࣓৔ϙςϯγϟϧ (ϕ, A) ͷදࣜΛ H2 ʹ༩͑ͯਖ਼४ํఔࣜ (1.7) Λղ͘͜ͱͰ͋Δɻ࢒೦ͳ͕Β͜ͷ໰୊͸͘͝‫ݶ‬ΒΕͨ৔߹Ͱ͔͠ղੳతʹղ͚ͳ͍ɻͦΕͲ͜Ζ͔ి࣓৔ϙςϯγϟϧ (ϕ, A) ͷղੳతͳද
ࣜ͢Β΋ཧ૝Խ͞Εͨ৚݅ԼͰ͔͠ಘΔ͜ͱ͕Ͱ͖ͳ͍ɻి࣓৔ϙςϯγϟϧ (ϕ, A) ͸Ұൠʹ͸ϏʔϜϥΠϯʹઃஔ͞Εͨి࣓ੴͳͲͷ૷ஔ͔Βͷ‫د‬༩ʹՃ͑ͯϏʔϜࣗମͷిՙ͕ൃੜ͢Δి࣓৔΋‫ؚ‬ΜͰ͍ΔɻຊߘͰ͸୯ཻࢠͷྗֶΛѻ͏ͷͰ͜ΕΒͷ‫د‬༩͸ແࢹ͢Δɻ͜ͷ৔߹Ͱ΋ɺྫ͑͹੩ిϨϯζ͕͋ΔϏʔϜϥΠϯͰ͸εΧϥʔϙςϯγϟϧ ϕ ͕ଘࡏ͢Δ͕ɺຊߘͰ͸؆୯ͷͨΊओʹి࣓ੴ͕ൃੜ͢Δ੩࣓৔Λߟ͑ ϕ = 0 ͱ͢ΔɻϋϛϧτχΞϯ H2 Ͱ ϕ = 0 ͱ͢Δͱཻ
ࢠͷΤωϧΪʔ E ͕ E = −pt Ͱ͋Δ͜ͱΛ༻͍ͯ৽͍͠ϋϛϧτχΞϯͱͯ͠

H3 (x, px , y, py , t, −E; s)  2xE
= −qAs −1+− m2 c2 − (px − qAx )2 − (py − qAy )2
ρ
c


x
= −qAs −1+p2 − (px − qAx )2 − (py − qAy )2(1.17)
ρΛಘΔɻ͜͜Ͱɺશྗֶతӡಈྔ p(E) ≡ E 2 /c2 − m2 c2 Λఆٛͨ͠ɻ


5

２－ 4

‫ͳີݫ‬ղ͕ಘΒΕΔ਺গͳ͍ྫͷҰ͕ͭࣗ༝ۭؒͰ͋Δɻઃ‫يܭ‬ಓ͸௚ઢͰ͋Δ͔Β 1/ρ = 0Ͱ͋ΓɺϕΫτϧϙςϯγϟϧ͸ A = 0 Ͱ͋Δɻ͜ͷΑ͏ͳ۠ؒΛυϦϑτεϖʔεͱ΋͘͠͸୯ʹυϦϑτশ͢ɻϋϛϧτχΞϯ H ͸ H3 ʹ ϕ = 0, A = 0, 1/ρ = 0 Λ୅ೖͯ͠
H = − p2 − p2x − p2y(1.18)
ͱͳΔɻਖ਼४ํఔࣜ͸

x′ =
y′ =



t′ =




px
p2 − p2x − p2ypyp2 − p2x − p2yE/c2

p2 − p2x − p2y,p′x = 0(1.19),

p′y = 0(1.20),− E′ = 0(1.21)

ैͬͯɺ௕͞ L ͷυϦϑτεϖʔεʹ࢝఺ s = s1 Ͱ (x1 , px1 , y1 , py1 , z1 , −E1 ) ͷཻࢠ͕ೖࣹ͞ΕΔͱऴ఺ s = s2 = s1 + L ʹ͓͍ͯpx
x 2 = x1 + × L,
px2 = px1 ,(1.22)
2p − p2x − p2ypyy2 = y1 + 

p2 − p2x − p2y

× L,

E/c2
× L,
t2 = t1 + 
p2 − p2x − p2ypy2 = py1 ,(1.23)− E2 = −E1 ,(1.24)

ͱͳΔɻࣜ (1.22)-(1.24) ͸ (x1 , px1 , y1 , py1 , t1 , −E1 ) ͔Β (x2 , px2 , y2 , py2 , t2 , −E2 ) ΁ͷ͋Δछͷࣸ૾ͱߟ͑Δ͜ͱ͕Ͱ͖Δɻਖ਼४ํఔࣜΛղ͘͜ͱ͸ϋϛϧτχΞϯʹΑܾͬͯ·ΔҐ૬ۭؒ࠲ඪ
x = (x, px , y, py , t, −E) ͷϏʔϜϥΠϯೖΓ‫ͱޱ‬ग़‫ޱ‬ͷؒͷࣸ૾ M

xs=s2 = M ◦ xs=s1(1.25)

Λ‫ٻ‬ΊΔ͜ͱͰ͋Δɻ͜͜Ͱະͩఆٛ͞Ε͍ͯͳ͍ԋࢉΛ ◦ ͱද‫ͨ͠ه‬ɻࣸ૾ (1.22)-(1.24) ͕ͦ͏Ͱ͋ΔΑ͏ʹҰൠʹ M ͸ඇઢ‫ܗ‬ͷࣸ૾Ͱ͋Δɻ࣍ͷઅͰ͸ϋϛϧτχΞϯʹઁಈల։Λࢪͯࣸ͠૾ M Λઢ‫͢ࣅۙʹ૾ࣸܗ‬Δɻ͜ͷۙࣅԼͰ͸ M ͸ϋϛϧτχΞϯͰܾ·Δ 6 × 6 ͷߦྻͱͳΓɺࣜ (1.25) ͸ߦྻԋࢉͱͳΔɻ͜Ε͸୯ཻࢠྗֶʹ͓͍ͯઢ‫ܗ‬ཧ࿦ɺઢ‫ͱͲͳֶྗܗ‬শ͞ΕΔ
ΞϓϩʔνͰ͋Δɻઢ‫ܗ‬ཧ࿦ͷఆࣜԽʹ༻͍ΒΕΔߟ͑ํ͸ඇઢ‫ֶྗܗ‬Λߟ͑Δ্Ͱ‫ج‬൫ͱͳΔɻ

1.4

ϋϛϧτχΞϯͷۙࣅઃ‫͔ࢠཻܭ‬ΒඍখʹͣΕཻͨࢠͷӡಈΛߟ͑ɺϋϛϧτχΞϯ (1.17) ͷฏํࠜΛల։ۙࣅ͢Δɻ·ͨઃ‫ࢠཻܭ‬ͷΤωϧΪʔ͸Ճ଎ۭಎͷޮՌʹΑͬͯҰൠʹ͸ s ͷؔ਺Ͱ͋Δ͕ɺຊߘͰ͸அΓͷͳ͍‫ݶ‬Γɺઃ‫ࢠཻܭ‬ͷΤωϧΪʔ͕৔ॴ s ʹґଘ͠ͳ͍͔ͦͷґଘੑ͕ແࢹͰ͖Δൣғʹٞ࿦Λ‫ݶ‬ఆ͢Δɻԣํ޲ͷྗֶతӡಈྔ͕શྗֶతӡಈྔΑΓे෼খ͍͞ɺ͢ͳΘͪɺϋϛϧτχΞ
ϯ H3 ͷฏํࠜͷதͷୈ 2,3 ߲͕ୈ߲̍ʹൺ΂ͯे෼ʹখ͍͞ͱԾఆͯ͠ɺ

H4 (x, px , y, py , t, −E; s) ∼





x
x 1 
− 1+
p +
1+
(px − qAx )2 + (py − qAy )2 − qAs
ρ
ρ 2p
6

２－ 5

(1.26)

ΛಘΔɻӡಈྔͷͣΕ ∆p = p − p0 ͸ΤωϧΪʔͷͣΕ ∆E = E − E0 Λ༻͍ͯۙࣅతʹ

1
∆E
∆p
− 2
= 2
p0
β0 E0 2γ0∆E
β02 E02+ ···(1.27)

ͱ͋Δ͜ͱΛ༻͍ͯɺਖ਼४ม਺ʹؔͯ̎࣍͠ͷ߲·ͰΛ࢒͢ͱ





x
p0
∆E
x ∆E
∆E 2
p0 − p0 2
H4 (x, px , y, py , t, −E; s) ∼ − 1 +
+ 2
− p0
ρ
ρ β02 E0
β0 E0 2γ0 β02 E01 (px − qAx )2 + (py − qAy )2 − qAs+
2p0(1.28)

ͱͳΔɻ
࣮༻తʹ͸ઃ‫͔ࢠཻܭ‬ΒͷζϨΛٞ࿦͢ΔͨΊʹਖ਼४࠲ඪΛՃ͑Δ͜ͱ͕ଟ͍ɻୠ͠ɺ౰વͳ
͕Βͦͷબͼํ͸ҰҙͰ͸ͳ͘ɺՃ଎‫ث‬ͷதͰ΋෼໺΍ਓʹΑͬͯ·ͪ·ͪͰ͋Δɻ·ͣਐߦํ޲ (t, −E) Λઃ‫( ࢠཻܭ‬t0 = s/β0 c, −E0 ) ͔ΒͷζϨʹม͑Δɻ฼ؔ਺s− t (E0 + ∆E)
F4 =(1.29)
β0 cʹΑͬͯઃ‫͔ࢠཻܭ‬ΒͷζϨ (−∆t, ∆E) ʹม‫͢׵‬Δɻ

−∆t = −(t − t0 ),∆E = E − E0

(1.30)

ม‫ޙ׵‬ͷϋϛϧτχΞϯ Ĥ4 = H4 + ∂F4 /∂s ͸ ∂F4 /∂s ͷ‫د‬༩͔Βࣜ (1.28) ͷୈೋ߲͕ফ͑Δɻ͞ΒʹҎԼͷεέʔϦϯάΛߦ͏ɻ

H5 = Ĥ4 /p0 ,p̂x = px /p0 ,p̂y = py /p0 ,ˆ = ∆E/p0∆E(1.31)

৽͍͠ϋϛϧτχΞϯ͸


2

ˆ
ˆ
x
x ∆E
∆E
1
ˆ s) ∼ − 1 ++ 2
H5 (x, p̂x , y, p̂y , −∆t, ∆E;−
ρρ β0 c2γ0 β0 c1+(p̂x − ax )2 + (p̂y − ay )2 − as2(1.32)

ͱͳΔɻ͜͜Ͱɺa ≡ qA/p0 ͱͨ͠ɻ͞Βʹਐߦํ޲ʹؔͯ͠ਖ਼४࠲ඪ ∆t ͷ࣍‫ݩ‬Λԣํ޲ͷ࣍‫ͱݩ‬ಉ͡ʹ͢ΔͨΊʹεέʔϦϯάΛࢪ͢ɻz = −β0 γ0 c∆t,

pz =

ˆ
∆E
1 ∆p
∆E
=
∼
β0 γ0 cγ0 β0 cp0
γ 0 p0(1.33)

ਖ਼४ม਺ΛվΊͯ (z, px , y, py , z, pz ) ͱද‫ࣅۙͯ͠ه‬ల։͞ΕͨϋϛϧτχΞϯ͸



x
1
γ0
− xpz +
(px − ax )2 + (py − ay )2 + p2z − as (1.34)
H(x, px , y, py , z, pz ; s) = − 1 +ρρ2ͱͳΔɻಛʹਐߦํ޲ͷਖ਼४ม਺ͷબͼํ͸༷ʑͰ͋Δɻ෼໺ԣஅతʹ‫ڀݚ‬Λͨ͠Γɺෳ਺ͷ਺஋γϛϡϨʔγϣϯίʔυΛ࢖͏৔߹ʹ͸ɺ࠲ඪͷఆٛʹ஫ҙ͞Ε͍ͨɻ

7

２－ 6

1.5

୅දతͳి࣓ੴͷ࣓৔

લઅ·Ͱʹઢ‫ܗ‬ཧ࿦ʹඞཁͳۙࣅ͞ΕͨϋϛϧτχΞϯ (1.34) Λಘͨɻ͜͜Ͱ͸୅දతͳϏʔϜϥΠϯͷߏ੒ཁૉʹରཻͯ͠ࢠͷӡಈΛఆੑతʹ‫͍ͯ͘ݟ‬ɻ·ͣɺσΧϧτ࠲ඪ (x, y, z) ʹ͓͍ͯࣗ༝ۭؒͷϚΫε΢ΣϧํఔࣜΛߟ͑ΔɻຊߘͰ͸ϏʔϜਐߦํ޲ͷ࣓৔ Bz = 0 ͱ͢Δɻ͜ͷ৔߹ɺ࣓৔͸ (x, y) ฏ໘ͷೋ࣍‫ݩ‬ͷ໰୊Ͱ‫ه‬ड़Ͱ͖Δɻ࣓৔ (Bx , By ) Λ‫਺ڏ‬୯Ґ i ͱͯ͠

By + iBx = B0∞(bn + ian ) (x + iy)n(1.35)

n=0

1 ∂By|x=y=0 ,bn =B0 n! ∂x1 ∂Bx
|x=y=0an =B0 n! ∂x

ͱ x, y ͷଟ߲ࣜͱͯ͠ද‫͢ݱ‬Δ͜ͱΛ࣓৔ͷଟॏ‫ۃ‬ల։ͱ‫Ϳݺ‬ɻ͜͜Ͱ bn ΛϊʔϚϧ੒෼ɺan ΛεΩϡʔ੒෼ͱ‫Ϳݺ‬ɻεΩϡʔ੒෼ͷ࣓৔ͱ͸ϊʔϚϧ੒෼ͷ࣓৔Λ‫఺ݪ‬पΓʹ 90/(n + 1) ౓ճసͨͨ͠͠΋ͷͰ͋ΔɻՃ଎‫Ͱث‬͸ಛఆͷ bn ΋͘͠͸ an ͷΈ͕θϩͰͳ͍࣓৔Λൃੜ͢Δి࣓ੴΛ࢖ͬͯϏʔϜϥΠϯΛઃ‫͢ܭ‬Δɻ1 ௨ৗɺεΩϡʔ࣓৔ͷΈൃੜ͢Δి࣓ੴΛεΩϡʔ࢛‫ిۃ‬࣓ੴɺεΩϡʔ࿡‫࣓ిۃ‬ੴͳͲͱ‫͕Ϳݺ‬ɺϊʔϚϧ੒෼ͷΈൃੜ͢Δి࣓ੴ͸୯ʹ࢛‫࣓ిۃ‬ੴɺ࿡‫࣓ిۃ‬ੴͱϊʔϚϧͱ͍͏‫ݴ‬༿Λলུ͢Δ͜ͱ͕ଟ͍ɻຊߘͰ΋ಛʹஅΓͷͳ͍‫ݶ‬Γɺ͜ͷ‫׳‬शʹै͏ɻ·ͨҎԼͰ͸ɺೋ‫ۃ‬੒෼ b0 ͕ b0 = 1 ͱͳΔΑ͏ʹ B0 ΛબͿ͜ͱʹ͠ɺిՙ q ͕ਖ਼ͷཻࢠͷӡಈΛߟ͑Δɻ

1.5.1

ภ޲ి࣓ੴ (n = 0)
(a)

(b)

1.00
0.75

Vertical position y

0.50
0.25

B

0.00



F

qv

−0.25
−0.50
−0.75
−1.00
−1.0

−0.5

0.0

0.5

1.0

Horizontal position x

ਤ 1.2: (a) ภ޲ి࣓ੴͷ࣓ྗઢɺཻࢠ͕‫͡ײ‬Δ࣓৔ B ‫࣓ͼٴ‬৔͔Βड͚Δྗ F ɻ(b)SuperKEKB
ཅిࢠμϯϐϯάϦϯϦϯάͷภ޲ి࣓ੴɻ
ਤ 1.2 ʹࣔͨ͠Α͏ʹೋͭͷ࣓‫ۃ‬Λ΋ͭి࣓ੴͰ͋ΓɺԖ௚ํ޲ʹҰ༷ͳ࣓৔Λൃੜ͢Δɻཧ૝
తͳภ޲ి࣓ੴ͸B = (Bx , By ) = B0 (0, 1)(1.36)
Ͱ͋ΓɻϏʔϜͷ‫ي‬ಓ͸ਫฏํ޲ʹ‫͛ۂ‬ΒΕΔɻ௨ৗɺઃ‫يܭ‬ಓ͸͜ͷภ޲ి࣓ੴͷύϥϝʔλͱͦͷฒͼʹΑܾͬͯ·Δɺਫฏํ޲ʹҰ༷࣓৔Λൃੜ͠ɺϏʔϜΛԖ௚ํ޲ʹ‫͛ۂ‬Δภ޲ి࣓ੴ΋ଘࡏ͢Δ (εΩϡʔೋ‫࣓ۃ‬৔)ɻ·ͨɺઃ‫يܭ‬ಓΛܾΊΔภ޲ి࣓ੴͷଞʹϏʔϜӡసʹ͓͍
1ແ࿦ɺ‫࣮ݱ‬తʹ͸ෆཁͳ੒࣓৔෼͕ൃੜͯ͠͠·͏ɻ

8

２－ 7

ͯϏʔϜ‫ي‬ಓΛ੍‫͢ޚ‬ΔͨΊʹ༻͍ΒΕΔิॿతͳภ޲ి࣓ੴ΋ଘࡏ͢Δɻ͜ͷΑ͏ͳௐ੔༻ͷখ‫ܕ‬ภ޲ి࣓ੴ͸εςΞϦϯά (Steering magnet) ͱ‫ݺ‬͹Εɺ࣮ࡍͷϏʔϜӡసʹ͸ඞཁෆՄܽͳ΋ͷͰ͋Δɻ

1.5.2

࢛‫࣓ిۃ‬ੴ (n = 1)
(a)

(b)

1.00

F
0.75

qv

Vertical position y

0.50

B

0.25

qv
0.00

B

F

−0.25
−0.50
−0.75
−1.00
−1.0

−0.5

0.0

0.5

1.0

Horizontal position x

ਤ 1.3: (a) ࢛‫࣓ిۃ‬ੴͷ࣓ྗઢ (B0 b1 > 0)ɺཻࢠ͕‫͡ײ‬Δ࣓৔ B ‫࣓ͼٴ‬৔͔Βड͚Δྗ F ɻ(b)SuperKEKB ཅిࢠμϯϐϯάϦϯάͷ࢛‫࣓ిۃ‬ੴɻ௨ৗϏʔϜΛߏ੒͢Δཻࢠ‫܈‬͸Ґ૬্ۭؒͰ෼෍Λ΋ͭͨΊɺภ޲ి࣓ੴΛฒ΂͚ͨͩͰ͸શͯͷߏ੒ཻࢠΛ҆ఆʹ༌ૹͰ͖ͳ͍ɻઃ‫͔ࢠཻܭ‬ΒҐஔ͕ͣΕཻͨࢠʹ෮‫ྗݩ‬Λ༩͑ΔͨΊʹ࢛‫࣓ిۃ‬ੴ͕࢖ΘΕΔɻཧ૝తͳ࢛‫࣓ిۃ‬ੴͷൃੜ͢Δ࣓৔͸

B = (Bx , By ) = B0 b1 (y, x)(1.37)

Ͱ͋Δɻਤ 1.3(a) ʹࣔͨ͠Α͏ʹઃ‫يܭ‬ಓ͔ΒζϨͯ௨ա͢Δཻࢠ͕ड͚ΔϩʔϨϯπྗΛߟ͑Δͱɺ΋͠ B0 b1 > 0 ͋Ε͹ɺx ํ޲ʹζϨཻͨࢠ͸෮‫ྗݩ‬Λɺy ํ޲ʹζϨͨ৔߹͸ൃࢄྗΛड
͚Δ͜ͱ͕Θ͔ΔɻҰํɺB0 b1 < 0 Ͱ͋Ε͹෮‫ྗࢄൃͱྗݩ‬͸൓స͢Δɻͭ·Γ࢒೦ͳ͕ΒͲͪΒͷํ޲ʹ΋෮‫ྗݩ‬Λಇ͔ͤΔ͜ͱ͸Ͱ͖ͳ͍ɻͱ͜Ζ͕‫ݟͰޙ‬ΔΑ͏ B0 b1 > 0 ͱ B0 b1 < 0 ͷ࢛‫࣓ిۃ‬ੴΛద੾ʹ഑ஔ͢Δ͜ͱͰ҆ఆʹϏʔϜΛ༌ૹͰ͖Δɻ͜Ε͸‫ڧ‬ऩଋͷ‫ݪ‬ཧͱ‫ݺ‬͹ΕΔՃ଎‫ث‬ՊֶΛࢧ͑Δॏཁͳ‫ݪ‬ཧͰ͋Δɻ‫ڧ‬ऩଋͷ‫ݪ‬ཧ͸Ϩϯζޫֶ‫͍͓ͯʹܥ‬ऩଋ‫ܕࢄൃͱܕ‬ͷϨϯζΛద੾ʹ഑ஔ͢Δ͜ͱͰࡱ૾‫ܥ‬Λ૊Ή͜ͱ͕Ͱ͖Δ͜ͱʹྨࣅ͍ͯ͠Δɻ͜ͷޫֶ‫ͱܥ‬ͷྨࣅੑ͔Β B0 b1 > 0, B0 b1 < 0 ͷి࣓ੴΛͦΕͧΕऩଋ‫ܕ‬ɺൃࢄ‫ܕ‬ͷ࢛‫࣓ిۃ‬ੴͱ‫͕ͱ͜Ϳݺ‬ଟ͍ɻ͞ΒʹϏʔϜϥΠϯͷઃ‫ܭ‬΍҆ఆੑΛٞ࿦͢Δ෼໺͸ϏʔϜޫֶ (Beam optics) ͱ‫ݺ‬͹ΕΔ͜ͱ͕͋Δɻ

1.5.3

࿡‫࣓ిۃ‬ੴ (n = 2)࢛‫࣓ిۃ‬ੴͰ͸Ґஔ͕ζϨཻͨࢠʹରͯ͠ϩʔϨϯπྗΛߟ͕͑ͨɺϏʔϜߏ੒ཻࢠ͸Ґஔ͚ͩͰͳ͘ΤωϧΪʔํ޲ʹ΋֦͕ΓΛ΋ͭɻैͬͯɺ౰વͳ͕ΒϩʔϨϯπྗͷେ͖͞͸ཻࢠͷӡಈྔʹ΋ґଘ͢Δɻৄࡉ͸‫͕͍ͯ͘ݟʹޙ‬࿡‫࣓ిۃ‬ੴΛ࢖͏͜ͱͰɺӡಈྔͷζϨʹ‫ى‬Ҽ͢Δ࢛‫࣓ిۃ‬ੴ͔Β‫͡ײ‬ΔϩʔϨϯπྗͷաෆ଍෼Λิ͏͜ͱ͕Ͱ͖Δɻਤ 1.4 ʹ࿡‫࣓ిۃ‬ੴͷ࣓ྗઢ

9

２－ 8

(a)

(b)

1.00
0.75

Vertical position y

0.50
0.25
0.00

−0.25
−0.50
−0.75
−1.00
−1.0

−0.5

0.0

1.0

0.5

Horizontal position x

ਤ 1.4: (a) ࿡‫࣓ిۃ‬ੴͷ࣓ྗઢɻ(b)SuperKEKB ཅిࢠμϯϐϯάϦϯϦϯάͷ࿡‫࣓ిۃ‬ੴɻͱ SuperKEKB ཅిࢠμϯϐϯάϦϯάͷ࿡‫࣓ిۃ‬ੴͷࣸਅΛࣔ͢ɻཧ૝తͳ࿡ి࣓ੴͷൃੜ͢Δ࣓৔͸
B = (Bx , By ) = B0 b2 (2xy, x2 − y 2 )(1.38)
Ͱ͋Δɻཻࢠͷ‫͡ײ‬ΔϩʔϨϯπྗ͸ҐஔͷζϨʹؔͯ͠ඇઢ‫ͳͱܗ‬ΔɻͦͷҙຯͰ͸ਖ਼֬ʹ͸ઢ‫ܗ‬ཧ࿦ͷ࿮Λ֎ΕΔ͕ϏʔϜϥΠϯʹෆՄܽͳߏ੒ཁૉͷҰͭͰ͋Δɻ

1.6

࣓৔ͷϕΫτϧϙςϯγϟϧ

લઅͰ୅දతͳϏʔϜϥΠϯͷߏ੒ཁૉͷ࣓৔Λ‫͕͖ͨͯݟ‬ɺզʑ͕ϋϛϧτχΞϯʹ༩͍͑ͨͷ͸࣓৔Ͱͳ͘ϕΫτϧϙςϯγϟϧ A Ͱ͋Δɻ‫ݹ‬యి࣓‫ֶؾ‬ͷ‫͑ڭ‬Δͱ͜ΖͰ͸ϕΫτϧϙ
ςϯγϟϧͷબͼํ͸Ұҙʹ͸ܾ·Βͣɺήʔδͷ෼͚ͩෆఆੑ͕͋Δɻ௨ৗɺήʔδͱͯ͠Ϋʔϩϯήʔδ ∇ · A = 0 Λ࠾༻͢Δ͜ͱ͕ଟ͍ɻΫʔϩϯήʔδΛ࠾༻͢Δ͜ͱͰࣜ (1.35) ͷ࣓৔Λ As ੒෼ͷΈͰද‫͖Ͱݱ‬ΔͨΊͰ͋Δɻ
·ͣภ޲ి࣓ੴͷϕΫτϧϙςϯγϟϧΛߟ͑ͯΈΔɻ࣓৔ͱϕΫτϧϙςϯγϟϧͷؔ܎͸




1
1
∂As
−∂As
ex +
ey
B =∇×A=(1.39)
1 + x/ρ ∂y
1 + x/ρ∂x
Ͱ͋Δ͔Βɺࣜ (1.36) ʹରԠ͢Δཧ૝తͳҰ༷࣓৔ B = B0 ey Λߟ͑Δͱ


1
∂As
B y = B0 = −1 + x/ρ ∂x
ͱͳΔɻैͬͯ As ͱͯ͠

As = −B0



x
1+
2ρ



x

(1.40)(1.41)

ΛಘΔɻಋग़͸লུ͢Δ͕ɺಛʹ‫ ཰ۂ‬1/ρ ͕θϩͷͱ͖͸ࣜ (1.35) ͷ࣓৔ʹରԠ͢ΔϕΫτϧϙ
ςϯγϟϧ͸
∞(x + iy)n+1(1.42)As = Az = −B0 ℜ(bn + ian )n+1
n=0

Ͱ͋Δ͜ͱΛࣔͤΔɻ

10

２－ 9

1.7

ϋϛϧτχΞϯͷ۩ମྫͱͦͷసૹߦྻ͜͜Ͱ͸۩ମతʹϋϛϧτχΞϯ (1.34) ʹϕΫτϧϙςϯγϟϧͷදࣜΛ༩͑ɺࣸ૾ʢసૹߦ
ྻʣΛ‫ٻ‬ΊΔɻ

1.7.1

υϦϑτεϖʔεϕΫτϧϙςϯγϟϧ͸θϩ (A = 0) Ͱɺઃ‫يܭ‬ಓ͸௚ઢͰ͋Δ͔Β 1/ρ = 0 Ͱ͋ΔɻϋϛϧτχΞϯ͸
p2x + p2y + p2z(1.43)
H = (x, px , y, py , z, pz ) =2௕͞ L ͷυϦϑτεϖʔεͷࣸ૾͸ҎԼͷ‫ͳʹܗ‬Δɻ





x
1 L 0 0 0 0
x





 px 
 0 1 0 0 0 0   px 





 y 
 0 0 1 L 0 0  y 





(1.44)
 p  =  0 0 0 1 0 0  p  y  y 





 z 
 0 0 0 0 1 L  z 
pz outpz in
0 0 0 0 0 1

1.7.2

ภ޲ి࣓ੴ

ઃ‫ي͕ࢠཻܭ‬ಓ௕ L Ͱ‫཰ۂ‬൒‫ ܘ‬ρ ͷ‫੻ي‬Λඳ͘ภ޲ి࣓ੴΛߟ͑ΔɻΛಘΔɻ͜ͷͱ͖ qB0 ρ = p0
Ͱ͋Δ͔ΒϕΫτϧϙςϯγϟϧγϟϧ as ͸


x
1
1+
x
(1.45)
as = −
ρ2ρ
Ͱ͋Δ͔ΒɺϋϛϧτχΞϯ͸

H = (x, px , y, py , z, pz ) =p2x + p2y + p2z
1 x 2 γ0
+
− xpz
2
2 ρ2
ρ

(1.46)

ͱͳΔɻ͜͜Ͱ͸ xpz ͱ͍͏߲͕͋Δ͜ͱ͕ॏཁͰ͋Δɻ‫͕͍ͯ͘ݟʹޙ‬ɺ͜Ε͸ਫฏํ޲ͷҐஔཻ͕ࢠͷӡಈྔͷζϨͱ૬ؔΛ΋ͭʢӡಈྔ෼ࢄʣ͜ͱʹ༝དྷ͢Δ߲Ͱ͋Δɻԣํ޲ͷӡಈํఔࣜ͸
γ01x′′ + 2 x = pz
(1.47)
ρρͱͳΔɻ͜Ε͸ඇ੪࣍ํఔࣜͰ͋Δ͔ΒɺͦͷҰൠղ͸੪࣍ํఔࣜͷҰൠղͱಛघղͷ࿨Ͱॻ͚Δɻൃ‫ݟ‬తͰ͸͋Δ͕
x = γ0 ρpz(1.48)
͕ಛघղͰ͋Δ͜ͱ͕෼͔Δɻ2 Ұํɺ੪࣍ํఔࣜ

x′′ +

1
x=0
ρ2

(1.49)

͸Α͘஌ΒΕͨ୯ৼಈͷӡಈํఔࣜͰ͋Δ͔ΒɺͦͷҰൠղ͸ A, B Λఆ਺ͱͯ͠

x(s) = A cos
2ss+ B sinρρ∵ pz ͸ఆ਺Ͱ͋Δɻ

11

２－ 10

(1.50)

Ͱ͋Δɻैͬͯɺࣜ (1.47) ͷҰൠղ͸

x(s) = A cosss+ B sin + γ0 ρpzρρͱͳΔɻॳ‫ظ‬৚݅ s = 0 Ͱ (x0 , px0 ) ͱͯ͠ఆ਺ A, B Λܾఆ͢Δͱɺ


s
s
s
γ0 p zx(s) = x0 cos + ρpx0 sin + ρ 1 − cosρρρ(1.51)(1.52)·ͨɺϏʔϜਐߦํ޲ʹؔͯ͠͸

∂H
γ0
=− x
∂pz
ρ


s
s
γ0
s
2
pz
= − x0 cos − γ0 px0 sin − γ0 1 − cos
ρ
ρ
ρ
ρ

z′ =

Ͱ͋Δ͔Βɺ͜ΕΛಉ༷ʹ s = 0 Ͱ (z0 , pz0 ) Λॳ‫ظ‬৚݅ͱͯ͠ղ͘ͱɺ




s
s
s
2
− γ0 s − ρ sin
pz0
z(s) = z0 − γ0 x0 sin − γ0 ρpx0 1 − cosρρρ(1.53)(1.54)

ͱͳΔɻԖ௚ํ޲ʹؔͯ͠͸υϦϑτεϖʔεͱಉ͡Ͱ͋Δ͔Βɺ݁‫ہ‬ɺసૹߦྻͱͯ͠





x
C
ρS
0 0 0 ργ0 (1 − C)
x





C
0 0 0
γ0 S
 px 
 −S/ρ  px 





 y 

 y 
0
1 L 0

 = 0(1.55)
 p  0 p 
0
0 1 0
0
 y 

 y 





 z 
 −γ0 S −γ0 ρ(1 − C) 0 0 1 −γ02 (L − ρS)   z 
pz out
0
0
0 0 0
1
pz in

ΛಘΔɻ͜͜ͰɺC ≡ cos (L/ρ) , S ≡ sin (L/ρ) ͱද‫ͨ͠ه‬ɻӡಈྔͷζϨ (∝ pz ) ʹΑͬͯ‫ܦ‬࿏௕͕ҟͳΓɺͦͷޮՌ͕௨ա࣌ؒ (∝ z) ͷζϨͱͯ͠‫ݱ‬ΕΔ͜ͱ͕Θ͔Δɻ

1.7.3

࢛‫࣓ిۃ‬ੴઃ‫يܭ‬ಓ͸௚ઢͰ͋Δ͔Β 1/ρ = 0 Ͱ͋ΔɻϕΫτϧϙςϯγϟϧ‫ͼٴ‬ϋϛϧτχΞϯ͸

as = −

k1  2x − y2 ,2k1 =qB0 b1p0(1.56)

 p2
p2x + p2y
1 
+ k1 x 2 − y 2 + z(1.57)
222ਫฏํ޲ͷӡಈ͸ภ޲ి࣓ੴͷ৔߹Ͱ pz = 0, k = 1/ρ2 ͱͨ͠΋ͷʹ౳͍͠ɻԖ௚ํ޲ͷӡಈ͸ෆ҆ఆͰɺͦͷసૹߦྻʹ͸ਫฏํ޲ʹ‫ݱ‬ΕΔ cos, sin ͷ୅ΘΓʹ૒‫ۂ‬ઢؔ਺ cosh, sinh ͕‫ݱ‬
ΕΔɻ





√
x
C
S/ k1
0
0
0 0
x



 √

C
0
0
0 0   px  px 
 − k1 S






 y 



Sh |k1 | 0 0 
0
0
Ch

 =
 y 

(1.58) p |k1 |Sh
Ch
0 0 
0
0
 y 

  py 





0
0
0
0
1 L  z  z pz out
pz in
0
0
0
0
0 1
H=

12

２－ 11



√
√
͜͜Ͱ k1 > 0 ͱ͠ɺC ≡ cos k1 L, S ≡ sin k1 L, Ch ≡ cosh |k1 |L, Sh ≡ sinh |k1 |L ͱද‫ه‬√√ͨ͠ɻk1 < 0 ͷ৔߹ɺసૹߦྻ͸ k1 Λ −k1 ͱม͑ͯ (x, px ) ͱ (y, py ) ʹରԠ͢ΔసૹߦྻΛೖΕସ͑Δɻk1 > 0 ͷ࢛‫࣓ిۃ‬ੴ͸ਫฏ (x) ํ޲ʹऩଋɺԖ௚ํ޲ (y) ํ޲ʹϏʔϜΛൃࢄͤ͞Δ
ޮՌ͕͋Δɻk1 > 0 ͷ৔߹͸ঢ়‫͕گ‬൓స͢Δɻ୯ಠͷ࢛‫࣓ۃ‬৔Ͱ x, y ͷ྆ํ޲ʹऩଋޮՌΛ΋ͨΒ͢͜ͱ͸Ͱ͖ͳ͍ʢEarnshaw ͷఆཧʣɻ

13

２－ 12

ୈ 2 ষ ԣํ޲ͷӡಈՃ଎‫ث‬தͷϏʔϜߏ੒ཻࢠ͕ઃ‫ࢠཻܭ‬ͷपΓʹߦ͏ࡾ࣍‫ݩ‬తͳӡಈͷ͏ͪɺϏʔϜਫฏํ޲‫ٴ‬ͼԖ௚ํ޲ͷӡಈΛϕʔλτϩϯӡಈͱ‫Ϳݺ‬ɻҰํɺϏʔϜਐߦํ޲ͷӡಈΛγϯΫϩτϩϯӡಈͱ‫Ϳݺ‬ɻγϯΫϩτϩϯӡಈʹؔͯ͠͸ୈ̏ষͰѻ͏ɻ͜͜Ͱ͸ϕʔλτϩϯӡಈʹ͍ͭͯड़
΂Δɻ

2.1

ϕʔλτϩϯӡಈͱͦͷఆࣜԽࠓɺΤωϧΪʔ͕ઃ‫ܭ‬ΤωϧΪʔͱ౳ཻ͍͠ࢠ (pz = 0) ͷਫฏํ޲ͷӡಈΛߟ͑Δɻୈ̍ষͰ
‫͖ͨͯݟ‬υϦϑτεϖʔεɺภ޲ి࣓ੴɺ࢛‫࣓ిۃ‬ੴͷϋϛϧτχΞϯ͸ k(s) ΛՃ଎‫ث‬தͷ৔ॴs ͷؔ਺ͱͯ͠
1
1
H = p2x + k(s)x2
(2.1)
2
2
ͱ͍͏‫ͱ·ʹܗ‬ΊΒΕΔɻ͜ͷϋϛϧτχΞϯ͔ΒಘΒΕΔӡಈํఔࣜ͸d2 x+ k(s)x = 0
ds2

(2.2)

ͱͳΔɻ͜ͷํఔࣜ͸ Hill ํఔࣜͱ‫ݺ‬͹ΕΔೋ֊ͷৗඍ෼ํఔࣜʹ‫·ؚ‬ΕΔɻࣜ (2.2) ͸όωఆ਺͕࣌ؒมԽ͢Δௐ࿨ৼಈࢠͱ਺ֶతʹ͸౳ՁͰ͋ΔɻΑ͘஌ΒΕ͍ͯΔΑ͏ʹ k ͕ఆ਺Ͱ͋Ε͹୯ৼಈͰ͋ΓɺͦͷҰൠղ͸ A, ϕ0 , ω Λఆ਺ͱͯ͠

x(s) = A cos(ωs + ϕ0 )(2.3)

Ͱ͋Δɻ͔͔͜͜Βྨਪͯ͠ w, ϕ Λ s ͷؔ਺ͱͯ͠

x(s) = Aw(s) cos [ϕ(s) + ϕ0 ](2.4)ͱ͍͏ҰൠղΛԾఆ͢Δɻ࣮ࡍʹ k(s) ͕प‫਺ؔظ‬ͷ৔߹ɺ͜ͷԾఆ͸ Floquet ͷఆཧͱ‫ݺ‬͹ΕΔఆཧʹΑΓਖ਼౰Խ͞ΕΔɻࣜ (2.4) Λࣜ (2.2) ʹ୅ೖ͢Δͱɺ

−A(2w′ ϕ′ + wϕ′′ ) sin(ϕ + ϕ0 ) + A(w′′ − wϕ′2 + kw) cos(ϕ + ϕ0 ) = 0(2.5)

ͱͳΔɻ౳ࣜΛ੒ཱͤ͞ΔͨΊʹࠨลͷ sin, cos ͷ܎਺͕ͦΕͧΕθϩͰ͋Δͱ͢Δɻ·ͣ sin ͷ܎਺͕θϩͰ͋Δ͜ͱ͔Β′
w(2w′ ϕ′ + wϕ′′ ) = w2 ϕ′ = 0(2.6)
ͱͳΓɺc1 , c2 Λఆ਺ͱͯ͠

ϕ=



c1
ds + c2
2
w (s)

(2.7)

Ͱ͋Δɻ͜ͷ݁ՌΛ cos ͷ܎਺͕θϩͱ͍͏ࣜʹ୅ೖͯ͠ɺ


w3 w′′ + kw = c21
14

２－ 13

(2.8)

ΛಘΔɻ͜͜Ͱ c1 = 1 ͱͯ͠ɺTwiss ύϥϝʔλʔͱ‫ݺ‬͹ΕΔՃ଎‫ث‬෼໺ಛ༗ͷؔ਺Λఆٛ͢Δɻ

β(s) ≡ w2 (s)1 dβ(s)α(s) ≡ −2 ds
1 + α2 (s)γ(s) ≡β(s)(2.9)(2.10)(2.11)

ؔ਺ β(s), ϕ(s) ͸ͦΕͧΕɺϕʔλؔ਺ɺϕʔλτϩϯҐ૬ͱ‫ݺ‬͹ΕΔɻϕʔλؔ਺Λ࢖͏ͱҰൠղ͸ J Λ೚ҙఆ਺ͱͯ͠
x(s) =2Jβ(s) cos(ϕ(s) + ϕ0 )(2.12)
1ds(2.13)ϕ =w2 (s)
Ͱ༩͑ΒΕΔɻୠ͠Ґ૬ͷ‫఺ݪ‬ͷ೚ҙੑʹ൐͏ఆ਺ c2 Λθϩͱͨ͠ɻҰൠղΛ s Ͱඍ෼͢Δ͜ͱͰɺ
2J[sin(ϕ(s) + ϕ0 ) + α(s) cos(ϕ(s) + ϕ0 )]x′ (s) = px (s) = −(2.14)β(s)ͱͳΔɻ͜͜ͰɺҎԼͷؔ܎͕੒ཱ͢Δ͜ͱ͕༰қʹࣔͤΔɻ

γx2 + 2αxpx + βp2x = 2J

(2.15)

͜͜Ͱ 2J ͸ Courant-Snyder ͷෆมྔͱ‫ݺ‬͹Εɺ৔ॴ s ʹґଘ͠ͳ͍อଘྔͰ͋Δɻ
Hill ํఔࣜʹै͏ཻࢠͷӡಈ͸౰વͳ͕Βཻࢠͷॳ‫ظ‬৚݅ʹΑͬͯҟͳΔɻ͜ͷॳ‫ظ‬৚݅ʹରԠ͢Δྔ͕ J ͱॳ‫ظ‬Ґ૬ ϕ0 Ͱ͋Δɻϕʔλؔ਺͸ඍ෼ํఔࣜ (2.8) Λղ͘͜ͱͰܾఆͰ͖Δ͕ɺ
Ұҙͳղ͕‫·ٻ‬Δͷ͸ k(s) ʹप‫ظ‬ੑΛ՝ͨ͠৔߹ͷΈͰ͋Δɻप‫ڥظ‬ք৚݅ͱͯ͠प௕ C ͷԁ‫ܕܗ‬Ճ଎‫ث‬Λߟ͑Δͱɺk(s) ͸ k(s + C) = k(s) Λຬͨ͢ɻ͜
ͷ৔߹ɺҰप͋ͨΓͷϕʔλτϩϯҐ૬ͷਐΈΛ 2π Ͱׂͬͨྔ ν Λఆٛ͢Δɻ

ϕ(s0 + C) − ϕ(s0 )
1
=
ν≡
2π
2π

 s0 +C
s0

ds
β(s)

(2.16)

͜ͷྔ͸Ұप͋ͨΓͷϕʔλτϩϯৼಈͷճ਺Ͱ͋Γɺϕʔλτϩϯৼಈ਺ɺ΋͘͠͸ϕʔλτϩϯνϡʔϯͱ‫Ϳݺ‬ɻࠓɺϏʔϜϥΠϯͷ͋Δ৔ॴ s1 ͔Β s2 ΁ͷసૹ




x2
x1
=M
(2.17)px2
px1
Λߟ͑Δɻ͜͜Ͱ

cos(ϕ(s1 ) + ϕ0 ) = x12Jβ(s1 ) s2s12Jβ(s2 )dsβ(s)α(s1 )β(s1 )
px1 + x12J2Jβ(s1 )α(s2 )β(s2 )
px2 + sin(ϕ(s1 ) + ∆ϕ + ϕ0 ) =x22J2Jβ(s2 )sin(ϕ(s1 ) + ϕ0 ) =x2

cos(ϕ(s1 ) + ∆ϕ + ϕ0 ) = 

∆ϕ =

,

,



(2.18)
15

２－ 14

px

slope = −α/β


−α



2Jγ

2J/β


2J/β



2Jβ

x

Particle

2J/γ

Area = 2πJ

ਤ 2.1: ࣜ (2.15) Ͱܾ·ΔପԁͰ͋Δ͜ͱ͔Βɺࡾ֯ؔ਺ͷՃ๏ఆཧΛ༻͍ͯߦྻͷ֤੒෼Λ‫ٻ‬ΊΔͱɺs1 ͔Β s2 ΁ͷసૹߦ
ྻM ͸
 

√
β2
β1 β2 sin ∆ϕβ1 (cos(∆ϕ) + α1 sin ∆ϕ) 
M =(2.19)(α2 −α1 )+(1+α1 α2 sin ∆ϕ)β1√−(cos(∆ϕ)
−
α
sin
∆ϕ)
1
β2
β β
1 2

ͱॻ͚Δɻ͜ͷߦྻ͸͞ΒʹҎԼͷΑ͏ʹ̏ͭͷߦྻͷ‫Ͱܗ‬ද‫͖Ͱݱ‬Δɻ

M=



−1 


√
√
1/ β2
cos ∆ϕ sin ∆ϕ
1/ β1
0
0
√
√
√
√
β2
β1
− sin ∆ϕ cos ∆ϕα2 / β2α1 / β1͜͜Ͱ৔ॴ s ʹґଘ͢Δ࠲ඪม‫׵‬


 


x
X
1/ β(s)
0


=
β(s)
px
P
α(s)/ β(s)(2.20)(2.21)

Λߟ͑ΔͱϏʔϜͷӡಈ͸ (X, P ) ฏ໘্ͰͷճసʹͳΔ͜ͱ͕෼͔Δɻ࠲ඪม‫׵‬Λߦ͏͜ͱͰઢ
‫ܗ‬ӡಈΛͳΔ΂͘୯७Խ͢Δ͜ͷछͷఆࣜԽ͸Ұൠͷ̏࣍‫ݩ‬ͷӡಈʹ֦ுͰ͖Δɻઢ‫ܗ‬ӡಈΛ୯७Խ͢Δ͜ͱͰඇઢ‫ͳܗ‬ӡಈΛઁಈతʹѻ͏ࡍʹ‫ݟ‬௨͕͠ྑ͘ͳΔɻ
ࠓɺs2 = s1 + C ͱ͢Δͱ Twiss ύϥϝʔλʔ͸Ұக͠ɺҐ૬ࠩ͸νϡʔϯʹͳΔͨΊసૹߦྻ͸
cos 2πν + α(s) sin 2πν
β(s) sin 2πν
M=(2.22)
γ(s) sin 2πν
cos 2πν − α(s) sin 2πνͱ͍͏‫ͳʹܗ‬Δɻ͜ͷ৔߹ͷసૹߦྻ M Λಛʹपճߦྻ”One turn matrix”ͱ‫Ϳݺ‬৔߹͕͋ΔɻՃ଎‫ث‬ͷ͋Δ৔
ॴ s ͰཻࢠΛ (x, px ) ฏ໘্ʹϓϩοτ͢Δͱɺਤ 2.1 ʹ༷ࣔͨ͠ʹपճ͝ͱʹࣜ (2.15) Ͱܾ·Δପԁ্ʹ৐ΔɻҰप‫ʹޙ‬ପԁ্ͷͲͷ఺ʹҠΔ͔͸νϡʔϯ ν ʹΑܾͬͯ·Δɻྫ͑͹νϡʔϯ͕੔਺Ͱ͋Ε͹લճͱಉ͡৔ॴʹདྷΔɻTwiss ύϥϝʔλʔ͕৔ॴ s ʹґଘ͢ΔͷͰɺ͜ͷପԁͷ‫ܗ‬͸৔ॴʹґଘ͢ΔɻҰํɺJ ͕ఆ਺Ͱ͋Δ͔Βପԁͷ໘ੵ͸৔ॴʹ͸ґଘ͠ͳ͍ɻ

16

２－ 15

Twiss ύϥϝʔλʔ α, β, γ ͕ s ʹґଘͯ͠ͲͷΑ͏ʹมԽ͍͔ͯ͘͠Λߟ͑Δɻࠓɺ৔ॴ s = s1
͔Β s = s2 ΁ͷཻࢠͷਖ਼४࠲ඪͷసૹ (x1 , px1 ) → (x2 , px2 ) Λ༩͑Δసૹߦྻ͕

 


x2
m11 m12x1=(2.23)
px2
m21 m22px1
Ͱ͋Δͱ෼͔͍ͬͯΔͱ͢Δɻ͜ͷͱ͖ɺCourant-Snyder ͷෆมྔͷࣜ (2.15) ͔Β

γ1 x21 + 2α1 x1 px1 + β1 p2x1 = γ2 x22 + 2α2 x2 px2 + β2 p2x2

(2.24)

Ͱ͋Δɻୈ̐ষͰ‫ݟ‬ΔΑ͏ʹӡಈ͕ϋϛϧτχΞϯͰࢧ഑͞Ε͍ͯΔ৔߹͸ det(M ) = 1 Ͱ͋Δɻ͜ͷ৔߹ɺࣜ (2.23) ͸సૹߦྻͷ‫ྻߦٯ‬Λ‫ٻ‬ΊΔ͜ͱͰ



x1
px1



=



m11 m12m21 m22

−1 

x2
px2



=



m22 −m12−m21 m11



x2
px2



(2.25)

ͱॻ͖௚ͤΔɻࣜ (2.25) Λ༻͍ͯࣜ (2.24) ͷࠨลΛ x2 , px2 Ͱॻ͖ද͠ɺ྆ลͷ܎਺Λൺֱ͢Δ͜
ͱͰ Twiss ύϥϝʔλʔͷసૹߦྻΛಘΔɻ



 
−2m11 m12
m212
β1
β2
m211



 
(2.26)
 α2  =  −m11 m21 m11 m22 + m12 m21 −m12 m22   α1 

γ2

m221

−2m21 m22

m222

γ1

࣍ʹɺपճߦྻ M ʹؔͯ͠ӡಈͷ҆ఆੑΛߟ͑ΔɻͦͷͨΊʹߦྻ M ͷ‫ݻ‬༗஋ λ Λ‫ٻ‬ΊΔɻ
ߦྻ P ͷର֯࿨‫ࣜྻߦͼٴ‬ΛͦΕͧΕ tr(P ), det(P ) ͱද‫ͯ͠ه‬λ2 − tr(M )λ + det(M ) = 0(2.27)

Ͱ͋Δ͔Β‫ݻ‬༗஋͸

tr(M )±λ± =2

tr(M )−1
2(2.28)

Ͱ͋Δɻୠ͠ det(M ) = 1 Λ༻͍ͨɻ͜ΕΒͷ‫ݻ‬༗஋ʹରԠ͢Δ‫ݻ‬༗ϕΫτϧΛ v+ , v− ͱ͢Ε͹N प‫ޙظ‬ͷ x ͸ a, b Λॳ‫ظ‬৚݅ʹରԠ͢Δ࣮ఆ਺ͱͯ͠
N
xN = M N (av+ + bv− ) = λN
(2.29)+ av+ + λ− bv−ैͬͯɺlimN →∞ xN ͕༗քͰ͋ΔͨΊʹ͸ |λ± | < 1 Ͱͳ͚Ε͹ͳΒͳ͍ɻ͜ͷ৚݅͸ࣜ (2.28)
ͷฏํࠜͷத਎͕θϩҎԼͰ͋Δ͜ͱΛҙຯ͢Δɻैͬͯɺपճߦྻ M ʹΑΔࣸ૾ͷ‫܁‬Γฦ͕҆͠ఆͰ͋ΔͨΊͷ৚݅ͱͯ͠

−2 ≤ tr(M ) ≤ 2(2.30)

͕ M ʹ՝ͤΒΕΔɻϏʔϜͷҐ૬ۭؒ෼෍ͱΤϛολϯε
͜͜·Ͱ͸ҰͭͷཻࢠͷӡಈΛ࿦͕ͨ͡ɺ࣮ࡍͷϏʔϜ͸ҟͳΔॳ‫ظ‬৚݅ (J, ϕ0 ) Λ΋ཻͭࢠͷूஂͰ͋Δɻ͜͜ͰɺϏʔϜ͕Ґ૬ฏ໘ (x, px ) Ͱ઎༗͢Δ໘ੵ
(2.31)
ε≡
dxdpx
A

17

２－ 16

Λఆٛ͢Δɻୠ͠ A ͸ϏʔϜ͕઎༗͢ΔྖҬΛද͢ɻ͜ͷྔ͸Τϛολϯεͱ‫ݺ‬͹ΕΔɻΤϛολϯε͕খ͍͞ϏʔϜ͸ɺཻࢠ‫܈‬ͷҐஔͱൃࢄ֯౓ͷ֦͕Γ͕খ͍ͨ͞ΊɺͦͷϏʔϜͷࢦ޲ੑ͕ߴ͍͜ͱΛҙຯ͢ΔɻͦͷͨΊΤϛολϯε͸ϏʔϜͷ඼࣭Λද͢ྔͱͯ͠Ճ଎‫ث‬ͷੑೳΛٞ࿦͢Δ্ͰෆՄܽͳ୯‫͋Ͱޠ‬Δɻ
ࠓɺཻࢠͷ෼෍ͱͯࣜ͠ (2.15) Ͱఆٛ͞ΕΔପԁ಺෦ͷҰ༷෼෍Λߟ͑Δɻ͜ͷ৔߹ɺΤϛολϯε͸ 2πJ Ͱ͋ΓɺอଘྔͰ͋Δ͜ͱ͸໌നͰ͋Δɻ࣮ࡍͷϏʔϜ͸Ұ༷෼෍Ͱ͸ͳ͘ɺΨ΢ε෼෍ɺ์෺ઢ෼෍ɺWaterbag ෼෍ͱ͍༷ͬͨʑͳ෼෍ΛͱΓಘΔɻ͜ΕΒ༷ʑͳ෼෍Λ࣋ͭϏʔϜͷҐ૬্ۭؒͷ‫ܗ‬΍େ͖͞Λ౷Ұతʹѻ͏ͨΊʹ͠͹͠͹෼෍ͷ̎࣍Ϟʔϝϯτʹண໨ͯٞ͠
࿦͢Δ͜ͱ͕ଟ͍ɻ1 ͜͜ͰΤϯϕϩʔϓϚτϦοΫεʢγάϚϚτϦοΫεʣΣ Λ⟨x2 ⟩ ⟨xpx ⟩(2.32)
Σ≡⟨xpx ⟩ ⟨p2x ⟩ͱఆٛ͢Δɻୠ͠ɺ⟨v⟩ ͸ v ͷฏ‫ۉ‬஋Ͱ͋Γɺີ౓෼෍ؔ਺ ρ(x, px ) ͔Βρ(x, px ) v dxdpx⟨v⟩ = ρ(x, px ) dxdpx(2.33)ͱಋ͔ΕΔɻࠓɺࣜ (2.31) ͷΤϛολϯεΛۙࣅͨ͠ྔͱͯ͠ೋ৐ฏ‫( ۉ‬RMS) Τϛολϯε εrms
Λ
εrms ≡ det(Σ) = ⟨x2 ⟩⟨p2x ⟩ − ⟨xpx ⟩2(2.34)
Ͱఆٛ͢ΔɻΤϯϕϩʔϓϚτϦοΫεͷసૹ͸ͦͷఆ͔ٛΒ

Σ2 = M t Σ1 M

(2.35)

Ͱ༩͑ΒΕΔɻ͜͜Ͱߦྻ P ͷసஔߦྻΛ P t ͱද‫ͨ͠ه‬ɻసૹલ‫ޙ‬ͷߦྻࣜΛ‫ͯ͠ࢉܭ‬ΈΔͱ

det(Σ2 ) = det(M t ) det(Σ1 ) (det(M )(2.36)

ͱͳΔɻୈ̐ষͰ‫ݟ‬ΔΑ͏ʹӡಈ͕ϋϛϧτχΞϯͰࢧ഑͞Ε͍ͯΔ৔߹͸ det(M ) = 1 Ͱ͋ΔɻҰൠʹ det(P ) = det(P t ) Ͱ͋Δ͔ΒɺRMS Τϛολϯε͸อଘ͢Δ͜ͱ͕෼͔Δɻ͢ͳΘͪՃ଎‫ݱʹث‬ΕΔඪ४తͳߏ੒ཁૉͷΈΛ࢖ͬͯΤϛολϯεΛॖখʢ඼࣭Λ޲্ʣͤ͞Δ͜ͱ͸࢒೦ͳ͕Β‫ݪ‬ཧతʹෆՄೳͰ͋ΔɻϚονϯάϏʔϜߏ੒ཻࢠͷϕʔλτϩϯҐ૬ͷ෼෍͕Ұ༷Ͱ͋Δ (⟨ϕ⟩ = 0) ͱԾఆ͢Δͱࣜ (2.12) ͓Αͼ (2.14) ͔ΒΤϯϕϩʔϓϚτϦοΫε͸ Twiss ύϥϝʔλʔΛ༻͍ͯβ−α(2.37)
Σ = εrms−α γͱͳΔɻϏʔϜϥΠϯͷ Twiss ύϥϝʔλʔ͸ࣜ (2.2) ʹ͓͍ͯप‫ڥظ‬ք৚݅Λ՝͢͜ͱͰܾఆ͠
ͨɻҰํɺࣜ (2.37) ͔ΒϏʔϜϥΠϯͱ͸ಠཱʹ೚ҙͷϏʔϜʹରͯ͠ϏʔϜͷ Twiss ύϥϝʔλʔ (βb , αb , γb ) Λఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ

βb ≡
1

⟨x2 ⟩
,
εrms

αb ≡ −⟨xpx ⟩
,
εrms

γb ≡

Ұ࣍Ϟʔϝϯτ͸͋Β͔͡Ίࠩ͠Ҿ͍͓ͯ͘͜ͱ͕ଟ͍ɻ

18

２－ 17

1 + αb2⟨p2x ⟩=
εrms
βb(2.38)

ϏʔϜϥΠϯͷ Twiss ύϥϝʔλʔ (β, α) ͱϏʔϜͷ෼෍͔Βࢉग़͞Εͨ Twiss ύϥϝʔλʔ(βb , αb ) Λ੔߹ͤ͞Δૢ࡞ΛϚονϯάͱ‫Ϳݺ‬ɻ௨ৗɺϏʔϜϥΠϯͷਅۭύΠϓͷ‫ܘ‬ɺϏʔϜίϦϝʔλͷҐஔͱ͍ͬͨύϥϝʔλ͸ϏʔϜϥΠϯʹ੔߹ͨ͠ Twiss ύϥϝʔλʔΛ΋ͭϏʔϜΛ૝ఆܾͯ͠ΊΒΕ͍ͯΔɻैͬͯɺՃ଎‫ث‬ͷӡసʹ͓͍ͯϚονϯά͸ॏཁͳௐ੔߲໨Ͱ͋Δɻ2ӡಈྔ෼ࢄઢ‫ܗ‬Խ͞Εͨภ޲ి࣓ੴͷϋϛϧτχΞϯʹ͓͍ͯ͸ γ0 xpz /ρ ͱ͍͏߲͕͋ΔͨΊʹਫฏํ޲ͱϏʔϜਐߦํ޲ͷӡಈ͕ಠཱͰ͸ͳ͘ͳΔɻ͜͜Ͱɺਫฏํ޲ͷਖ਼४࠲ඪ (x, px ) ʹશྗֶతӡಈྔͷ૬ରతͳζϨ δ = ∆p/p0 = (p − p0 )/p0 ΛՃ͑ͯɺసૹߦྻΛ 3 × 3 ʹ֦ு͢ΔɻͦΕ͸δ = γ0 pz Ͱ͋Δ͔Βɺ̏ͭ໨ͷཁૉͱͯ͠ਐߦํ޲ͷਖ਼४ӡಈྔΛՃ͑Δ͜ͱʹରԠ͢Δɻԁ‫ܗ‬Ճ଎‫ث‬ΛҰप͢Δલ‫ޙ‬ͷసૹ͸





x
x
m11 m12 m16





=  m21 m22 m26   px 
(2.39)
 px 

pz

s=s0 +C

0

0

1

pz

s=s0

ͱॻ͚Δɻୠ͠ɺmij ͸पճߦྻ M ͷ i ߦ j ྻ໨ͷ੒෼Ͱ͋Δɻ͜͜Ͱ pz Λఆ਺ͱͯ͠ɺࣜ (2.39)
ͷप‫ظ‬ղɺ͢ͳΘͪฏߧ‫ي‬ಓ (x̃, p̃x , pz ) Λ‫ٻ‬ΊΔͱɺ



x̃
p̃x



=



1 − m11 −m12
−m21 1 − m22

−1 

m16
m26



pz

ͱͳΔɻ͜͜Ͱӡಈྔ෼ࢄؔ਺ Dx (ͱͦͷ s ඍ෼ Dx′ ) Λ࣍ࣜͰఆٛ͢Δɻ






∂
∂
x̃
x̃
Dx
=
=
∂(∆p/p0 )∂(γ0 pz )
Dx′
p̃x
p̃x

(2.40)(2.41)

͜Ε͸ࣜ (2.40) ͔Β



Dx
Dx′



=



1 − m11 −m12
−m21 1 − m22

−1 

m16 /γ0m26 /γ0(2.42)

ͱॻ͚Δɻ΋ࣜ͠ (2.42) ͷ‫͕ྻߦٯ‬ଘࡏ͢Ε͹ D, D′ Λܾఆ͢Δ͜ͱ͕Ͱ͖Δɻ෼ࢄؔ਺͸ͦͷఆ͔ٛΒɺӡಈྔ͕ઃ‫ܭ‬஋͔Β 100%ͣΕཻͨࢠͷด‫ي‬ಓͱ‫͖Ͱ͕ͱ͜͏ݴ‬Δɻ3 ෼ࢄؔ਺ͷ৔ॴs1 ͔Β s2 ΁ͷసૹ͸ਫฏํ޲ͷసૹͱಉ༷ʹ







Dx
Dx
m11 m12m16 /γ0(2.43)=+
Dx′
m21 m12Dx′
m26 /γ0s=s2

s=s1Ͱ༩͑ΒΕΔɻྫ͑͹ɺภ޲ి࣓ੴʹΑΔ෼ࢄؔ਺ͷసૹ͸ภ޲ి࣓ੴͷసૹߦྻ (1.55) ͔Β







Dx
C ρS
Dx
ρ(1 − C)/γ0(2.44)
=
+
Dx′
Dx′
S/γ0
S/ρ C
s=s2

s=s1

Ͱ͋Δɻ
2
Ϛονϯάͱ͍͏‫ݴ‬༿͸΋͏গ͠޿͍ҙຯͰ࢖ΘΕΔ͜ͱ΋͋Δɻྫ͑͹ϏʔϜϥΠϯͷઃ‫Ͳͳܭ‬ͷࡍʹి࣓ੴͷύϥϝʔλʔΛ࠷దԽͯ͠ॴఆͷ৔ॴͰ Twiss ύϥϝʔλʔͳͲΛॴ๬ͷ஋ʹ͢Δ͜ͱ΋Ϛονϯά΋͘͠͸Ϛονϯϯά‫Ϳݺͱࢉܭ‬৔߹͕͋Δɻ
3
ແ࿦ɺҰൠʹӡಈྔ͕ 100%ζϨཻͨࢠͷӡಈ͸զʑͷۙࣅ͞ΕͨϋϛϧτχΞϯͰ͸ਖ਼͘͠‫ه‬ड़Ͱ͖ͳ͍͕ɻ

19

２－ 18

(a)

(b)

QF

QD

QF

K

Beam

K −1
L

L

ਤ 2.2: (a) ബ೑ۙࣅͱ (b)FODO ηϧͷ‫܁‬Γฦ͠ӡಈྔͷΈ͕ઃ‫͔ࢠཻܭ‬ΒζϨ͍ͯΔཻࢠͷ‫ي‬ಓ͸౰વ࣓৔͔ΒͷϩʔϨϯπྗ͕ҧ͏ͷͰภ޲ి࣓ੴΛਐΉؒʹঃʑʹਫฏํ޲ʹઃ‫يܭ‬ಓ͔Βԕ͔͍ͬͯ͘͟ɻҰํɺΤωϧΪʔ͕ಉ͡Ͱ
ਫฏํ޲ͷਖ਼४ม਺͕ζϨཻͨࢠ΋౰વઃ‫يܭ‬ಓͱ͸ҧ͏‫੻ي‬Λඳ͘ɻͭ·Γภ޲ి࣓ੴ௨ա‫ޙ‬ͷਫฏํ޲ͷਖ਼४࠲ඪ (x, px ) ͸ਫฏํ޲ͷਖ਼४࠲ඪͷζϨͱΤωϧΪʔͷζϨʹ‫ى‬Ҽ͢ΔޮՌͷ଍͠߹ΘͤͱͳΔɻ෼ࢄؔ਺ͷಋೖʹΑΓɺਖ਼४࠲ඪ (x, px ) ͸ਫฏํ޲ͷζϨʹ‫ى‬Ҽ͢Δ੒෼Λ(xb , pxb )ɺӡಈྔͷζϨʹ‫ى‬Ҽ͢Δ੒෼Λ (xd , pxd ) ͱද‫ͯ͠ه‬x = x β + x δ = x β + D x γ 0 pz
px = pxβ + pxδ = pxβ + Dx′ γ0 pz(2.45)

ͱॻ͚Δɻ͢ͳΘͪɺӡಈྔͷζϨཻͨࢠ͸෼ࢄؔ਺ͱӡಈྔͷζϨͰɹܾ·Δด‫ي‬ಓͷपΓͰϕʔλτϩϯৼಈ͢Δɻ෼ࢄؔ਺ͷಋೖ͸‫ݟͰޙ‬ΔΑ͏ʹϋϛϧτχΞϯ͔Β xpz ͱ͍͏ࣗ༝౓
ؒͷ݁߹߲Λফ͢Α͏ͳ৔ॴ s ʹґଘ͢Δ࠲ඪม‫׵‬Λࢪ͢͜ͱΛҙຯ͢Δɻ4
҆ఆੑղੳͷྫɿFODO ηϧ࠷΋୯७ͳྫͱͯ͠ਤ 2.2 ʹࣔͨ͠ऩଋ‫ܕ‬ɺൃࢄ‫࣓ిۃ࢛ܕ‬ੴ (QFɺQD) ͱυϦϑτεϖʔεͷ‫܁‬Γฦ͔͠ΒͳΔϏʔϜϥΠϯʹ͓͍ͯਫฏํ޲ͷӡಈΛߟ͑Δɻ͜ͷΑ͏ͳ୯Ґऩଋߏ଄Λ
FODO ηϧͱ‫Ϳݺ‬ɻ5 ҰൠʹϏʔϜϥΠϯ͸͋Δ୯Ґऩଋߏ଄ͷ‫܁‬Γฦ͔͠ΒͳΔ෦෼Λ΋ͭ͜ͱ͕ଟ͍ɻͦͷΑ͏ͳ୯Ґऩଋߏ଄Ληϧͱ‫Ϳݺ‬ɻզʑ͕͜͜Ͱߟ͍͑ͯΔͷ͸ FODO ηϧΛ୯Ґऩଋߏ଄ͱ͢Δແ‫ʹݶ‬ଓ͘ϏʔϜϥΠϯͰ͋Δɻ
͜͜Ͱٞ࿦Λ؆୯ʹ͢ΔͨΊʹ௕͞θϩͷి࣓ੴΛߟ͑Δɻ͜ͷΑ͏ͳۙࣅΛബ೑ (Thin lens)√ۙࣅͱ‫Ϳݺ‬ɻബ೑ۙࣅʹ͓͚Δ QF ͓Αͼ QD ͷసૹߦྻ͸ࣜ (1.58) Ͱ k1 l = K = const. ͷ৚݅ԼͰ l → 0 ͷ‫ݶۃ‬ΛͱΔ͜ͱͰಘΒΕΔɻ
10
MQF,QD =(2.46)∓K 1
ബ೑ۙࣅ͸ҰൠͷྗֶͰ‫͜ͱ͏ݴ‬ΖͷܸྗۙࣅͰ͋Δɻͭ·ΓϏʔϜҐஔ͸มԽͤͣ͞ɺਖ਼४ӡಈྔͷΈมԽͤ͞ΔۙࣅͰ͋Δɻബ೑ۙࣅ͸ͦͷ؆ศ͔͞Βཧ࿦‫͠ʹࢉܭ‬͹͠͹༻͍ΒΕΔɻ͜
ͷϏʔϜϥΠϯͷసૹߦྻ͸֤ཁૉͷసૹߦྻͷੵΛ‫ͯ͠ࢉܭ‬





1
0
1 L
1 0
1 L
M =
(2.47)
−K 1
0 1
K 1
0 1


1 + KL2L + KL2=(2.48)−K 2 L 1 − KL − K 2 L24సૹߦྻΛ۠෼ର֯Խ͢Δ͜ͱʹ૬౰͢Δɻ
FODO ͷ F ͸ Focusing magnetɺD ͸ Defocusing magnet Λࢦ͢ɻFODO ͷ O ͸࣓৔͕θϩͷυϦϑτεϖʔ
εΛࢦ͢ɻ͜Ε͸ӳ‫Ͱޠ‬͸ 0(θϩ) Λ O(Φʔ) ͱಡΉ͜ͱʹ༝དྷ͍ͯ͠ΔͷͰ͋Ζ͏ɻ
5

20

２－ 19

25

180

20
135

15

β/L

Phase advance µ [deg.]

βQD /LβQF /L

90

10

45

5

0
0.0

0
0.5

1.0

2.0

1.5

20

40

60

80

100

120

140

160

Phase advance µ [deg.]KLਤ 2.3: ϕʔλτϩϯҐ૬ͷਐΈͱ KLɻਤ 2.4: βQD,QF ͱҐ૬ͷਐΈ µɻ

ͱͳΔɻ͜ͷϏʔϜϥΠϯʹ͓͚Δཻࢠͷӡಈ͕҆ఆͰ͋Δ৚݅͸ࣜ (2.30) ͔Β

−2 ≤ 2 − K 2 L2 ≤ 2 → L < 2/K(2.49)

ͱͳΔɻয఺‫཭ڑ‬͸ K −1 Ͱ͋Δ͔Βɺ͜ͷ৚݅͸࢛‫࣓ిۃ‬ੴͷִؒΛয఺‫཭ڑ‬ͷ̎ഒΑΓ‫͘͠ڱ‬ͳ͚Ε͹ͳΒͳ͍͜ͱΛ͍ࣔͯ͠Δɻࣜ (2.48) Λࣜ (2.22) ͱൺֱ͢Δ͜ͱͰ Twiss ύϥϝʔλʔͱ̍ηϧลΓͷϕʔλτϩϯҐ૬ͷਐΈ µ Λ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δɻ۩ମతʹ͸
(2.50)
2 cos µ = 2 − K 2 L2β sin µ = L + KL2(2.51)

Λղ͚͹ྑ͍ɻୠ͠ɺµ = 2πν Ͱ͋Δɻ·ͣɺୈ͔̍ࣜΒҐ૬ͷਐΈ µ ͸

sin2

K 2 L2µ=24(2.52)

ͱ‫͖Ͱࢉܭ‬Δɻਤ 2.3 ʹ KL ͷؔ਺ͱͯ͠ µ Λϓϩοτͨ͠ɻ͜ͷ݁ՌΛୈ̎ࣜʹ༻͍Δ͜ͱͰQF த৺Ͱͷϕʔλؔ਺ βQF ͸

βQF =

2L 
µ
1 + sinsin µ2(2.53)

Ͱ͋ΔɻQD த৺Ͱͷϕʔλؔ਺ βQD ͸͜͜·Ͱͷٞ࿦ʹ͓͍ͯ K → −K ͱஔ‫͢׵‬ΔΕ͹Α͘ɺ
ͦͷ݁Ռ͸

βQD =

2L 
µ
1 − sinsin µ2(2.54)

ͱͳΔɻਤ 2.4 ʹ βQD,QF ͱҐ૬ͷਐΈ µ ͷؔ܎Λࣔ͢ɻ௨ৗ µ ͷ஋ͱͯ͠͸ϕʔλؔ਺͕‫ʹ୺ۃ‬େ͖͘ͳΒͣɺµ ʹର͢Δґଘੑ͕ൺֱత؇΍͔ͳྖҬ͕બ͹ΕΔɻՃ଎‫ث‬ͷઃ‫ܭ‬΍ӡసͰ͸ϕʔλؔ਺΍νϡʔϯΛॴఆͷ஋ʹ͢ΔͨΊʹి࣓ੴͷ࣓৔ͷ‫͞ڧ‬΍഑ஔΛ࠷దԽ͢Δඞཁ͕͋Δ͕ɺͦΕ͸Ұൠʹ੍໿৚͖݅ͭͷඇઢ‫࠷ܗ‬దԽ໰୊ͱͳΔɻै࣮ͬͯࡍ্͸ԿΒ͔ͷՃ଎‫ࢉܭث‬ίʔυ͕ෆՄܽͰ͋Δɻ

21

２－ 20

2.2

࣓৔‫ࠩޡ‬ͷӨ‫ڞͱڹ‬໐ෆ҆ఆੑ

͜͜·Ͱશͯͷి࣓ੴ͕ઃ‫ܭ‬௨Γͷ࣓৔Λൃੜ͢Δ৔߹Λߟ͖͑ͯͨɻ‫࣮ݱ‬ͷՃ଎‫Ͱث‬͸ɺి࣓ੴͷઃஔ‫ࠩޡ‬ɺి‫ݯ‬ͷઃఆਫ਼౓ͷ‫ݶ‬քɺి࣓ੴͷ੡࡞‫ʹࠩޡ‬༝དྷ͢Δཧ૝తͳ࣓৔͔Βͷ࿪Έɺͱ͍༷ͬͨʑͳཧ༝ʹΑΓϏʔϜ͕‫͡ײ‬Δ࣓৔͕ઃ‫ܭ‬஋͔ΒζϨͯ͠·͏ɻ͜͜Ͱ͸ઃ‫ܭ‬ӡಈྔΛ΋ཻͭࢠͷԣํ޲ͷӡಈํఔࣜd2 x+ k(s)x = 0(2.55)
ds2
ʹରͯ͠प‫ ظ‬C ͷप‫ڥظ‬ք৚݅Λߟ͑ɺ࣓৔‫ࠩޡ‬ͷӨ‫ڹ‬Λ࿦ͣΔɻ

2.2.1

ೋ‫࣓ۃ‬৔‫ͱࠩޡ‬੔਺‫ڞ‬໐࣓৔‫ͯ͠ͱࠩޡ‬৔ॴ s = s0 ʹബ೑ͷೋ‫࣓ۃ‬৔͕͋Δ৔߹Λߟ͑Δɻ࣓৔‫ʹࠩޡ‬ΑΓཻࢠ͸




x0
x0
→
(2.56)
px 0
px0 + ∆k0ͷΩοΫΛड͚Δɻೋ‫࣓ۃ‬৔ͷଘࡏԼͰप‫ ظ‬C ͷप‫ڥظ‬ք৚݅Λຬͨ͢ (x0 , px0 ) Λ‫ٻ‬ΊΔɻपճߦྻΛ M ͱͯ͠

 

x0
x0
(2.57)=
M
px0 + ∆k0
px 0
Λ (x0 , px0 ) ʹؔͯ͠ղ͘͜ͱͰ


x0
p x0



= (I − M )−1



0
∆k0



(2.58)

ͱͳΔɻ͜͜Ͱ I ͸୯ҐߦྻͰ͋Δɻ۩ମతʹ੒෼Λॻ͖Լ͢ͱɺ

x0 =
px0 =

β0
∆k0 cos πν
2 sin πνxβ0
∆k0 (sin πνx − α0 cos πν)
2 sin πνx(2.59)(2.60)

Ͱ͋Δɻ೚ҙͷ৔ॴ s ͷด‫ي‬ಓͷ࿪Έ͸࣓৔‫ࠩޡ‬ͷ͋Δ৔ॴ s0 ͔Β৔ॴ s ΁ͷసૹߦྻʢࣜ (2.19)
ΛࢀরʣΛ༻͍ͯ

β(s)β0∆k0 cos(|ϕ(s) − ϕ0 | − πν)(2.61)
x(s) =
2 sin πνxͱͳΔɻ͜ͷ͔ࣜΒ෼͔Δॏཁͳ͜ͱ͸ϕʔλτϩϯνϡʔϯ ν ͕੔਺ͷ࣌ɺ‫ي‬ಓ͕ൃࢄ͢Δ͜
ͱͰ͋Δɻ͜Ε͸੔਺‫ڞ‬໐ͱ‫ݺ‬͹ΕΔෆ҆ఆੑͰ͋ΔɻՃ଎‫ʹث‬ෳ਺ͷೋ‫࣓ۃ‬৔ ∆ki ͕ଘࡏ͢Δ৔߹ɺͦΕΒͷ࿨Λͱͬͯ
β(s)  x(s) =
βi ∆ki cos(|ϕ(s) − ϕi | − πν)(2.62)
2 sin πνxiͰ͋Δɻ΋͘͠͸ੵ෼‫Ͱࣜܗ‬

 
β(s)
x(s) =
β(s′ )∆k(s′ ) cos(|ϕ(s) − ϕ(s′ )| − πν)ds′
2 sin πνx

22

２－ 21

(2.63)

ͱද‫͖Ͱݱ‬ΔɻҰൠʹԿΒ͔ͷ࣓৔‫ࠩޡ‬΍֎ཚʹΑͬͯൃੜ͢Δด‫ي‬ಓͷཚΕΛ”Closed orbit
distortion”ɺུͯ͠ COD ͱ‫Ϳݺ‬ɻ͜͜Ͱɺෳ਺ͷೋ‫࣓ۃ‬৔͕ϥϯμϜʹଘࡏ͢Δ৔߹ɺ౷‫ܭ‬తʹ͸ͲͷΑ͏ͳ COD ͕‫؍‬ଌ͞ΕΔ
͔Λߟ͑ͯΈΔɻࣜ (2.62) ʹ͓͍ͯೋ৐ฏ‫ۉ‬Λߟ͑Δɻ

β(s)⟨x(s)2 ⟩ =4 sin2 πνx



i

βi ∆ki cos(|ϕ(s) − ϕi | − πν)2 (2.64)

͜͜Ͱ࣓৔‫ࠩޡ‬ͷ෼෍ʹ૬͕ؔͳ͍ɺ͢ͳΘͪɺ͍͔ͭ͘ͷ࣓৔‫ࠩޡ‬͸Ө‫ʹ͍ޓ͕ڹ‬ଧͪফ͋͠
͏ɺͱ͍͏͜ͱΛԾఆ͢Δͱɺ



β(s)
⟨x(s)2 ⟩ ∼βi ∆ki2 cos2 (|ϕ(s) − ϕi | − πν)
4 sin2 πνxi(2.65)β(s)βi ∆ki2=
8 sin2 πνxi
௨ৗɺβi ͱ ∆ki ͷؒʹ΋૬͕ؔͳ͍ͷͰɺNerror Λ࣓৔‫ࠩޡ‬ͷ૯਺ͱͯ͠

⟨x(s)2 ⟩ =β(s)Nerror ⟨βi ⟩⟨∆ki2 ⟩2
8 sin πνx(2.66)

ͱͳΔɻ༨෼ͳೋ‫࣓ۃ‬৔Λ΋ͨΒ͢୅දతͳ‫ݪ‬Ҽͱ࢛ͯ͠‫࣓ిۃ‬ੴͷઃஔ‫͕͋ࠩޡ‬Δɻࣜ (1.37)
ʹ͓͍ͯ x → ∆x ͱ͢Δͱ͔֬ʹ࢛‫࣓ۃ‬৔ʹՃ͑ͯೋ‫࣓ۃ‬৔੒෼ −k1 ∆x ͕ੜ͡Δ͜ͱ͕෼͔Δɻ͜ͷ৔߹ɺࣜ (2.66) ͸⟨x(s)2 ⟩ =β(s)2Nerror ⟨βi ⟩⟨k1i⟩⟨∆x2i ⟩8 sin2 πνx(2.67)

ͱͳΔɻ͜ͷࣜΛ࢖͏͜ͱͰ
COD ͷେ͖͞Λॴ๬ͷ஋ҎԼʹ཈͑ΔͨΊʹඞཁͳ࢛‫࣓ిۃ‬ੴͷઃஔਫ਼౓ σx = ⟨∆x2i ⟩ ͷ‫ڐ‬༰஋Λେࡶ೺ʹ‫ੵݟ‬΋Δ͜ͱ͕Ͱ͖Δɻ͜͜Ͱ͸༨෼ͳೋ‫࣓ۃ‬৔‫ʹࠩޡ‬Αͬͯ COD ͕ൃੜ͢Δ͜ͱΛ‫͕͖ͨͯݟ‬ɺεςΞϦϯά࢖ͬͯ‫ނ‬ҙʹ COD Λൃੜͤ͞Δ͜ͱͰՃ଎‫ث‬ͷૉੑΛղੳ͢Δ͜ͱ͕Ͱ͖Δɻ͜Ε͸ࣜ (2.62) ͔Β
COD ʹ͸ Twiss ύϥϝʔλʔͷ৘ใ͕‫·ؚ‬Ε͍ͯΔͨΊͰ͋ΔɻͦͷҙຯͰ͸ Twiss ύϥϝʔλʔͱ͸ด‫ي‬ಓͷઁಈʹର͢ΔԠ౴ͷඍ܎਺Ͱ͋Δͱ‫͖Ͱ͕ͱ͜͏ݴ‬Δɻਤ 2.5 ʹ SuperKEKB ిࢠஷଂ‫ܕ‬Ϧϯάʹ͓͍ͯଌఆͨ͠ CODɺϞσϧՃ଎‫͔ث‬Βͷ༧ଌ஋‫ऀ྆ͼٴ‬ͷࠩ෼Λࣔ͢ɻCOD
Λղੳ͢Δ͜ͱͰӡసʹ࢖͍ͬͯΔՃ଎‫ث‬ͷϞσϧͱ࣮‫ػ‬ͷဃ཭ͷ౓߹͍ਪఆ͢Δ͜ͱ͕Ͱ͖Δɻ
ิਖ਼ి࣓ੴͳͲͷௐ੔ʹΑͬͯͦͷࠩҟΛຒΊΔ͜ͱͰ‫࣮ݱ‬ͷՃ଎‫ث‬ͷϏʔϜޫֶΛઃ‫ܭ‬஋ʹ͚ۙͮΔ͜ͱ͕Ͱ͖Δɻ࣍ͷখઅͰ࢛‫࣓ۃ‬৔ͷ‫ࠩޡ‬Λѻ͏͕ɺࣜ (2.62) Ͱ‫ͨݟ‬Α͏ʹҰൠʹԁ‫ܗ‬Ճ଎Ճ଎‫࣓͍͓ͯʹث‬৔‫ࠩޡ‬ͷӨ‫Ͳͳڹ‬ͷԿΒ͔ͷ֎ཚ͔ΒͷӨ‫ڹ‬Λཧ࿦తʹௐ΂͍ͯ͘ͱɺ͠͹͠͹νϡʔϯ͕͋Δ
৚݅Λຬͨ࣌͢ʹӡಈ͕ෆ҆ఆʹͳΔɺͱ͍͏݁࿦ʹߦ͖ண͘ɻ͜Ε͸ϏʔϜ͕‫͡ײ‬Δྗ͕प‫ظ‬
తͰ͋Δ͕‫ʹނ‬ɺຊ࣭తʹ֎ཚͷӨ‫ڞ͕ڹ‬໐తɺࢦ਺ؔ਺తʹ૿େ͢ΔՄೳੑΛ΋ͭͨΊͰ͋Δɻ
ैͬͯνϡʔϯͱ͍͏෺ཧྔ͸Ճ଎‫ث‬ͷઃ‫ܭ‬ɺӡసʹ͓͍ͯඇৗʹॏཁͳ΋ͷͰ͋Δɻ

2.2.2

࢛‫࣓ۃ‬৔‫ͱࠩޡ‬൒੔਺‫ڞ‬໐࣍ʹ৔ॴ s = s0 ͷബ೑ͷ࢛‫࣓ۃ‬৔‫ࠩޡ‬Λߟ͑Δɻ
1
0−∆k1 123

２－ 22(2.68)

(b) Measurement

2

2

1

1

∆x [mm]∆x [mm](a) Model

0
−1
−2
−1500

0
−1
−2

−1000

0

−500

500

1000

1500

−1500

−1000

Postion s [m]

−500

0

500

1000

1500

Postion s [m](c) Measurement - Modelδ(∆x) [µm]

100

0

−100

−200

−1500

−1000

0

−500

500

1000

1500

Postion s [m]ਤ 2.5: SuperKKEK ͷిࢠϦϯάʹ͓͍ͯਫฏํ޲ͷεςΞϦϯάʹΑͬͯൃੜͤͨ͞ CODɻ(a) ࣜ (2.67) ʹΑΔ༧ଌ஋ɺ(b) ଌఆ஋ɺ(c) ྆ऀͷࠩ෼ɻিಥ఺͸ s = 0 ʹҐஔ͢Δɻ
पճߦྻ M Λߟ͑Δͱ
cos 2πν + α(s) sin 2πν
β(s) sin 2πν
M =(2.69)
γ(s) sin 2πν
cos 2πν − α(s) sin 2πν



1
0
cos 2πν0 + α0 sin 2πν0β0 sin 2πν0=(2.70)−∆k1 1
γ0 sin 2πν0
cos 2πν0 − α0 sin 2πν0
cos 2πν0 + α0 sin 2πν0β0 sin 2πν0=−∆k1 (cos 2πν0 + α0 sin 2πν0 ) + γ0 sin 2πν0 −∆k1 β0 sin 2πν0 + cos 2πν0 − α0 sin 2πν0(2.71)

ͱͳΔɻैͬͯɺ

1
1
cos 2πν = tr(M ) = cos 2πν0 − ∆k1 β0 sin 2πν022ͱͳΔɻࣜ (2.72) ͸ ∆k1 > 0 Ͱ͋Ε͹ n Λ੔਺ͱͯ͠ ν0 ͕nn
− W < ν0 < ,
2
2

W =

1
∆k1 β0tan−1π2(2.72)(2.73)

ͷൣғͷͱ͖ʹղΛ΋ͨͳ͍ɻ͢ͳΘཻͪࢠͷӡಈ͸ෆ҆ఆͰ͋ΔɻҰํɺ∆k1 < 0 Ͱ͋Ε͹൒੔਺ͷ্ଆʹಉ༷ͷྖҬ͕ੜ͡Δɻ͜ͷΑ͏ʹಈ࡞ෆೳͳνϡʔϯͷൣғͷ͜ͱΛετοϓόϯυ (‫ࢭې‬ଳ) ͱ‫Ϳݺ‬ɻҰൠʹετοϓόϯυͷ෯͸࣓৔‫͕ࠩޡ‬େ͖͍΄Ͳ޿͍ɻ͢ͳΘ࣮ͪӡసʹ͓͍ͯετοϓόϯυ͕খ͍͞Ճ଎‫Ͳ΄ث‬ΑΓௐ੔͞ΕͨϚγϯͱ‫͑ݴ‬Δɻਤ 2.6 ʹࣜ (2.72) ʹΑΓ‫ٻ‬Ί࣓ͨ৔‫͋ࠩޡ‬Γͷνϡʔϯ ν ͱ ν0 ͷؔ܎Λ͓ࣔͯ͘͠ɻ
ࠓɺν = ν0 + ∆ν ͱද‫∆ ͯ͠ه‬ν ͕े෼ʹখ͍͞৔߹͸ࣜ (2.72) ͸ۙࣅతʹ∆ν =1∆k1 β0
4π
24

２－ 23

(2.74)

1.0
0.9
0.8
0.7

tan−1 (∆k1 β0 /2)/π

ν

0.6
0.5
0.4
0.3
0.2
∆k1 > 0
∆k1 < 0

0.1
0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ν0
ਤ 2.6: ࣜ (2.72) ͷղ ν ͱ‫͍ͳ͕ࠩޡ‬৔߹ͷ ν0 ͷؔ܎ͱͳΔɻෳ਺ͷ৔ॴ s = si ʹ෼෍͍ͯ͠Δ৔߹ɺͦΕΒͷ࿨Λͱͬͯ1 ∆ν =∆k1i βi
4π(2.75)i·ͨ͸ੵ෼‫Ͱࣜܗ‬
1∆k1 (s′ )β(s′ )ds′∆ν =(2.76)
4π
ͱͳΔɻཻࢠ͸ઃ‫ܭ‬஋ʹൺ΂ͯෆཁͳ࢛‫࣓ۃ‬৔Λ‫͡ײ‬ΔΘ͚Ͱ͋Δ͔Βɺ౰વͳ͕Βνϡʔϯ͕ઃ‫ܭ‬஋͔ΒζϨΔɻ͜ͷζϨΛνϡʔϯγϑτͱ‫Ϳݺ‬ɻ6࣍ʹ‫࣓ࠩޡ‬৔ʹΑΔϕʔλؔ਺ͷมԽΛߟ͑ΔɻͦͷͨΊʹपճߦྻ M (s → s + C) Λ
10
M0 (s → s0 )M (s → s + C) = M0 (s0 → s + C)−∆k1 1
(2.77)
b11 b12a11 a1210=
b21 b22a21 a22−∆k1 1ͱॻ͖௚͢ɻӈลͷߦྻੵΛ‫ͯ͠ࢉܭ‬Ұߦೋྻ໨੒෼ʹண໨͢Δͱ‫͕͋ࠩޡ‬Δ৔߹ͷϕʔλؔ਺Λ β = β0 + ∆β ͱද‫ͯ͠ه‬[β0 (s) + ∆β(s)] sin [2π (ν0 + ∆ν)] = b11 a12 + b12 a22 − b12 a12 ∆k1= β0 sin 2πν0 − b12 a12 ∆k1(2.78)

ͱͳΔɻ͜͜Ͱࣜ (2.74) Λ༻͍ͯ̍࣍ͷඍখྔͷΈߟྀ͢Δͱ∆β(s)∆k1 β0 (s0 )=−cos [2|ϕ0 (s) − ϕ0 (s0 )| − πν]
β0 (s)
2 sin 2πν0(2.79)

ͱͳΔɻ࣓৔‫͕ࠩޡ‬ෳ਺Օॴ s = si ʹ͋Δ৔߹͸∆β(s)  ∆k1i β0 (si )=−cos [2|ϕ0 (s) − ϕ0 (si )| − πν]
β0 (s)
2 sin 2πν0(2.80)i6Ұൠʹཧ༝͸ԿͰ͋ΕɺϏʔϜଌఆ΍਺஋γϛϡϨʔγϣϯͷղੳͳͲʹΑΓಘΒΕͨνϡʔϯ͕͋Β͔͡Ί૝ఆ͍ͯͨ͠஋͔ΒζϨͨ৔߹΋ಉ༷ʹɺ”ʙͷӨ‫Ͱڹ‬νϡʔϯγϑτ͕ੜͨ͡”ͳͲͱද‫͢ݱ‬Δɻ

25

２－ 24

ੵ෼‫Ͱࣜܗ‬͸

1
∆β(s)
=−
β0 (s)
2 sin 2πν0∆k1 (s′ )β0 (s′ ) cos 2|ϕ0 (s) − ϕ0 (s′ )| − πν ds′(2.81)

Ͱ͋Δɻ͜ͷΑ͏ͳϕʔλؔ਺ͷཚΕ͸ϕʔλϏʔτͱ‫ݺ‬͹ΕΔɻϕʔλϏʔτ͕ଘࡏ͢Δͱ༧‫ظ‬ͤ͵৔ॴͰϏʔϜαΠζ͕๲Έɺͦͷ݁Ռͱͯ͠ෆཁͳϏʔϜϩεΛൃੜͤ͞Δ͜ͱ͕͋Δɻࣜ
(2.81) ͔Βνϡʔϯ͕൒੔਺ͷͱ͖ϕʔλϏʔτ͕ൃࢄ͢Δɻͭ·Γཻࢠͷӡಈ͸ෆ҆ఆԽ͢Δɻ͜Ε͸൒੔਺‫ڞ‬໐ͱ‫ݺ‬͹ΕΔ‫ݱ‬৅Ͱ͋Δɻ

2.2.3

ϕʔλτϩϯ‫ڞ‬໐ͱ‫ڞ‬໐৚݅

͜͜·Ͱɺೋ‫͓ۃ‬Αͼ࢛‫࣓ۃ‬৔ͷ‫ࠩޡ‬ͷӨ‫ڹ‬Λௐ΂͖ͯͨɻͦͷ݁Ռ෼͔ͬͨ͜ͱ͸‫࣮ݱ‬ͷՃ଎‫ث‬ͷಈ࡞఺ͱͯ͠νϡʔϯΛ੔਺΋͘͠͸൒੔਺෇ۙʹબͿͱϏʔϜ͕ෆ҆ఆʹͳΓ͑Δͱ͍
͏͜ͱͰ͋Δɻ࣮ࡍɺ௨ৗ͸͜ͷΑ͏ͳνϡʔϯ͕ӡస৚݅ʹબ͹ΕΔ͜ͱ͸ͳ͍ɻ
͜͜Ͱ·Ͱٞ࿦͔Βਪ࡯Ͱ͖Δ͔΋͠Εͳ͍͕ɺ࿡‫࣓ۃ‬৔‫ʹࠩޡ‬ରͯ͠͸ n Λ੔਺ͱͯ͠ਫฏ
ํ޲ͷνϡʔϯ͕ ν = n/3 ͱ͖ʹෆ҆ఆੑ͕‫͜ى‬Δɻ͞Βʹߴ࣍ͷ࣓৔‫ʹࠩޡ‬ରͯ͠΋‫ڞ‬໐‫ݱ‬৅͸ੜ͡ɺਫฏ (x) ํ޲ͱԖ௚ (y) ํ޲ͷӡಈΛ݁߹ͤ͞Δ࣓৔੒෼΋ߟྀ͢ΔͱɺҰൠʹ k, l, N Λ੔਺ͱͯ͠ԣํ޲ͷνϡʔϯ (νx , νy ) ͕kνx + lνy = N(2.82)

Λຬͨ͢ͱ͖‫ڞ‬໐‫ݱ‬৅͕‫͜ى‬Δɻ͜ͷΑ͏ͳ‫ڞ‬໐Λ૯শͯ͠ϕʔλτϩϯ‫ڞ‬໐ͱ‫Ϳݺ‬ɻ·ͨ |k|+|l| =N ͸‫ڞ‬໐ͷ࣍਺ͱ‫ݺ‬͹ΕɺҰൠʹ࣍਺ N ͕খ͍͞΄Ͳ‫ڞ͍ڧ‬໐ෆ҆ఆੑ͕‫͖ى‬Δɻਤ 2.7 ʹ (νx , νy )ฏ໘ʹࣜ (2.82) Ͱඳ͔ΕΔઢ (N ≤ 3) Λϓϩοτͨ͠ɻ͜ͷΑ͏ͳਤ͸νϡʔϯμΠΞάϥϜͱ‫ݺ‬͹ΕɺՃ଎‫ث‬ͷઃ‫ܭ‬΍ಈ࡞఺Λ࿦ͣΔࡍʹ༻͍ΒΕΔɻ
͜͜Ͱ·Ͱ࣓৔‫ࠩޡ‬ͷӨ‫ڹ‬Λബ೑ۙࣅΛ༻͍Δ͜ͱͰߦྻԋࢉͷ‫͔఺؍‬Βٞ࿦ΛਐΊ͖ͯͨɻ͜
Ε͸࣓৔‫ࠩޡ‬ͷ෼෍‫ܗ‬ঢ়͕प௕ C Λप‫͢ͱظ‬Δप‫ظ‬తσϧλؔ਺Ͱ͋Δ͜ͱʹରԠ͢Δɻྫ͑͹৔ॴ s = s0 ʹೋ‫࣓ۃ‬৔‫͕͋ࠩޡ‬Δ৔߹ͷӡಈํఔࣜ͸d2 x+ k(s)x = ∆k0 δp (s)(2.83)
ds2
Ͱ͋Δɻ͜͜Ͱɺྗֶͷߨٛͷ୊ࡐͱͯ͠Α͘ѻΘΕΔ‫ৼ੍ڧ‬ಈΛߟ͑Δɻόωఆ਺ k ͷόωʹͭͳ͛ΒΕ࣭ͨྔ m ͷ࣭఺Λ֯प೾਺ ω ͷ֎ྗͰৼಈͤ͞Δɻӡಈํఔࣜ͸kd2 x+ x = F cos(ωt)
2dtmͰ͋ΓɺͦͷҰൠղ͸ ω0 =


k/m ͱະఆ܎਺ A, ϕ Λ༻͍ͯx(t) = A cos(ω0 t + ϕ) +F[cos(ωt) − cos(ω0 t)]
ω02 − ω 2

(2.84)(2.85)

Ͱ͋Δɻैͬͯ‫ܥ‬ͷ‫ݻ‬༗ͷप೾਺ ω0 ͱ֎ྗͷप೾਺ ω ͕Ұக͢Δͱ͖‫ڞ‬໐తʹৼಈ͕੒௕͢Δɻ
͜͜Ͱप‫ ظ‬C ͷप‫ظ‬తσϧλؔ਺͕ n Λ೚ҙͷ੔਺ͱͨ͠શͯͷߴௐ೾ cos(2πns/C) ͷॏͶ߹
Θͤɺ


∞
1
2 
2πns
+
(2.86)
δp (s) =
cos
C
C
C
n=1

Ͱ͋Δ͜ͱΛߟ͑ΔͱɺՃ଎‫ث‬ͷ৔߹͸ബ೑ۙࣅͷ࣓৔‫ࠩޡ‬͸प௕ C Λ‫ج‬ຊ೾ͱͨ͠શͯͷߴௐ೾Λ‫ۦ‬ಈྗͱͯ͠΋ͭɻैͬͯ೚ҙͷ੔਺Ͱ‫ڞ‬໐͕‫͜ى‬Δ͜ͱʹͳΔɻͦ͏͍͏ҙຯͰ͸ബ೑ۙࣅ͞Ε࣓ͨ৔‫ࠩޡ‬͸‫ࠩޡ‬ͷ෼෍ͷ‫ܗ‬ঢ়ͱͯ͠͸࠷΋‫͋Ͱྗڧ‬Δͱ‫͑ݴ‬Δɻ࣮ࡍͷՃ଎‫Ͱث‬͸Ͳͷ‫ڞ‬໐ઢ͕‫؍͘ڧ‬ଌ͞ΕΔ͔͸‫ࠩޡ‬ͷେ͖ͦ͞ͷ΋ͷ͚ͩͰͳ͘ɺϦϯάશपʹ౉Δ‫ࠩޡ‬ͷ෼෍ͷಛੑʹ΋ґΔɻͦͷͨΊ‫ڞ‬໐ෆ҆ఆੑͷղੳͰ͸ɺ͠͹͠͹प೾਺ྖҬʹҠߦͯٞ͠࿦͢Δ͜ͱ
͕͋Δɻ

26

２－ 25

1.0

Fractional vertical tune {νy }

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Fractional horizontal tune {νx }ਤ 2.7: νϡʔϯμΠΞάϥϜ (N = 1, 2, 3)

2.3

৭ऩࠩ (ΫϩϚςΟγςΟ) ͱͦͷ੍‫ޚ‬ӡಈྔ͕ઃ‫ܭ‬஋͔ΒζϨ͍ͯΔཻࢠ͕ઃ‫ͱࢠཻܭ‬ҟͳΔೋ‫࣓ۃ‬৔Λ‫͡ײ‬Δ͜ͱ͔Β෼ࢄؔ਺Λಋೖͨ͠ɻ͜Ε͸ཻࢠ͕‫͡ײ‬Δೋ‫࣓ۃ‬৔͕ӡಈྔʹґଘ͢ΔͨΊͰ͋Δ͕ɺͦ΋ͦ΋ϩʔϨϯπྗ͕ӡಈྔʹґଘ͢ΔͷͰ͋Δ͔Βɺ౰વͳ͕Β࢛‫ۃ‬Ҏ্ͷߴ࣍ͷ࣓৔ʹؔͯ͠΋ಉ༷ͷ‫ݱ‬৅͕‫͖ى‬Δɻྫ͑͹ɺ࢛‫࣓ిۃ‬ੴཻ͕ࢠʹ༩͑ΔӨ‫ڹ‬͸ӡಈྔʹґଘ͢Δɻͦͷ݁Ռɺϕʔλؔ਺΍νϡʔϯͱ͍ͬͨզʑ͕͜Ε·Ͱ‫͖ͨͯݟ‬छʑͷύϥϝʔλʹӡಈґଘੑ͕‫ݱ‬ΕΔɻ͜ͷΑ͏ͳՃ଎‫ث‬ύϥϝʔλʔͷӡಈྔґଘੑ͸৭ऩࠩʢΫϩϚςΟγςΟʣͱ‫ݺ‬͹ΕΔɻ͜Ε͸Ϩϯζޫֶʹ͓͍ͯϨϯζͷ۶ં཰͕ޫͷ೾௕ʹΑͬͯมΘΔ͜ͱʹྨਪ͞Ε͚ͯͭΒΕͨ‫ݺ‬শͰ͋ΔͱࢥΘΕΔɻ͜͜Ͱ͸ΫϩϚςΟγςΟͷఆٛͱͦͷௐ੔ํ๏ʹ͍ͭͯड़΂Δɻ

2.3.1

ΫϩϚςΟγςΟ

ΫϩϚςΟγςΟ͸Ұൠʹӡಈྔʹରͯ͠ඇઢ‫ޮͳܗ‬ՌͰ͋ΔͨΊɺզʑ͕ઢ‫ܗ‬ཧ࿦Λ࿦ͣΔ
ͨΊʹ༻͍ͨۙࣅ͞ΕͨϋϛϧτχΞϯ (1.34) ʹ͸‫·ؚ‬Εͳ͍΋ͷͰ͋Δɻ͜ͷӨ‫ڹ‬Λ࿦ͣΔͨ
Ίʹ͸ϋϛϧτχΞϯͷۙࣅల։ʹ͓͍ͯల։ͷ࣍਺Λ্͛Δඞཁ͕͋Δɻ͜͜Ͱ͸؆୯ͷͨΊɺӡಈྔͷζϨ͕ఆ਺Ͱ͋Δͱͯ͠ɺ࢛‫࣓ۃ‬৔͔Βड͚Δྗ͕Ͳͷ༷ʹมΘΔ͔Λߟ͑Δɻࣜ (1.26) ʹ͓͍ͯ ρ = 0, (Ax , Ay ) = 0 ͱͯ͠ As ʹ࢛‫࣓ۃ‬৔ͷϕΫτϧϙςϯγϟϧΛ୅ೖ͢Δͱ

p2x + p2y
p0
+ k1 (x2 − y 2 )H ∼ −p +2p2(2.87)
͜͜ͰӡಈྔͷζϨ δ = (p − p0 )/p0 Λಋೖ͠ɺϋϛϧτχΞϯͱਖ਼४ӡಈྔΛ p0 Ͱআͯ͠H = −(1 + δ) +p2x + p2y1
+ k1 (x2 − y 2 )2(1 + δ) 2
(2.88)27

２－ 26

ΛಘΔɻԣํ޲ͷӡಈํఔࣜ͸

k1
d2 x
x=0
+
2
ds
1+δ
k1
d2 y
y=0
−
2
ds
1+δ

(2.89)(2.90)

ͱͳΔɻ͢ͳΘͪɺӡಈྔ͕ δ ͣΕཻͨࢠʹରͯ͠͸࢛‫࣓ۃ‬৔ͷӨ‫͕ڹ‬Ҽࢠ 1/(1 + δ) ͚ͩมԽ͢
Δ͜ͱΛҙຯ͢Δɻಛʹ δ ≪ 1 ͷ৔߹d2 x+ (1 − δ)k1 x = 0
ds2
d2 y− (1 − δ)k1 y = 0
ds2(2.91)(2.92)

ͱۙࣅ͞ΕΔɻ͜ͷӨ‫ڹ‬͸࢛‫࣓ۃ‬৔‫͕͋ࠩޡ‬Δ͜ͱͱ਺ֶతʹ͸౳ՁͰ͋Δ͔Βɺࣜ (2.76) ͔Βνϡʔϯγϑτ
1∆νx = − δ k1 (s)βx (s)ds4π(2.93)1
δ k1 (s)βy (s)ds∆νy =4πΛൃੜͤ͞ΔɻಛʹΫϩϚςΟγςΟͷ͏ͪ δ ʹରͯ͠ઢ‫ͳܗ‬෦෼ΛϦχΞΫϩϚςΟγςΟͱ‫ݺ‬ͿɻϏʔϜϥΠϯʹଘࡏ͢Δ࢛‫࣓ۃ‬৔ʹ‫ى‬Ҽ͢ΔνϡʔϯͷϦχΞΫϩϚςΟγςΟ͸ࣜ (2.93)
͔Βɺ

1
∂νx
=−
k1 (s)βx (s)ds
ξx =
∂δ
4π

(2.94)∂νy1=
k1 (s)βy (s)ds
ξy =
∂δ
4π
Ͱ༩͑ΒΕΔɻ‫ڧ‬ऩଋͷ‫ݪ‬ཧʹ‫ͮ͘ج‬య‫ܕ‬తͳՃ଎‫Ͱث‬͸ऩଋ‫࣓ిۃ࢛ܕ‬ੴ (k1 > 0) ͷ৔ॴͰ βx ͕େ͖͘ βy͸খ͍͞ɻҰํɺൃࢄ‫࣓ిۃ࢛ܕ‬ੴ (k1 < 0) ͷ৔ॴͰ͸‫ ʹٯ‬βx ͕খ͘͞ βy ͸େ͖͍ɻैͬͯࣜ
(2.94) ͷඃੵ෼ؔ਺ͷத਎͸ෛͷ੒෼͔Βͷ‫د‬༩͕େ͖͘ ξx , ξy ͸௨ৗͲͪΒ΋ෛͷ஋ʹͳΔɻ

2.3.2

ΫϩϚςΟγςΟͷίϯτϩʔϧ௨ৗνϡʔϯΫϩϚςΟγςΟΛ੍‫͢ޚ‬ΔͨΊʹ͸࿡‫࣓ిۃ‬ੴ͕࢖ΘΕΔɻ࿡‫࣓ిۃ‬ੴ͕ൃੜ͢ΔϕΫτϧϙςϯγϟϧ͸ࣜ (1.42) ͔Β

As = −
Ͱ͋Δ͔ΒϋϛϧτχΞϯ͸

H=


B 0 b2  3
x − 3xy 23


p2x + p2y
1 
+ k2 x3 − 3xy 22
6(2.95)(2.96)

Ͱ͋Δɻୠ͠ɺk2 = 2qB0 b2 /p0 ͱͨ͠ɻैͬͯԣํ޲ͷӡಈํఔࣜ͸d 2 x k2 2
+ (x − y 2 ) = 0
ds2
2
d2 y
− k2 xy = 0
ds2
28

２－ 27

(2.97)(2.98)

Ͱ͋Δɻ͜͜Ͱɺബ೑ۙࣅΛ͢Δͱཻࢠͷԣํ޲ӡಈྔ px,y ͸࿡‫࣓ిۃ‬ੴΛ௨ա͢Δલ‫Ͱޙ‬k2 2(x − y 2 )∆s2= k2 xy∆s∆px = −

(2.99)∆py(2.100)͚ͩมԽ͢Δɻ࠲ඪ x ͸ӡಈྔͷͣΕʹΑΔ੒෼ Dx δ ͱͦͷपΓͷϕʔλτϩϯৼಈ xβ ͷ࿨Ͱ͋Δ͔Βɺx = xβ + Dx δ Λ୅ೖͯ͠k2∆s(x2β + 2Dx δxβ + Dx2 δ 2 − y 2 )2∆py = k2 ∆s (xβ + Dx δ) y

∆px = −

(2.101)ୠ͠ɺ؆୯ͷͨΊԖ௚ํ޲ͷ෼ࢄؔ਺͸θϩͱͨ͠ɻࣜ (2.101) ͔Βཻࢠ͸ k2 Dx δ∆s ͷ࢛‫࣓ۃ‬৔੒෼Λ‫͡ײ‬Δ͜ͱ͕෼͔Δɻ͜ͷ࢛‫࣓ۃ‬৔੒෼ʹΑΔνϡʔϯγϑτ͸ࣜ (2.76) ͔Β1
sextδ k2 (s)Dx (s)βx (s)ds
∆νx =
4π

(2.102)1sext∆νy = − δ k2 (s)Dx (s)βy (s)ds4πͱͳΔɻ·ͨ͸ϦχΞΫϩϚςΟγςΟʹͯ͠
1sext
k2 (s)Dx (s)βx (s)ds
ξx =
4π

1
k2 (s)Dx (s)βy (s)ds
ξysext = −4π(2.103)ͷมԽΛ༩͑Δ͜ͱ͕Ͱ͖ΔɻνϡʔϯͷϦχΞΫϩϚςΟγςΟ͸ࣜ (2.94) ͱࣜ (2.103) ʹՃ͑ͯͦͷଞͷඇઢ‫࣓ిܗ‬ੴʹΑΔޮՌͷ଍͠߹ΘͤͱͳΔɻಛʹภ޲ి࣓ੴͱ࢛‫࣓ిۃ‬ੴʹ‫ى‬Ҽ͢ΔϦχΞΫϩϚςΟγςΟ(2.94) ΛφνϡϥϧΫϩϚςΟγςΟͱ‫͕͋ͱ͜Ϳݺ‬Δɻ
࣮ࡍͷϏʔϜ͸ӡಈྔͷζϨ δ ͕ҟͳΔཻࢠͷू߹Ͱ͋Δ͔Βϕʔλτϩϯνϡʔϯ͸ཻࢠʹΑͬͯҧ͏͜ͱʹͳΔɻैͬͯΑΓଟ͘ͷཻࢠΛ҆ఆʹपճͤ͞ΔͨΊʹ͸ɺӡಈྔ͕ζϨཻͨࢠʹରͯ͠΋ϕʔλτϩϯ‫ڞ‬໐͕‫͍ͳ͖ى‬Α͏ʹνϡʔϯΫϩϚςΟγςΟΛ੍‫͢ޚ‬Δඞཁ͕͋Δɻ͜͜Ͱ͸ӡಈྔͷζϨ δ ͕ҰఆͰ͋Δͱ͍͏ԾఆΛ͓͍ͨ͜ͱʹ஫ҙ͢Δ΂͖Ͱ͋Δɻ࣮ࡍʹ͸ཻࢠ͸γϯΫϩτϩϯӡಈΛߦ͏ͷͰ δ ͸ఆ਺Ͱ͸ͳ͍ɻैͬͯɺӡಈྔͷζϨΛ‫ݻ‬ఆ֤ͯ͠ӡಈྔʹରͯ͠νϡʔϯΛఆٛͯ͠‫ڞ‬໐ෆ҆ఆੑΛ࿦ͣΔඳ૾͸෼͔Γқ͍͕ɺඞͣ͠΋ਖ਼͍݁͠࿦Λ༩͑Δͱ͸‫ݶ‬Βͳ͍ɻ
ͱ͜ΖͰɺࣜ (2.101) ʹ͸ෆཁͳ߲΋‫·ؚ‬Ε͍ͯΔɻ͜Ε͸࿡‫࣓ిۃ‬ੴΛಋೖͨ͜͠ͱʹΑΔ͍
Θ͹෭࡞༻Ͱ͋Δɻ͜ΕΒͷ߲͸ཻࢠʹඇઢ‫ྗܗ‬Λ༩͑ɺҰൠʹཻࢠͷӡಈΛෆ҆ఆԽͤ͞Δ‫ݪ‬ҼʹͳΓಘΔɻ͜ΕΒͷ߲ͷ͏ͪӡಈྔͷζϨ δ ʹґଘ͠ͳ͍߲Λ”Geometrical abberation term”ɺ
ӡಈྔͷζϨ δ ʹґଘ͢Δ߲ͷӨ‫ڹ‬Λ”Chromatic abberation term”ͱশ͢Δ͜ͱ͕͋ΔɻಛʹφνϡϥϧΫϩϚςΟγςΟ͕େ͖ͳՃ଎‫Ͱث‬͸ɺͦͷ੍‫ޚ‬ͷͨΊʹΑΓ‫͍ڧ‬࿡‫࣓ిۃ‬ੴ͕ඞཁͱ
ͳΔ͕ɺͦͷ෼͜ΕΒͷ෭࡞༻͕‫͏·ͯͬ͠ͳ͘ڧ‬ɻ͜ΕΒ෭࡞༻ΛͳΔ΂͘খ͘͢͞Δํ๏ͱͯ̎͠୆ͷಉ͡‫͞ڧ‬ͷ࿡‫࣓ిۃ‬ੴΛͦͷؒͷసૹߦྻ M ͕ਫฏํ޲ɺਨ௚ํ޲ͱ΋ʹ


−1 0
M=
(2.104)0 −1
ͱΔΑ͏ʹϏʔϜϥΠϯΛ૊Ήํ๏͕͋Δɻ͜ͷ࣌ɺ࿡‫࣓ిۃ‬ੴ͕े෼ʹബ͚Ε͹̎୆ͷ࿡‫ిۃ‬࣓ੴʹΑΔඇઢ‫ܗ‬ΩοΫ͕ͪΐ͏Ͳଧͪফ͠߹͏ɻࣜ (2.104) ͷΑ͏ͳసૹߦྻ͸ೖ‫ͱޱ‬ग़‫ޱ‬ͷ
Twiss ύϥϥϝʔλʔ͕౳͘͠ɺϕʔλτϩϯҐ૬ͷਐΈ͕ 180 ౓ͷ੔਺ഒͱͳΔΑ͏ͳ۠ؒΛҙ
ຯ͢Δɻ͜ͷछͷख๏͸ SuperKEKB ʹ΋࠾༻͞Ε͍ͯΔɻ

29

２－ 28

ୈ 3 ষ ਐߦํ޲ͷӡಈຊষͰ͸ϏʔϜਐߦํ޲ͷӡಈ (γϯΫϩτϩϯӡಈ) Λ࿦ͣΔɻγϯΫϩτϩϯӡಈ͸ओʹߴप೾Ճ଎ۭಎʹΑͬͯੜ͡ΔཻࢠͷΤωϧΪʔͱ࣌ؒํ޲ͷӡಈͰ͋Δɻ

3.1‫ܦ‬࿏௕ͱඈߦ࣌ؒ
dC
x = Dx δ

ds

ρ

dθ

ਤ 3.1: ઃ‫ͱࢠཻܭ‬ӡಈྔ͕ζϨཻͨࢠͷ‫ي‬ಓߴप೾Ճ଎ۭಎ͕ൃੜ͢Δి࣓৔͸࣌ؒมԽ͢ΔͨΊɺཻࢠͷपճ࣌ؒͱӡಈྔͷؒͷؔ܎͕ॏཁͱͳΔɻ·ͣӡಈྔ͕ઃ‫ܭ‬஋͔Β δ ͚ͩζϨཻͨࢠͷ‫ܦ‬࿏௕Λ C ɺઃ‫ࢠཻܭ‬ͷ‫ܦ‬࿏௕Λ C0 ͱͯ͠∆C
C − C0== αc δ + αc2 δ 2 + αc3 δ 3 + · · ·
C0
C0

(3.1)

ͱల։͢Δɻ͜͜Ͱ αc Λӡಈྔѹॖ཰ (ϞʔϝϯλϜίϯύΫγϣϯ) ͱ‫Ϳݺ‬ɻୈೋ߲Ҏ߱͸͍Θ͹ӡಈྔѹॖ཰ͷΫϩϚςΟγςΟͰ͋ΔɻຊߘͰ͸͜ΕΒߴ࣍ͷ߲͸ແࢹ͢Δɻͯ͞ɺਤ 3.1 ʹ༷ࣔͨ͠ʹภ޲ి࣓ੴʹ͓͍ͯӡಈྔͷͣΕཻͨࢠͷ‫ي‬ಓ௕ dC Λߟ͑Δɻ‫ز‬ԿֶΛߟ͑Δͱ




1
Dx
dC = 1 + x ds = 1 +
δ ds
(3.2)
ρ
ρ
Ͱ͋Δ͔Βӡಈѹॖ཰ αc ͸ҎԼͰ༩͑ΒΕΔ͜ͱ͕෼͔Δɻ1
Dx (s)
ds
αc =
C0
ρ(s)

(3.3)

࣍ʹඈߦ࣌ؒͷζϨ ∆T /T0 Λߟ͑Δɻඈߦ࣌ؒ T ͸ T = L/(cβ) Ͱ͋Δ͕ɺӡಈྔ͕ζϨͨ৔߹ɺ‫ܦ‬࿏௕͕มԽͨ͠ޮՌͱ଎౓͕มԽͨ͠ޮՌͷ࿨ͱͯ͠

∆C
∆β
∆T
=
−
T0
C0
β0
30

２－ 29

(3.4)

ͱॻ͚Δɻ͜Ε͸ӡಈྔѹॖ཰ αc Λ༻͍Δͱ∆T= αc − γ02 δ = ξδT0
ξ ≡ αc − γ02

(3.5)
(3.6)

ͱͳΔɻ͜͜Ͱ ξ ͸ϑΣΠζεϦοϓϑΝΫλʔͱ‫ݺ‬͹ΕΔɻϑΣΠζεϦοϓϑΝΫλʔ ξ ͕
ਖ਼ (ෛ) ͷ৔߹ɺӡಈྔ͕ઃ‫ܭ‬஋ʹൺ΂ͯߴ͍ (௿͍) ཻࢠͷඈߦ࣌ؒ͸௕͘ͳΔɻ·ͨ ξ = 0 ͷͱ͖ඈߦ࣌ؒ͸ӡಈྔʹґΒͳ͘ͳΔɻ͜ͷΑ͏ͳ৚݅Λຬͨ͢ϏʔϜϥΠϯ͸”ΞΠιΫϩφεʹ
ͳ͍ͬͯΔ”ͱ͍ͬͨ‫ํ͍ݴ‬Λ͢ΔɻϑΣΠζεϦοϓϑΝΫλʔ͕θϩͱͳΔͱ͖ͷϏʔϜΤωϧΪʔΛભҠΤωϧΪʔͱ‫Ϳݺ‬ɻ͜ͷͱ͖ͷϩʔϨϯπҼࢠ γ Λ γt ͱද‫͢ه‬ΔͱͦΕ͸1γt ≡ √αc(3.7)Ͱ༩͑ΒΕΔɻSuperKEKB ͳͲͷߴΤωϧΪʔͷԁ‫ܗ‬Ճ଎‫Ͱث‬͸ 1/γ02 ≪ 1 Ͱ͋Δ͔Β ξ ͷූ߸͸΄΅ αc ͷූ߸ͷΈͰܾ·͍ͬͯΔɻ௨ৗ αc > 0 Ͱ͋Δ͔Β ξ > 0 ͱͳΔɻ͜Ε͸ϏʔϜͷ଎౓͕΄΅ޫ଎Ͱ͋Δ͜ͱͷ‫͋Ͱ݁ؼ‬Δɻ
ຊߘͰ͸ओʹ γ0 ͕Ұఆͷ৔߹Λߟ͍͑ͯΔ͕ɺભҠΤωϧΪʔ͸γϯΫϩτϩϯ΍αΠΫϩτ
ϩϯͳͲͷϏʔϜΛपճͤ͞ͳ͕ΒՃ଎ͤ͞ΔՃ଎‫Ͱث‬ॏཁʹͳΔɻ͜ͷΑ͏ͳՃ଎‫Ͱث‬͸ϏʔϜ͸ೖࣹ͞Ε͔ͯΒ໨తͷΤωϧΪʔʹՃ଎͞ΕΔ·Ͱͷؒʹ ξ < 0 ͷྖҬ͔Β ξ = 0 Λ‫Ͱ͍ލ‬ξ > 0 ͷྖҬʹભҠ͢Δ͜ͱ͕͋ΔɻભҠΤωϧΪʔΛ‫ʹࡍ͙ލ‬ϏʔϜෆ҆ఆੑ͕༠ൃ͞ΕΔ৔߹
͕͋ΔɻભҠΤωϧΪʔͷԣஅΛආ͚Δํࡦͱͯ͠ภ޲֯ͷූ߸͕ෛ (ρ < 0) Ͱ͋Δภ޲ి࣓ੴΛ࢖͏ͳͲͯ͠ αc ΛෛʹͳΔΑ͏ʹઃ‫͠ܭ‬ɺભҠΤωϧΪʔΛ‫͢ʹ਺ڏ‬Δํ๏͕͋Δɻ

3.2

Ґ૬҆ఆੑͷ‫ݪ‬ཧ·ͣγϯΫϩτϩϯӡಈΛఆੑతʹઆ໌͢ΔͨΊʹҐ૬҆ఆੑͷ‫ݪ‬ཧͱ‫ݺ‬͹ΕΔՃ଎‫ث‬ՊֶΛࢧ͑Δॏཁͳ‫ݪ‬ཧΛઆ໌͢Δɻࠓɺԁ‫ܗ‬Ճ଎‫ث‬ͷ͋Δ৔ॴʹՃ଎ۭಎ͕͋Γ࣌ؒมԽ͢Δి৔

V (t) = Vrf sin(ωrf t + ϕs )(3.8)

Higher energy particle

Acceleration

Synchronous particle

Vrf [arb.unit]

Vrf1

Lower energy particle

0

ξ<0

ξ>0

−V
−1
rf
Deceleration−π/2−1.571

π/2
0
0.000

Deceleration

π
π/2
1.571
3.142
RF
phase
φ
[rad]
RF phase [rad]3π/2

ਤ 3.2: ઃ‫( ܭ‬ಉ‫ࢠཻ )ظ‬ɺΤωϧΪʔ͕ߴ͍‫ͼٴ‬ΤωϧΪʔ͕௿ཻ͍ࢠͷి৔೾‫( ܗ‬3.8) ͱͷҐ૬ؔ܎Λࣔ͢໛ࣜਤɻ

31

２－ 30

Λൃੜ͍ͯ͠Δͱ͢ΔɻҐ૬ ϕs ͸γϯΫϩτϩϯҐ૬ͱ‫ݺ‬͹ΕΔɻ͋ͨ͞͠Γ ϕs = π ͱ͠ɺ࣌ࠁ
t = 0 Ͱઃ‫͕ࢠཻܭ‬௨ա͢Δͱ͠Α͏ɻ·ͨ֯प೾਺ ωrf ͸ઃ‫ࢠཻܭ‬ͷपճप೾਺ ωs ͱ ωrf = hωsͷؔ܎ʹ͋Δͱ͢Δɻୠ͠ɺh ͸ਖ਼ͷ੔਺ͱ͢Δɻ͜ͷ੔਺ h ΛϋʔϞχοΫε਺ͱ‫Ϳݺ‬ɻࠓͷ৔߹ɺઃ‫ࢠཻܭ‬͸ৗʹి৔͕θϩͷ࣌ʹՃ଎ۭಎΛ௨ա͢ΔͨΊՃ‫ݮ‬଎͞Εͳ͍ɻ͜͜Ͱਤ 3.2 ʹ༷ࣔͨ͠ʹిѹ೾‫( ܗ‬3.8)ɺཻࢠɺΤωϧΪʔ͕ζϨͨઃ‫ࢠཻܭ‬ͷӡಈΛߟ͑ΔɻϑΣΠζεϦοϓϑΝΫλʔ͕ਖ਼ͷ৔߹ɺΤωϧΪʔͷߴཻ͍ࢠ͸ઃ‫ʹࢠཻܭ‬ൺ΂ͯपճ͕࣌ؒ
௕͍ͨΊɺͦΕ͚ͩ஗͘Ճ଎ۭಎʹ౸ண͢Δɻैͬͯిѹ೾‫ܗ‬ͷ‫ݮ‬଎Ґ૬ʹ৐ΓɺΤωϧΪʔΛࣦ͏ɻҰํɺΤωϧΪʔ͕௿ཻ͍ࢠ͸पճ͕࣌ؒ୹͍ͨΊɺઃ‫ʹࢠཻܭ‬ൺ΂ͯՃ଎ۭಎʹ౸ண͢Δͷ͕ૣ͘ͳΓɺՃ଎ۭಎ͔ΒΤωϧΪʔΛಘΔɻ͜ͷΑ͏ʹՃ଎ۭಎ͸ΤωϧΪʔ͕ߴཻ͍ࢠɺ
௿ཻ͍ࢠͷ྆ํʹରͯͦ͠ͷΤωϧΪʔΛઃ‫ܭ‬ΤωϧΪʔʹଗ͑Δํ޲ʹ࡞༻͢Δɺ͢ͳΘͪ෮‫ྗݩ‬ɺऩଋ࡞༻Λ༩͑Δɻ
Ճ଎ۭಎʹΑΔऩଋ࡞༻ͷ݁ՌɺϏʔϜ͸Ճ଎‫ث‬தʹ h ‫͞ू܈ʹݸ‬ΕΔ͜ͱʹͳΔ (Bunching)ɺ
͜ͷմΛόϯνͱ‫Ϳݺ‬ɻͭ·ΓϋʔϞχοΫε਺ h ͸ͦͷՃ଎‫ʹث‬ੜ੒Ͱ͖Δόϯν਺ͷ࠷େ஋
Ͱ͋ΔɻϑΣΠζεϦοϓϑΝΫλʔ͕ෛͷ৔߹͸पճ࣌ؒͱΤωϧΪʔେখͷؔ܎͕൓స͢Δɻ͜ͷ৔߹΋ઃ‫ࢠཻܭ‬ͷҐ૬Λ 180 ౓‫ٯ‬ଆͷ (ϕs − π) ʹબ΂͹ಉ༷ʹऩଋ࡞༻ΛಘΒΕΔɻϏʔϜશମΛՃ଎͍ͨ͠৔߹ɺಉ‫ظ‬Ґ૬ ϕs ΛՃ଎Ґ૬ʹબͼɺՃ଎ʹ൐͏ϏʔϜͷपճ࣌ؒͷมԽʹಉ‫ۭͯ͠ظ‬ಎͷप೾਺ ωrf Λม͑Ε͹ϏʔϜΛਐߦํ޲ʹ‫͕ͳͤ͞ू܈‬ΒՃ଎͢Δ͜ͱ͕Ͱ͖Δɻ͞Βʹԣํ޲ͷϏʔϜऩଋΛߦ͏ͨΊʹՃ଎ʹԠͯ͡ి࣓ੴͷ࣓৔΋‫͍ͯ͘͘͠ڧ‬ɻ͜Ε͕γϯΫϩτϩϯͱ‫ݺ‬͹ΕΔ͘͝య‫ܕ‬తͳԁ‫ܗ‬Ճ଎‫͚͓ʹث‬ΔϏʔϜՃ଎ͷ‫ݪ‬ཧͰ͋ΔɻҰൠʹిࢠγϯΫϩτϩϯͷ৔߹ɺೖࣹ࣌఺Ͱ‫ʹط‬ϏʔϜͷ଎౓͸΄΅ޫ଎Ͱ͋Δ͔Βۭಎͷप೾਺͸ҰఆͰྑ͍ɻҰํɺՃ଎தʹ଎౓ β0 c ͕େ͖͘มԽ͢Δཅࢠ΍ॏΠΦϯͷγϯΫϩτϩϯ͸ۭಎͷप೾਺ ωrf ΛϏʔϜՃ଎ʹ൐͍ߴ͍ͯ͘͘͠ඞཁ͕͋ΔɻຊߘͰ͸ిࢠ΍ཅిࢠϏʔϜΛ૝ఆ͠ωrf ͸Ұఆͱ͢Δɻ

3.3

γϯΫϩτϩϯӡಈ

3.3.1

ӡಈํఔࣜͷ௚઀తͳಋग़

ߴप೾Ճ଎ۭಎͷՃ଎ిѹ V (t) ΛվΊͯॻ͖Լ͢ɻ

V (t) = Vrf sin(hωs t + ϕs ),ωs = ωrf /h(3.9)

ઃ‫ࢠཻܭ‬͸Ճ଎ۭಎΛ௨ա͢Δͨͼʹৗʹ∆Es = qVrf sin ϕs(3.10)

ͷΤωϧΪʔΛಘΔɻपճप೾਺͕ fs = ωs /2π Ͱ͋Δ͔Βɺ୯Ґ࣌ؒ౰ͨΓͷΤωϧΪʔརಘ͸

dEs
ωs
=
qVrf sin ϕs(3.11)dt
2π
Ͱ͋Δɻ͜͜Ͱɺઃ‫͔ࢠཻܭ‬ΒΤωϧΪʔ͕ ∆E = E − Es ͚ͩͣΕཻͨࢠΛߟ͑Δɻ୯Ґ౰ͨΓͷΤωϧΪʔརಘ͸पճप೾਺‫ͼٴ‬Ґ૬ΛͦΕͧΕ ω, ϕ ͱͯ͠dEω=qVrf sin ϕdt
2π͜͜Ͱɺࣜ (3.11) ͱ (3.12) ͷ྆ลΛͦΕͧΕ ωs , ω Ͱআͯࠩ͠ΛͱΔɻࠨล͸
1 dEs
1 d(∆E) ω − ωs dE
1 dE
−
=
−
ω dt
ωs dt
ωs dt
ω 2 dt s
1 d(∆E)dE ∆(1/ωs )+∆E + · · ·
∼
ωs dt
dt ∆E


d ∆E
=
dt ωs
32

２－ 31

(3.12)(3.13)

Ͱ͋Δ͔ΒɺΤωϧΪʔํ޲ͷӡಈํఔࣜͱͯ͠


qVrf
d ∆E
(sin ϕ − sin ϕs )
=
dt ωs
2π

(3.14)

ΛಘΔɻ͜͜Ͱɺཻࢠ͕Ճ଎‫ث‬ΛҰप͢Δ͜ͱͰ 2π ਐΉม਺ θ Λಋೖ͢Δͱɺઃ‫͔ࢠཻܭ‬ΒͷҐ૬ࠩ͸ θ Λ༻͍ͯ ∆ϕ = ϕ − ϕs = −hθ Ͱ͋Δɻ͜ͷͱ͖पճ֯प೾਺ͷͣΕ ∆ω ͸

1 d
1 dϕ
d
∆θ = −∆ϕ = −dt
h dt
h dt
ͱॻ͚ΔɻҰํɺपճप೾਺ͷͣΕ͸ࣜ (3.5) ͔Β∆ω =∆ω = −ξδω0(3.15)(3.16)

Ͱ͋Δ͔Βɺ

dϕ= hω0 ξδ
(3.17)dt
ͱͳΔɻैͬͯɺγϯΫϩτϩϯӡಈΛ‫ه‬ड़͢Δ࿈ཱඍ෼ํఔࣜ͸ࣜ (3.14) ͱࣜ (3.17) Λ·ͱΊͯdϕ= hωs ξδ
(3.18)

dt 

qVrf
d βs2 Es δ=
(3.19)(sin ϕ − sin ϕs )

dt
ωs
2π

ͱͳΔɻୠ͠ɺδ = ∆Es /(βs2 Es ) Λ༻͍ͨɻ͜͜Ͱ ϕ ͱ δ Ҏ֎ͷύϥϝʔλ͕ҰఆͱΈͳͤΔ୹͍࣌ؒͷൣғΛߟ͑Δ͜ͱʹͯ͠ɺ͜ΕΒΛఆ਺ͱΈͳ͢ͱ (ϕ, δ) ʹؔ͢Δ࿈ཱඍ෼ํఔࣜdϕ= hωs ξδ
(3.20)
dtdδqωs Vrf=(sin ϕ − sin ϕs )(3.21)
dt
2πβs2 Es

ͱͳΔɻࣜ (3.21) ͸ྗֶͷߨٛͷ୊ࡐͱͯ͠ѻΘΕΔ୯ৼΓࢠͷӡಈํఔࣜͱಉ͡‫ܗ‬Λ͍ͯ͠Δɻ
ैͬͯγϯΫϩτϩϯӡಈ͸୯ৼΓࢠͱಉ༷ʹҰൠʹ͸୯ৼಈͱ͸ͳΒͳ͍ɻ͜͜Ͱ͸ඍখৼಈʹ͍ͭͯߟ͑ͯΈΔɻཻࢠͷҐ૬͕ઃ‫ࢠཻܭ‬ͷҐ૬ʹे෼ʹ͍ۙ (|∆ϕ| = |ϕ − ϕs | ≪ 1) ͱԾఆ͢Δͱɺࣜ (3.20) ‫( ࣜͼٴ‬3.21) ͔Βhqωs2 Vrf ξ cos ϕs
d2 ∆ϕ
=
∆ϕ
dt2
2πβs2 Es

(3.22)

ΛಘΔɻैͬͯ qVrf > 0 Ͱ͋ΔͳΒɺξ cos ϕs < 0(3.23)

ͷͱ͖ɺͦͷӡಈ͸ઃ‫ࢠཻܭ‬पΓͷ୯ৼಈͱͳΔɻ͜ͷ৚݅͸ಉ‫ظ‬Ґ૬ ϕs ͱભҠΤωϧΪʔ γt ʹ
ର͢Δ৚݅ͱͯ͠

(3.24)
0 ≤ ϕs ≤ π/2 for γs < γt or ξ < 0

π/2 ≤ ϕs ≤ π

forγs > γt or ξ > 0(3.25)

Ͱ͋Δɻਤ 3.2 Ͱ‫ͨࣔ͠ʹط‬Α͏ʹϏʔϜΤωϧΪʔ͕ભҠΤωϧΪʔΑΓখ͍͔͞େ͖͍͔ʹΑͬͯ҆ఆྖҬ͕มΘΔɻ͜Ε͸ભҠΤωϧΪʔΛ‫ʹڥ‬ΤωϧΪʔͷζϨ͕पճ࣌ؒʹ༩͑ΔӨ
‫͕ڹ‬൓స͢ΔͨΊͰ͋Δɻ҆ఆ৚݅ΛΈͨ͢ͱ͖ɺ͜ͷ୯ৼಈͷ֯प೾਺Λઃ‫ࢠཻܭ‬ͷपճ֯प೾਺ ωs Ͱআͨ͠ྔ

hqVrf |ξ cos ϕs |νs =(3.26)
2πβs2 EsΛϕʔλτϩϯνϡʔϯʹରͯ͠γϯΫϩτϩϯνϡʔϯͱ‫Ϳݺ‬ɻ

33

２－ 32

3.3.2ϋϛϧτϯ‫ࣜܗ‬લઅͰγϯΫϩτϩϯৼಈͷӡಈํఔࣜΛಘ͕ͨɺͦ͜Ͱ͸ϋϛϧτχΞϯʹ‫͍ͨͮج‬Ξϓϩʔν͸ͱΒͳ͔ͬͨɻཧ༝͸ൺֱత௚‫ײ‬తʹཧղ͠΍͍ͨ͢ΊͰ͋Δɻ͜͜Ͱ͸ϋϛϧτχΞϯΛ
࢖͏͜ͱͰಉ༷ʹγϯΫϩτϯӡಈͷํఔ͕ࣜಘΒΕΔ͜ͱΛࣔ͢ɻͦͷաఔͰӡಈํఔࣜ (3.20)‫( ͼٴ‬3.21) ͸ಛघͳ৚݅ԽͰ੒ཱ͢Δ΋ͷͰ͋Δ͜ͱΛ஌Δɻ
·ͣɺ‫֨ن‬Խ͢ΔલͷϋϛϧτχΞϯ (1.28) ͔Βग़ൃ͢Δɻ

H(x, px , y, py , t, −E; s) =




ps
∆E
x ∆E
x
∆E 2
ps − ps 2
− 1+
+ 2
− ps
2
ρ
βs Es 2γs βs Esρ βs2 Es1(px − qAx )2 + (py − qAy )2 − qAs+
2ps(3.27)ୠ͠ (β0 , γ0 , p0 , E0 ) → (βs , γs , ps , Es ) ͱॻ͖‫ͨ͑׵‬ɻ͜͜Ͱ Ax = Ay = 0 ͱͯ͠ As ͱͯ͠ਫฏ
ํ޲ͷภ޲ి࣓ੴͱ࢛‫࣓ిۃ‬ੴ͓ΑͼՃ଎ۭಎ͔Βͷ‫د‬༩ Arf ͷΈΛߟ͑Δͱɺ

H( x, px , y, py , t, −E; s) =


ps
ps
∆E
∆E x p2x + p2y∆E 2−ps − ps 2
+
+ 2
−
p
+ K1 (x2 − y 2 ) − Arfs22
βs Es 2γs βs Es
β s Es ρ
2ps
2(3.28)

Ԗ௚ํ޲ͷӡಈ͸໌Β͔ʹਫฏํ޲‫ͼٴ‬ਐߦํ޲ͱ͸ಠཱͰ͋Δ͔ΒɺҎ߱Ͱ͸Ԗ௚ํ޲ͷਖ਼४ม਺͸ল͘ɻ
ࠓɺ৔ॴ s = 0 ʹஔ͔Εͨे෼ʹബ͍Ճ଎ۭಎ͕ϏʔϜਐߦํ޲ͷి৔ Erf ͱͯ͠

Erf = −

∂As
= Vrf δp (s) sin(ωrf t + ϕs )∂t

(3.29)

Λൃੜ͍ͯ͠Δͱ͢Δɻ͜͜Ͱ δp ͸प௕ C = 2πR Λप‫͢ͱظ‬Δσϧλؔ਺Ͱ͋ΔɻΤωϧΪʔEs ͷઃ‫ࢠཻܭ‬͸ t = 0 ͰҐ૬ ϕ = ϕs ͰۭಎΛ௨ա͢Δɻ͜ͷి৔ͷϕΫτϧϙςϯγϟϧ͸

Arf =

Vrf
δp (s) cos(ωrf t + ϕs )
ωrf

(3.30)

Ͱ༩͑ΒΕΔɻ࣍ʹΤωϧΪʔ͕ઃ‫ܭ‬ΤωϧΪʔ Es ͔Β ∆E ζϨͨ‫ي‬ಓʹ࠲ඪ‫ܥ‬Λ৐Γ‫͑׵‬Δɻ͜ͷม‫׵‬͸฼ؔ਺ˆ =F1 (x, p̂x , t, −∆E)



ˆ
∆E
x−D 2
β s Esˆˆ t + D′ ps x ∆E − 1 DD′ psp̂x − Es + ∆E
2
β s Es 2ˆ∆E2
β s Es2(3.31)

ʹΑ࣮ͬͯ‫͖Ͱݱ‬Δɻ͜ͷ฼ؔ਺͔Β৽͍͠ਖ਼४ม਺͸p̂x = px − ps D′ˆ∆E
βs2 Esˆ∆E
βs2 EsDD′x̂t̂ = t + 2 p̂x −β s Esβs cx̂ = x − D,

ˆ = E − Es ,
∆E

34

２－ 33

(3.32)(3.33)

ͱͳΔɻ͜͜Ͱ D ͸ s ͷཅͳؔ਺ͱ͢Δɻ৽͍͠ϋϛϧτχΞϯ͸


2


ˆ
ˆ 1
ˆ
ˆ
ps
∆E
∆E
∆E
∆E
x̂ + D 2+
H 1 = − ps − ps 2
− ps 2βs Es 2γs2 βs2 Es
β s Es ρ
β s Es





2
2 
ˆ
ˆ
ˆ
ˆ
1  2
∆E
∆E
 + ps K1 x̂2 + 2Dx̂ ∆E + D2 ∆E+p̂ + 2D′ ps p̂x 2+ D′2 p2s
2ps x
β s Esβs2 Es2βs2 Esβs2 Es− qArf (t̂; s) +∂F1
∂s

(3.34)2ˆͱͳΔɻҰൠʹ ∂F1 /∂s ͷ‫ʹࢉܭ‬͸ D, D′ , ps ‫ͼٴ‬ɺ∆E/β
s Es ͷ s ඍ෼͕൐͏͕ɺҎ߱Ͱ͸ઃ‫ཻܭ‬ࢠͷΤωϧΪʔ Es ‫ ͼٴ‬D, D′ ͷΈ͕ s ʹཅʹґଘ͢ΔͱԾఆ͢Δɻ͜ͷԾఆͷ΋ͱͰ ∂F1 /∂s ͸




2
ˆ
ˆ
ˆ
ˆ
 ∆E
ps  ′2
∂F1
∆E
∆E
′′
′′
′ ∆E
= −D 2 p̂x + D ps x̂ + D 2
D + DD−∂sβ s Esβs Es βs2 Es2
βs2 EsDD′x̂ sin ϕs− qVrf δp (s) t̂ − 2 p̂x +β s Esβs c(3.35)

ͱͳΔɻୠ͠ɺ

∂Es= qVrf δp (s) sin ϕs∂s(3.36)

Λ༻͍ͨɻ͜ΕΛࣜ (3.34) ʹ୅ೖͯ͠੔ཧ͢Δͱˆ
1 2 ps∆E+p̂ + K1 x̂22
βs Es 2ps x
2






ˆ
1
ps
2
1
∆E
′′
′′
−D D + K1 D −− 2 − ps D + K 1 D +
x̂ 2
−
2
ρ
γs
ρ
β s EsDD′− qVrf δp (s) t̂ − 2 p̂x +x̂ sin ϕs − qArf (t̂; s)β s Esβs c

H2 = −ps − ps

ˆ ʹൺྫ͢Δ߲͕θϩͱͳΔΑ͏ʹબͿɻ͢ͳΘͪඍ෼ํఔࣜ
͜͜Ͱؔ਺ D(s) Λ x̂∆E
D′′ + K1 D =1ρ(3.37)

Λຬͨ͢प‫ظ‬ղͱ͢Δɻ࣮͸ D(s) ͸ԣํ޲ͷӡಈͷఆࣜԽͰಋೖͨ͠෼ࢄؔ਺ͦͷ΋ͷͰ͋Δɻ
ԣํ޲ͷӡಈΛఆࣜԽͨ͠ࡍʹ͸ӡಈྔ͕ζϨཻͨࢠʹରͯ͠ӡಈํఔࣜΛߟ͑Δ͜ͱͰ෼ࢄؔˆ ͷ߲Λফ͢ɺͱ͍͏໨తͰಋೖ͞ΕΔɻ
਺Λಋೖͨ͠ɻҰํɺ͜͜Ͱ͸ϋϛϧτχΞϯ͔Β x̂∆E
ࣜ (3.37) Λ༻͍ΔͱɺϋϛϧτχΞϯ (3.37) ͸  ˆ 2
ˆ
ps D
p̂2x
ps
∆E
1
∆E
− 2
H3 = −ps 2−++ K1 x̂2
β s Es2 ργsβs2 Es
2ps
2




 
D
D
D′
qVrf
D′
x̂ sin ϕs −x̂ + ϕs− qVrf δp (s) t̂ − 2 p̂x +δp (s) cos ωrf t̂ − 2 p̂x +β s Esβs c
ωrf
β s Esβs c(3.38)
ͱॻ͚Δɻୠ͠ɺఆ਺߲ −ps Λল͍ͨɻ

35

２－ 34

͜ͷϋϛϧτχΞϯ͔Βਫฏํ޲ͷӡಈͱਐߦํ޲ͷӡಈ͸Ճ଎ۭಎͰͷ෼ࢄؔ਺Λ௨ͯ݁͠
߹͍ͯ͠Δ͜ͱ͕෼͔Δɻ͜ͷΑ͏ͳӡಈͷ݁߹ʢΧοϓϦϯάʣ͸γϯΫϩτϩϯνϡʔϯͱ
ϕʔλτϩϯνϡʔϯ͕ಛఆͷ৚݅Λຬͨ࣌͢ʹཻࢠͷӡಈΛෆ҆ఆԽͤ͞ΔՄೳੑ͕͋ΔɻʢγϯΫϩϕʔλ‫ڞ‬໐ʣ ैͬͯ௨ৗɺՃ଎ۭ۠ಎ͸෼ࢄؔ਺͕θϩͰ͋Δ௚ઢ෦ʹઃஔ͞ΕΔɻ‫ʹٯ‬ԿΒ͔ͷ໨తͷͨΊʹੵ‫ۃ‬తʹγϯΫϩτϩϯӡಈͱϕʔλτϩϯӡಈΛ݁߹͍ͤͨ͞৔߹ʹ͸෼ࢄؔ਺ͷ͋Δ৔ॴʹՃ଎ۭಎΛઃஔ͢Ε͹ྑ͍ɻͯ͞ɺՃ଎ۭಎͰ෼ࢄؔ਺θϩͰ͋Δ৔߹ɺϋϛϧτχΞϯ (3.38) ͸

H3 = H x + Hsp̂2xps
+ K1 x̂2
Hx =
2ps
2
Hs

(3.39)(3.40)



ˆ
1
ps D
∆E
− 2
= −ps 2−β s Es
2 ρ
γs



ˆ
∆E
2
β s Es2− qVrf t̂δp (s) sin ϕs −



qVrf
δp (s) cos ωrf t̂ + ϕs
ωrf

(3.41)

ͱͳΓɺਫฏํ޲ͷϋϛϧτχΞϯ Hx ͱਐߦํ޲ͷϋϛϧτχΞϯ Hs ʹ෼཭Ͱ͖Δɻ͜͜Ͱ͸γϯΫϩτϩϯӡಈΛ࿦ͣΔͨΊ Hs ͷΈߟ͑Δɻ
·ͣɺ฼ؔ਺
hs+ ϕs W, ωrf = hωsF2 = ωrf t̂ −(3.42)
R
Λ༻͍ͯਖ਼४ม‫׵‬Λߦ͏ɻ৽‫چ‬ͷਖ਼४ม਺ͷؒͷؔ܎͸

W =−

ˆ
ˆ
R ∆E
∆E
= − ps 2 ,ωrf
h β s Es

ϕ = ωrf t̂ −

hs
+ ϕs
R

(3.43)

Ͱ͋Γɺ৽͍͠ϋϛϧτχΞϯ͸






2
ωrf
qVrf
qVrf
hs
hs
1
D
2
− 2 W −
+ ϕs sin ϕs −Hs1 = − 3δp (s) ϕ +δp (s) cos ϕ +2βs cEs ρ
γs
ωrf
R
ωrf
R
(3.44)
࣍ʹ Hs1 ΛϦϯάҰपͰฏ‫ۉ‬Խͯ͠ਖ਼४ม਺Λ‫߲͍ͳ·ؚ‬Λল͘ͱĤs ∼ −

2ξ
ωrf
qVrf
qVrf sin ϕs
cos ϕ −
ϕW2 −32βs cEs
2πωrf R2πωrf R

ͱͳΔɻ͜͜Ͱɺξ ͸ࣜ (3.5) Ͱఆٛͨ͠ϑΣΠζεϦοϓϑΝΫλʔ

 
1
D
1
− 2 dsξ=
2πRργsͰ͋Δɻ͜ͷϋϛϧτχΞϯ͔Βਖ਼४ํఔࣜ͸


ω2 ξ
dϕ
∂ Ĥs


=
= − 3 rf Wds∂W
βs cEs

dW
∂ Ĥs
qVrf



=−
=−
(sin ϕ − sin ϕs )ds
∂ϕ
2πωrf R

ͱͳΔɻ͜͜Ͱɺs = βs ct, W = −βs2 Eδ/ωrf , ωrf = hωs ͳͲΛ༻͍Δͱ࠷ऴతʹdϕ= hωs ξδdtdδ
ωs qVrf=(sin ϕ − sin ϕs )dt
2πβs2 Es

(3.45)(3.46)(3.47)(3.48)(3.49)(3.50)

ΛಘΔɻ͜ͷ݁Ռ͸ϋϛϧτχΞϯΛ࢖Θͣʹ‫ٻ‬Ίͨࣜ (3.20) ‫( ࣜͼٴ‬3.21) ͱ͔֬ʹҰக͢Δɻ

36

２－ 35

(a) φs = 0 deg.

(c) φs = 30 deg.

(b) φs = 10 deg.

3

3

3

2

2

2

1

1

1

0

0

0

−1

−1

−1

−2

−2

−2

−3

−3

−3

U (φ)

−2π

−π

0

π

2π −2πPhase coordinate φ [rad]

0

−π

π

2π −2π−πPhase coordinate φ [rad]

0

π

2π

Phase coordinate φ [rad]ਤ 3.3: γϯΫϩτϩϯҐ૬ ϕs ͕ (a) 0 ౓ɺ(b) 10 ౓ɺ(c) 30 ౓ͷ৔߹ͷϙςϯγϟϧΤωϧΪʔU (ϕ) ͷ‫ܗ‬

3.3.3

ΤωϧΪʔΞΫηϓλϯε

‫ʹ༷ͨݟʹط‬γϯΫϩτϩϯӡಈ͸ண໨ཻࢠͱઃ‫ࢠཻܭ‬ͷζϨ͕ඍখͳ৔߹ʹ͸୯ৼಈͱͳΔɻ
͜͜Ͱ͸ζϨ͕ඍখͰͳ͍৔߹΋‫ؚ‬Ίͯߟ͑Δɻࠓɺਖ਼४ม਺Ҏ֎ͷύϥϝʔλʔ͕ఆ਺Ͱ͋Δ
ͱ͢Ε͹ Ĥs ͸ s ʹཅʹґଘ͠ͳ͍ɻैͬͯ A Λఆ਺ͱͯ͠

−

2ξ
ωrf
qVrf
qVrf sin ϕs
cos ϕ −
ϕ=AW2 −3
2βs cEs
2πωrf R2πωrf R

(3.51)

ਖ਼४ํఔࣜΛ༻͍ͯ W Λফ‫͢ڈ‬Δͱ

qVrf sin ϕs
β 3 cEsqVrf
cos ϕ −
ϕ=A
− s 2 ϕ̇2 −2πωrf R2πωrf R2ωrf ξ

(3.52)

ͱ͍͏ॻ͚Δɻ͜͜Ͱ ϕ̇ = dϕ/ds Ͱ͋Δɻ͞Βʹ྆ลʹద౰ͳ਺Λֻ͚ͯࣜ (3.26) ͷ νs Λ༻͍ͯ
੔ཧ͢Δͱ J Λఆ਺ͱͯ͠
1  νs 2
1 2
ϕ̇ −(cos ϕ + ϕ sin ϕs ) = J(3.53)
2cos ϕs Rͱ͍͏ؔ܎ࣜΛಘΔɻҎ্ʹΑΓγϯΫϩτϩϯৼಈ͸ϙςϯγϟϧΤωϧΪʔ1  νs 2
U (ϕ) = −(cos ϕ + ϕ sin ϕs )cos ϕs R(3.54)

ʹ͓͚Δ࣭఺ͷӡಈͱΈͳͤΔɻਤ 3.3 ʹ ϕs ͕ 0 ౓ɺ10 ౓ɺ30 ౓ͷ৔߹ͷϙςϯγϟϧΤωϧ
Ϊʔ U (ϕ) Λࣔ͢ɻཻࢠ͕҆ఆʹৼಈӡಈΛߦ͏ʹ͸௚ۙͷϙςϯγϟϧͷࢁʹൺ΂ͯΤωϧΪʔͷζϨ͕খ͘͞ͳ͚Ε͹ͳΒͳ͍ɻ࣍ʹࣜ (3.53) Ͱඳ͔ΕΔ౳ߴઢΛਤ 3.4 ʹࣔ͢ɻཻࢠͷॳ‫ظ‬৚݅ʹΑܾͬͯ·Δ J ͕͋Δᮢ஋
Jmax ΑΓখཻ͍͞ࢠ͸෮‫ʹྗݩ‬Αͬͯ ϕs ͷपΓͰৼಈ͢Δ͕ɺᮢ஋ Jmax Λӽ͑Δͱෆ҆ఆԽ͢Δɻ͜ͷ҆ఆͱෆ҆ఆΛܾΊΔᮢ஋ Jmax ʹΑͬͯඳ͔ΕΔ‫ۂ‬ઢΛηύϥτϦοΫεͱ‫Ϳݺ‬ɻηύϥτϦοΫεΛ‫ٻ‬ΊΔͨΊʹ U (ϕ) ͷ‫ ఺ۃ‬ϕ = ϕsp Λ‫ٻ‬ΊΔɻ৚݅ dU/dϕ = 0 Λղ͘͜ͱͰ
ϕsp = ϕs , π − ϕs ͷೋͭͰ͋Δ͜ͱ͕෼͔Δɻϕsp = π − ϕs , ϕ̇ = 0 Λࣜ (3.53) ʹ୅ೖͯ͠

Jmax = −1  νs 2[cos(π − ϕs ) + (π − ϕs ) sin ϕs ]cos ϕs R

ΛಘΔɻैͬͯ (ϕ, ϕ̇) ฏ໘্ͰηύϥτϦοΫε͸
2  ν s 22[cos ϕ + cos ϕs + (ϕ + ϕs − π) sin ϕs ]ϕ̇ =cos ϕs R

37

２－ 36

(3.55)(3.56)

3

2

2

φ˙

0

1
0

φ˙

φ̇ ∝ ∆E/Es(∆E/Es )max

3

1

−1

−1

−2

−2

−3

−2π

φ = φ s φ = π − φs(b) φs = 30 deg.

(∆E/Es )max(a) φs = 0 deg.

−3

−π
0
π
Phasecorrdinate
coordinate φφ[rad]
Phase
[rad]

2π −2π

0

−π

π

2π

Phasecorrdinate
coordinate φφ[rad]
Phase
[rad]

ਤ 3.4: γϯΫϩτϩϯҐ૬ ϕs ͕ (a) 0 ౓ɺ(b) 30 ౓ͷ৔߹ͷࣜ (3.53) Ͱඳ͔ΕΔ౳ߴઢɻ੺ͷ఺ઢ͸ࣜ (3.56) Ͱ༩͑ΒΕΔηύϥτϦοΫεͰ͋ΔɻͰ༩͑ΒΕΔɻ·ͨɺ͜ͷͱ͖ ϕ = ϕs Ͱ ϕ̇2 ͕࠷େ஋ϕ̇2max =
4  νs 2 
πcos ϕs + ϕs −sin ϕscos ϕs R
2(3.57)

ʹͳΔɻ͜Ε͸ΤωϧΪʔͷζϨͷ૬ର஋ ∆E/Es ʹ‫͢ࢉ׵‬Δͱ



∆E
Es

2

4
=
cos ϕsmax=βs3 cνsωrf |ξ|R

2 



π
cos ϕs + ϕs −sin ϕs22βs2 qVrf
2βs2 qVrf[2 cos ϕs + (2ϕs − π) sin ϕs ] =G(ϕs )
πhEs |ξ|πhEs |ξ|(3.58)G(ϕs ) = 2 cos ϕs + (2ϕs − π) sin ϕs

ͱͳΔɻ͜Ε͸͢ͳΘͪॳ‫ʹظ‬ઃ‫ͱࢠཻܭ‬ಉ͡Ґ૬Ͱཻ͋ͬͨࢠ͕҆ఆʹपճ͢ΔͨΊʹ‫ڐ‬༰͞ΕΔΤωϧΪʔͷζϨͷ࠷େ஋Ͱ͋Δɻ
ਤ 3.5 ʹࣜ (3.58) ͷؔ਺ G(ϕs ) Λ‫ܗ‬Λࣔ͢ɻ‫ڐ‬༰஋͸γΫϩτϩϯҐ૬ ϕs = 0ɺ͢ͳΘͪՃ‫ݮ‬଎ͳ͠ͷͱ͖࠷େͱͳΔɻ·ͨɺγϯΫϩτϩϯҐ૬͕‫ݻ‬ఆͰ͋ΔͳΒ͹ɺ‫ڐ‬༰஋Λେ͖ͨ͘͠
͍৔߹͸Ճ଎ۭಎͷిѹΛେ͖͘͢Δඞཁ͕͋ΔɻՃ଎‫ث‬Λ҆ఆʹपճͰ͖ΔΤωϧΪʔͷζϨ
15.0
12.5

G(φs )

10.0
7.5
5.0
2.5
0.0
−π

0

−π/2

π/2

φs

ਤ 3.5: ࣜ (3.58) ͷؔ਺ G(ϕs )

38

２－ 37

π

ͷ࠷େ஋ΛΤωϧΪʔΞΫηϓλϯεͱ‫Ϳݺ‬ɻΑΓҰൠʹ͸ɺΤωϧΪʔํ޲ʹ‫ݶ‬ΒͣɺՃ଎‫ث‬্Λ҆ఆʹपճͰ͖ΔϏʔϜͷ޿͕Γͷ࠷େ஋ΛͦͷՃ଎‫ث‬ͷΞΫηϓλϯεͱ‫Ϳݺ‬ɻ

39

２－ 38

ୈ 4 ষ γϯϓϨΫςΟοΫม‫ࢠཻͱ׵‬τϥοΩϯά‫͖ͨͯݟʹط‬Α͏ʹཻࢠͷӡಈΛࢧ഑͢Δӡಈํఔࣜ͸ຊ࣭తʹඇઢ‫͋Ͱܗ‬ΓɺҰൠͷ৔߹ʹղੳతʹղ͘͜ͱ͸Ͱ͖ͳ͍ɻैͬͯɺՃ଎‫ث‬ͷઃ‫ܭ‬΍࣮‫ػ‬ͷղੳʹ͸‫ʹػࢉܭ‬ΑΔ਺஋γϛϡϨʔγϣϯ͕ෆՄܽͰ͋ΔɻಛʹɺϏʔϜϥΠϯதͷཻࢠͷ‫ي‬ಓΛ௥੻ (τϥοΩϯά) ͢Δ਺஋γϛϡϨʔγϣϯ͸τϥοΩϯάγϛϡϨʔγϣϯɺ΋͘͠͸୯ʹτϥοΩϯάͱ‫ݺ‬শ͞ΕΔɻ͜͜Ͱ͸ɺτϥοΩϯάʹ͓͍ͯॏཁͱͳΔγϯϓϨΫςΟοΫม‫͍ͯͭʹ׵‬؆ܿʹड़΂ͨ‫ޙ‬ɺτϥοΩϯάͷख๏Λͦͷ۩ମྫΛަ͑ͯࣔ͢ɻ࠷‫ʹޙ‬ඇઢ‫ֶྗܗ‬ͷಋೖͱͯ͠ Lie ୅਺Λ༻͍ͨඇઢ‫૾ࣸܗ‬Λ༻͍ͨτϥοΩϯάͷख๏Λ֓આ͢Δɻ

4.1

γϯϓϨΫςΟοΫม‫׵‬؆୯ͷͨΊ̍࣍‫( ݩ‬q, p) ͷਖ਼४ํఔࣜΛߟ͑Δɻ

dq
∂H
=
,
ds
∂p

dq
∂H
=−
ds
∂q

(4.1)

͜͜Ͱ

z = (z1 , z2 ) = (q, p)


0 1
J =
−1 0
Λಋೖ͢Δͱਖ਼४ํఔࣜ͸

dzi
∂H
= Jij
dt
∂zj

(4.2)
(4.3)

(4.4)

ͱॻ͚Δɻ͜͜Ͱਖ਼४ม਺ z Λ z → Z = (Z1 (z1 , z2 ), Z2 (z1 , z2 )) ʹม਺ม‫͢׵‬Δ͜ͱߟ͑Δɻ͜ͷ࣌ɺม‫ޙ׵‬ͷม਺ʹؔͯ͠΋
dZi
∂H
= Jij
(4.5)
dt
∂Zj
͕ຬͨ͞ΕΔͱ͖ɺ͜ͷม‫׵‬Λਖ਼४ม‫׵‬΋͘͠͸γϯϓϨΫςΟοΫม‫Ϳݺͱ׵‬ɻ͜ͷ৚݅Λ۩ମతʹߟ͑ͯΈΔɻ

dZa
dt
ैͬͯɺ

=

∂Za ∂Zb ∂H∂Za dzi∂Za ∂H=Jij=
Jij∂zi dt∂zi∂zj∂zi∂zj ∂Zb∂Za ∂Zb
Jij
= Jab
∂zi
∂zj

(4.6)

(4.7)

Ͱ͋Ε͹ɺ͜ͷม‫׵‬͸ਖ਼४ํఔࣜΛຬ଍͢Δɻ͜ͷ৚݅͸‫͑׵͍ݴ‬Δͱม਺ม‫׵‬ͷϠίϏΞϯ



M=

∂Z1
∂z1
∂Z2
∂zi

∂Z1
∂z2
∂Z2
∂z2

40

２－ 39

(4.8)

͕

M JM t = J(4.9)

Λຬͨ͢͜ͱͰ͋Δɻ͜ͷ৚݅ΛγϯϓϨΫςΟοΫ৚݅ͱ‫Ϳݺ‬ɻ1 ‫ܥ‬ͷࣗ༝౓͕̎΍̏ͷ৔߹΋ಉ༷ͷ৚݅ΛಘΔɻઢ‫ܗ‬ม‫׵‬ͷ৔߹ɺϠίϏΞϯ M ͸ਖ਼४ม਺Λ‫ͳͱྻߦ͍ͳ·ؚ‬Δɻ͞Βʹࣗ༝౓͕̍Ͱ͋Δ৔߹ɺγϯϓϨΫςΟοΫ৚݅ (4.9) ͸det(M ) = 1(4.10)

ͱ਺ֶతʹ౳ՁͰ͋Δ͜ͱΛࣔͤΔɻ2
ͱ͜ΖͰϋϛϧτϯ‫ܥ‬ͷ࣌ؒൃలͱ͸ແ‫ݶ‬খͷਖ਼४ม‫׵‬ͷ‫܁‬Γฦ͠ͱղऍͰ͖Δʢ෇࿥ࢀরʣɻ
ҰํɺτϥοΩϯάγϛϡϨʔγϣϯͱ͸ӡಈํఔࣜΛ཭ࢄԽͯ͠৔ॴ s = si Ͱͷཻࢠͷ৘ใ͔Β s = si+1 Λ‫ٻ‬ΊΔɺͱ͍͏ม਺ม‫׵‬Λ‫܁‬Γฦ͢͜ͱͰ͋Δɻैͬͯ͜ͷม਺ม‫ͯ͠ͱ׵‬γϯϓϨΫςΟοΫม‫׵‬ΛબͿ͜ͱ͕ϋϛϧτϯ‫ܥ‬ͷ࣌ؒൃలΛۙࣅతʹ‫ٻ‬ΊΔ্Ͱຊ࣭తʹॏཁͰ͋Δ
ͱߟ͑ΒΕ͍ͯΔɻ

4.2

τϥοΩϯάख๏ͱͦͷ۩ମྫ͜͜Ͱ͸۩ମతͳྫΛ༻͍ͯτϥοΩϯάख๏ͱγϯϓϨΫςΟοΫม‫׵‬ͷಛ௃Λ‫͍ͯ͘ݟ‬ɻ͜͜Ͱѻ͏୊ࡐ͸จ‫[ ݙ‬6] ͔Βआ༻ͨ͠ɻΑΓਂֶ͘ͼ͍ͨํ͸ͦͪΒΛࢀর͞Ε͍ͨɻ࢛‫࣓ిۃ‬ੴͷϋϛϧτχΞϯ
1
1
H = p2x + kx2 , k > 0(4.11)
22
Λߟ͑Δɻͦͷసૹߦྻ M ͸‫ʹط‬ΈͨΑ͏ʹ

√
√
√ 
cos kLsin kL/ k√√√M=
cos kL− k cos kL(4.12)

Ͱ͋ͬͨɻ͜ͷ৔߹ɺզʑ͸ղੳతͳసૹߦྻΛಘΒΕΔΘ͚Ͱ͋Δ͕ɺ཭ࢄԽʹΑͬͯԿ͕‫ى‬
͜Δ͔Λ‫͍ͯ͘ݟ‬ɻసૹߦྻͷ֤੒෼Λςʔϥʔల։͢Δɻ






1
0
0
1
−k/2
0
+ L1
+ L2
+ ···
(4.13)
M = L0
0 1
−k 0
0
−k/2
࢛‫࣓ిۃ‬ੴ͕े෼ʹ୹͍ͱͯ͠ L ͷೋ࣍Ҏ্ͷ߲Λແࢹ͢Δͱ


1
L
M=
+ O(L2 )−kL 1(4.14)

͜ͷసૹߦྻɺ͢ͳΘͪม਺ม‫׵‬͸໌Β͔ʹ det(M ) ̸= 1 ͰγϯϓϨςΟοΫͰ͸ͳ͍ɻ͜͜Ͱɺ
̎ߦ̐ྻ໨ʹਓҝతʹ −kL2 ΛՃ͑ΔͱγϯϓϨΫςΟοΫͳม‫׵‬Λ༩͑Δۙࣅਫ਼౓ O(L1 ) ͷࣸ૾1
L+ O(L2 )M=(4.15)−kL 1 − kL2ͱͳΔɻ1ӳ‫ޠ‬ද‫Ͱه‬͸”Symplectic”Ͱ͋Δɻ‫ݩ‬ʑ͸਺ֶͷ෼໺ʹ͓͍ͯ”Complex”ͱද‫͞ه‬Ε͍͕ͯͨෳૉ਺ͱͷࠞಉΛආ͚ΔͨΊʹ਺ֶऀ Hermann Weyl(1885-1955) ͕‫ݹ‬୅ΪϦγϟ‫ͱޠ‬ϥςϯ‫͔ޠ‬Β଄‫ͨ͠ޠ‬ɻ
2ࣗ༝౓͕̎΋͘͠͸̏ͷ৔߹ det(M ) = 1 ͸γϯϓϨΫςΟοΫ৚݅ (4.9) ͷඞཁ৚݅Ͱ͋Δ͕े෼৚݅Ͱ͸ͳ͍ɻ

41

２－ 40

(b)

(c)

2.0

1.5

1.5

1.5

1.0

1.0

1.0

0.5

0.5

0.5

0.0

2.0

px

px

px

(a)
2.0

0.0

0.0

−0.5

−0.5

−0.5

−1.0

−1.0

−1.0

−1.5

−1.5

−1.5

−2.0
−2

2.0

−1

(d)

0

1

−2.0
−2

2

x
2.0

−1

0

1

−2.0
−2

2

x

(e)

0

−1

1

2

x

(f)

1.0

1.0

0.5

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

−1.5

−1.5

−2.0
−2

−2.0
−2

−1

0

x

1

2

0.4

((H − Hexact )/Hexact

1.5

px

px

(d)

1.5

(e)

0.2

0.0

−0.2

−0.4

−1

0

1

2

0

100

x

200

300

400

500

# of steps

ਤ 4.1: τϥοΩϯάγϛϡϨʔγϣϯʹΑΓಘΒΕͨ (x, px ) ͷ‫੻ي‬ɻͦΕͧΕɺ(a) ࣜ (4.12)ɺ
(b) ࣜ (4.14)ɺ(c) ࣜ (4.15)ɺ(d) ࣜ (4.16)ɺ(e) ࣜ (4.17) Λ༻͍ͯ‫ͨ͠ࢉܭ‬ɻϋϛϧτχΞϯͷ૬ର‫ࠩޡ‬Λ (f) ʹࣔ͢ɻۙࣅਫ਼౓্͛ΔͨΊʹςʔϥల։ (4.13) ʹ͓͍ͯ L2 ͷ߲·Ͱߟྀ͢Δͱɺ
1 − 12 kL2
L+ O(L3 )M=−kL
1 − 12 kL2(4.16)

ͱͳΔɻ͜ͷม‫׵‬΋໌Β͔ʹγϯϓϨΫςΟοΫͰ͸ͳ͍͕ɺࠓ౓͸̍ߦ̎ྻ੒෼ʹ −kL3 /4 ΛՃ͑Δ͜ͱͰۙࣅਫ਼౓ O(L2 ) ͷγϯϓϨΫςΟοΫͳࣸ૾
3
1 − 12 kL2 L − kL4+ O(L3 )M=(4.17)−kL
1 − 12 kL2ΛߏஙͰ͖Δɻਤ 4.1(a)-(e) ͸ࣜ (4.12)ɺࣜ (4.14)ɺࣜ (4.15)ɺࣜ (4.16)ɺࣜ (4.17) ʹରԠ͢Δτ
ϥοΩϯάγϛϡϨʔγϣϯʹΑͬͯಘΒΕͨҐ૬ฏ໘ (x, px ) Ͱͷཻࢠͷ‫͋Ͱ੻ي‬ΔɻγϯϓϨΫ
ςΟοΫͰͳ͍ม‫׵‬Λ࢖͏ͱঃʑʹ‫͕੻ي‬ղੳղ͔Β֎Ε͍ͯ͘͜ͱ͕෼͔ΔɻҰํɺγϯϓϨΫ
ςΟοΫͳม‫׵‬͸‫੻ي‬͸ղੳղͱζϨΛ΋͕ͭͦͷζϨ͸੒௕͠ͳ͍ɻ·ͨɺࣜ (4.16)ɺࣜ (4.17)
Λ༻͍ͨ৔߹ͷϋϛϧτχΞϯͷ‫ີݫ‬஋͔Βͷ૬ର‫ࠩޡ‬ͷ࣌ؒൃలΛਤ 4.1(f) ʹࣔ͢ɻγϯϓϨΫςΟοΫͳख๏Ͱ͋Δࣜ (4.17) Λ༻͍ͨ৔߹ɺอଘྔͰ͋Δ΂͖ϋϛϧτχΞϯͷζϨ΋ৼಈ͸͢Δ͕੒௕͠ͳ͍͜ͱ͕෼͔Δɻ
͜͜Ͱɺࣜ (4.15) ʹΑΔۙࣅ͕΋ͭ෺ཧతҙຯΛߟ͑ͯΈΔɻࣜ (4.15) ͸



1
0
1 L
M=
(4.18)
−kL 1
0 1

42

２－ 41

(a)

(b)

(c)

kL

kL

k
L

L

L

ਤ 4.2: ࢛‫࣓ిۃ‬ੴͷϞσϧɺ(a) ‫ີݫ‬ղ (4.12)ɺ(b) ۙࣅࣜ (4.15)ɺ(c) ۙࣅࣜ (4.17)ɻ
ʹ෼ղͰ͖Δ͜ͱ͕෼͔Δɻ͜Ε͸ਤ 4.2(a) ʹ࢛ࣔͨ͠‫࣓ిۃ‬ੴΛਤ 4.2(b) ͷΑ͏ʹ௕͞ L ͷυϦϑτεϖʔεͱ‫ ͞ڧ‬kL ͷബ೑࢛‫ʹۃ‬ΑΔΩοΫͰ͋Δɻ·ͨɺࣜ (4.17) ʹؔͯ͠͸




1
0
1 L2
1 L2
(4.19)
M=
0 1
−kL 1
0 1
ͱ෼ղͰ͖Δɻͭ·Γɺਤ 4.2(c) ʹࣔͨ͠௕͞ L/2 ͷυϦϑτεϖʔεɺ‫ ͞ڧ‬kL ͷബ೑ΩοΫɺ
L/2 ͷυϦϑτεϖʔεͷ߹੒Ͱ͋Δɻ࢛‫࣓ۃ‬৔ʹ‫ݶ‬ΒͣɺҰൠʹബ೑ΩοΫ͸γϯϓϨΫςΟο
Ϋͳม‫͋Ͱ׵‬Δɻैͬͯɺ௕͞ͷ͋ΔϏʔϜϥΠϯߏ੒ཁૉʹରͯ͠τϥοΩϯάΛߦ͏৔߹ɺ
ΩοΫͱυϦϑτεϖʔεͷ‫܁‬Γฦ͠Ͱۙࣅ͢Δํ๏͸ৗʹγϯϓϨΫςΟοΫͳม‫ͳͱ׵‬Δͨ
Ίɺଟ͘ͷγϛϡϨʔγϣϯͰ࠾༻͞ΕΔɻͯ͞ɺࣜ (4.17) ͷख๏Ͱ O(L2 ) ͷۙࣅਫ਼౓Λୡ੒Ͱ͖͕ͨɺΩοΫͷ਺Λ̎ͭʹ෼ׂ͠ɺυϦϑτεϖʔεΛ̏ͭʹ͢Δ͜ͱͰۙࣅਫ਼౓Λ͞Βʹ޲্Ͱ͖ΔͰ͋Ζ͏͔ɻਤ 4.3(a) ͷۙࣅϞσϧΛߟ͑ͯΈΔɻ͜͜Ͱ α, β ͸

2α + β = 1(4.20)

Λຬ࣮ͨ͢਺Ͱ͋Δɻ͜ͷͱ͖ɺసૹߦྻ͸






1 βL
1
0
1 αL
1 αL
1
0
M =
1
0 1
− kL
1
0 1
0 1
− kL
2
2


1 − 12 kL2 + 14 αβk 2 L4 L − α(α + β)kL3 + 41 α2 βk 2 L5=
−kL + 14 βk 2 L3
1 − 12 kL2 + 14 αβk 2 L4

(b)

(a)
kL/2

αL

kL/2

βL

γkL

αL

αL

ਤ 4.3: ࢛‫࣓ిۃ‬ੴͷۙࣅϞσϧ

43

２－ 42

δkL

βL βL

γkL

αL

(4.21)(4.22)

(6)

(2)
(7)

(1)

(3)

(5)

(4)
ਤ 4.4: ࣜ (4.30) ͕ O(L4 ) ͷۙࣅਫ਼౓ͱͳΔͱ͖ͷ࢛‫࣓ిۃ‬ੴͷۙࣅϞσϧ֤੒෼Λղੳղͷςʔϥʔల։ͱൺֱ͢Δͱɺۙࣅਫ਼౓Λ্͛ΔͨΊʹ͸1α(α + β) =



6
1
1
β=


4
6

2α + β = 1(4.23)(4.24)(4.25)

ͱͳΔඞཁ͕͋Δ͕ɺ࢒೦ͳ͕Β͜ͷ࿈ཱํఔࣜʹ͸ղ͕ͳ͍ɻͦ͜Ͱະఆ܎਺Λ૿΍ͨ͢ΊʹΩοΫΛ̏ͭʹ૿΍͠ɺ͞ΒʹΩοΫͷ‫͞ڧ‬ͷ෼഑ൺ཰΋ύϥϝʔλʔͱͨ͠ਤ 4.3(b) ͷۙࣅϞσϧΛߟ͑Δɻ͜͜Ͱ͜͜Ͱ α, β, γ, δ ͸

2α + 2β = 1(4.26)

2γ + δ = 1(4.27)

Λຬ࣮ͨ͢਺Ͱ͋Δɻ͜ͷͱ͖సૹߦྻ͸





1 αL
1
0
1 βL
1
0
M =
(4.28)
0 1−γkL 10 1−δkL 1




1 βL
1
0
1 αL
×
(4.29)
0 1−γkL 1
0 1


1
 3
1 2
1
2 4
L − 4 δ + αγ + 2αβγ kL
 1 − 2 kL + βγ(α + 2 δ)k L





2
2
3
6

−αβ γ δk L
+2αβγ αγ + 12 δ k 2 L5 − α2 β 2 γ 2 δk 3 L7 


= 







−kL + βγ(1 + δ)k 2 L3
1 − 12 kL2 + βγ α + 12 δ k 2 L2


3
5
2
2
3
6
2
2
−αβ γ δk L
−β γ δk L(4.30)

ͱͳΔɻैͬͯɺ




1
1


βγ α + δ =



2
24


1
βγ (1 + δ) =


6



1

 δ + αγ + 2αβγ = 1
4
6
44

２－ 43

(4.31)(4.32)(4.33)

0.20

O(L1 ) map
O(L2 ) map((H − Hexact )/Hexact

0.15

O(L4 ) map

0.10
0.05
0.00
−0.05
−0.10
−0.15
−0.20

0

100

200
300
# of steps

400

500

ਤ 4.5: ࣸ૾ (4.15)(O(L1 )), (4.17)(O(L2 )) ‫( ͼٴ‬4.30)(O(L4 )) ʹΑΓಘΒΕͨϋϛϧτχΞϯͷ
૬ର‫ࠩޡ‬ͷ࣌ؒൃలɻΛຬ଍͢Δ α, β, γ, δ Λ‫ٻ‬ΊΕ͹ྑ͍ɻͦͷղ͸

1 − 21/3= −0.1756 · · ·2(2 − 21/3 )
1
1
α =
−β =
= 0.6756 · · ·22(2 − 21/3 )
1
1
γ =
=
= 1.3512 · · ·224β
2 − 21/3
21/3
δ = 1 − 2γ = −= −1.7024 · · ·2 − 21/3
β =(4.34)(4.35)(4.36)(4.37)

ͱͳΓɺ͜ͷͱ͖ۙࣅਫ਼౓͸ O(L4 ) ͱͳΔɻ͜͜Ͱ β, δ ͕ෛͰ͋Δ͜ͱ͔Βɺ͜ͷͱ͖ࣜ (4.30)
ʹΑΔม‫׵‬͸෺ཧతʹ͸ਤ 4.4 ͷΑ͏ͳ̓ͭͷεςοϓͷ߹੒Ͱ͋ΔͱղऍͰ͖Δɻ΋͸΍୯७ʹ౳ִؒʹΩοΫΛฒ΂ͨ΋ͷͰ͸ͳ͍ɻਤ 4.5 ʹγϯϓϨΫςΟοΫͳࣸ૾ (4.15), (4.17) ‫ͼٴ‬
(4.30) Λ༻͍ͨ৔߹ͷϛϧτχΞϯͷ‫ີݫ‬஋͔Β૬ର‫ࠩޡ‬ͷ࣌ؒൃలΛࣔ͢ɻτϥοΩϯάγϛϡϨʔγϣϯʹ͓͍ͯਫ਼౓Λ্͛Δํ๏ͱͯ͠ɺ୯७ʹΩοΫͱυϦϑτΛ
౳ִؒʹฒ΂Δ͕ͦͷ਺Λ૿΍͢ํ਑ͱɺզʑ͕ࣜ (4.15) ͔Βࣜ (4.30) ʹਫ਼౓Λ޲্ͤͨ͞Α͏ʹͳ͑Δ΂͘গͳ͍ΩοΫͰਫ਼౓Λ্͛Δํ਑͕͋Δɻ‫ऀޙ‬͸γϯϨΫςΟοΫ਺஋ੵ෼๏ͱΑ͹ΕΔγϛϡϨʔγϣϯख๏Ͱ͋ΓɺՃ଎‫ث‬෼໺ʹؔΘΒͣɺॏྗଟମ‫ܥ‬ɺੜମ෼ࢠ‫Ͳͳܥ‬ͷγϛϡϨʔγϣϯͰ༻͍ΒΕ͍ͯΔख๏Ͱ͋Δɻ

45

２－ 44

4.3

ඇઢ‫ ͱֶྗܗ‬Lie ม‫׵‬

͜͜·ͰΫϩϚςΟγςΟͷ෦෼Λআ͍ͯɺ‫ج‬ຊతʹ୯ཻࢠྗֶͷ͏ͪઢ‫ֶྗܗ‬Λѻ͖ͬͯͨɻ
͜͜Ͱ͸࠷‫ʹޙ‬ඇઢ‫ܗ‬෦෼ͷऔΓѻ͍ʹؔͯ͠؆୯ʹड़΂Δɻ‫ͨݟʹط‬Α͏ʹཻࢠͷӡಈΛ௥੻͢Δࡍʹඞཁͳ͜ͱ͸ϏʔϜϥΠϯͷ͋Δ৔ॴ s1 ͔Β s2 ΁ͷసૹxs=s2 = M ◦ xs=s1(4.38)

ʹ͓͚Δࣸ૾ M Λ‫ٻ‬ΊΔ͜ͱͰ͋Γɺઢ‫ͳܗ‬ӡಈͰ͸ M ͸ཻࢠͷਖ਼४࠲ඪʹґଘ͠ͳ͍ߦྻʹ
ͳΔɻඇઢ‫ܗ‬ӡಈΛѻ͏৔߹ɺM ͸ҰൠʹߦྻͰ͸ͳ͍ɻ͜ͷΑ͏ͳඇઢ‫૾ࣸͳܗ‬ΛఆࣜԽ͢ΔͨΊʹ਺ֶతಓ۩ͱͯ͠ Lie ୅਺͕޿͘༻͍ΒΕΔɻ
ϙΞιϯ‫ࣜހׅ‬ਖ਼४ม਺ (qi , pi ) ͷؔ਺ f (qi , pi ), g(qi , pi ) ʹରͯ͠ϙΞιϯ‫ݺͱࣜހׅ‬͹ΕΔೋ߲ԋࢉࢠ [, ] Λఆٛ͢Δɻ[f, g] =

n 

∂f ∂g
i=1

∂f ∂g
−
∂qi ∂pi ∂pi ∂qi(4.39)

ୠ͠ɺ੔਺ n ͸‫ܥ‬ͷࣗ༝౓Ͱ͋ΔɻϙΞιϯ‫ࣜހׅ‬͸ͦͷఆ͔ٛΒҎԼͷੑ࣭Λ΋ͭ͜ͱ͕ࣔͤΔɻ

[f, g] = − [f, g]
ddfdg[f, g] = [ , g] + [f, ]dt
dt
dt[f g, h] = f [g, h] + [f, h] g

(4.41)[αf + βg, h] = α [f, h] + β [g, h](4.43)[f, [g, h]] + [h, [f, g]] + [g, [h, f ]] = 0(4.40)(4.42)(4.44)

·ͨɺϋϛϧτχΞϯΛ H ͱͯ͠

∂H
dxi
=
∂pi
ds
dpi
∂H
=
[pi , H] = −∂qids[qi , H] =(4.45)(4.46)

Ͱ͋ΔɻҰൠʹ (qi , pi ) ͓Αͼಠཱม਺ s ͷؔ਺ f ʹରͯ͠df
∂f
= [f, H] +ds∂s

(4.47)

ͱॻ͚Δɻैͬͯɺf ͕ಠཱม਺ s ʹཅʹґଘ͠ͳ͍ (∂f /∂s = 0) ৔߹ɺ[f, H] = 0 Ͱ͋Ε͹df /ds = 0 Ͱ͋Δ͔Β f ͸อଘྔͰ͋Δɻྫ͑͹ f ͱͯ͠ϋϛϧτχΞϯΛબͿͱɺϋϛϧτχ
Ξϯ͕ಠཱม਺ s ʹཅʹґଘ͠ͳ͍৔߹͸dH= [H, H] = 0dsͱͳΔͨΊɺ͔֬ʹϋϛϧτχΞϯ͸อଘྔͱͳΔɻ

46

２－ 45

(4.48)

·ͨɺ৔ॴ s = s1 Ͱͷਖ਼४ม਺ (qi1 , pi1 ) ͔Β s = s2 ΁ͷਖ਼४ม਺ (qi2 , pi2 ) ΁ͷม‫͕׵‬γϯϓϨΫςΟοΫͰ͋Δ৚݅ࣜ (4.9) ͸ϙΞιϯ‫ࣜހׅ‬Λ༻͍ͯ[q2i , qj2 ] = 0,[q2i , pj2 ] = − [p2i , qj2 ] = δij ,(4.49)[p2i , pj2 ] = 0

n 

∂f ∂g
∂f ∂g
[f, g] =−∂q1i ∂p1i ∂p1i ∂q1ii=1

ͱॻ͚Δɻ͜͜Ͱɺδij ͸ΫϩωοΧʔͷσϧλͰ͋Δɻ
1 (i = j)δij =
0 (i ̸= j)(4.50)

Lie ԋࢉࢠ

Lie ԋࢉࢠ : f : Λ࣍ͷΑ͏ʹఆٛ͢Δɻ

n 

∂f ∂
∂f ∂
−
: f : g = [f, ] =∂qi ∂pi ∂pi ∂qi(4.51)i=1͢ͳΘͪɺؔ਺ g ʹ࡞༻ͤ͞Δͱ f ͱ g ͷϙΞιϯ‫ࣜހׅ‬: f : g = [f, g](4.52)

ʹͳΔɻҎ߱Ͱ͸ओʹࣗ༝౓ n = 1 ͷ৔߹Λߟ͑ਖ਼४ม਺Λ (x, p) ͱද‫͢ه‬Δɻ
Lie ԋࢉࢠΛ༻͍ΔͱϋϛϧτχΞϯ H ʹର͢Δਖ਼४ํఔࣜ͸dx= − : H : x,dsdp=−:H:pds(4.53)

ͱද‫͖Ͱݱ‬ΔɻLie ԋࢉࢠͷੑ࣭Λ͍͔ͭ͘ྻ‫͢ڍ‬Δɻ∂∂p∂2(: x :)2 = 2∂p∂: x2 := 2x
∂p
∂
∂
−x
: xp := p∂p∂x,: x :=

,
,
,

∂
,
∂x
∂2
(: p :)2 =,∂x2∂: p2 := −2p ,∂x: p := −: x :: p :=: p :: x := −(4.54)(4.55)(4.56)∂2∂x∂p(4.57)

͞Βʹ͜ΕΒͷ݁ՌΛ࢖͏ͱɺ: p2 : x = −2p,(4.58)

2: p : p = 0,(4.59)

2

2

2

2

2

(4.60)

2

2

2

2

2

(4.61)(: p :) x = : p : (: p : x) =: p : (−2p) = 0,(: p :) p = : p : (: p : p) =: p : (0) = 0
Ͱ͋Δ͜ͱ͕෼͔Δɻ

47

２－ 46

ࢦ਺ԋࢉࢠ -Lie ม‫׵‬ਖ਼४ํఔࣜ (4.53) Λ‫ݟ‬Δͱɺ‫ࣜܗ‬తʹ͸ x(s) = e−s:H: x(0) ͷΑ͏ͳ‫ܗ‬Λͨ͠ղΛ‫ظ‬଴Ͱ͖Δɻ
ͦ͜Ͱɺ·ͣ Lie ԋࢉࢠͷ΂͖৐Λఆٛ͢Δɻྫ͑͹ f Λਖ਼४ม਺ (q, p) ͷؔ਺ͱͯ͠

(: f :)2 g =: f : (: f : g) = [f, [f, g]](: f :)3 g =: f : (: f : (: f : g)) = [f, [f, [f, g]]](4.62)···
ͳͲͰ͋Δɻ͜ͷ΂͖৐Λ༻͍ͯࢦ਺ԋࢉࢠ e:f : Λ

e:f : =∞

1
1
1(: f :)m = 1+ : f : + (: f :)2 + (: f :)3 + · · ·
m!
2!
3!

(4.63)

m=0

ͱఆٛ͢Δɻࢦ਺ԋࢉࢠ e:f : ʹΑΔม‫׵‬Λ Lie ม‫ͼݺͱ׵‬ɺf Λม‫׵‬ͷੜ੒ࢠͱশ͢ɻLie ม‫׵‬͸ৗʹγϯϓϨΫςΟοΫͳม‫׵‬Λ༩͑Δ͜ͱ͕ূ໌͞Ε͍ͯΔɻ

Lie ม‫׵‬ͷ۩ମྫ۩ମྫͱͯ͠ੜ੒ࢠ

L
f = − p22(4.64)

Λߟ͑ɺ(x, p) ʹ࡞༻ͤͯ͞ΈΔɻ11/e:−Lp 2: x = x − L : p2 : x + L2 (: p2 :)2 x + · · · = x + Lp
2
8
1:−Lp/ 2:2ep = x − L : p : p + ··· = p
2(4.65)(4.66)

ैͬͯɺੜ੒ࢠ (4.64) ͸υϦϑτεϖʔεͷࣸ૾Λੜ੒͢Δɻ࣍ͷʹੜ੒ࢠ f ͱͯ͠L
f = − (kx2 + p2 ),2(k > 0)(4.67)

Λߟ͑ͯΈΔɻม‫( ׵‬: f :)m x ͸(: f :)0 x = x,
∂f ∂x ∂f ∂x
1
−
= Lp,
(: f :) x =
∂x ∂p
∂p ∂x


∂f ∂p ∂f ∂p
2(: f :) x = L : f : p = L−= −kL2 x,∂x ∂p ∂p ∂x
∂f ∂x ∂f ∂x(: f :)3 x = −kL2 : f : x = −kL2−
= −kL3 p,
∂x ∂p
∂p ∂x


∂f ∂p
4
3
3 ∂f ∂p(: f :) x = −kL : f : p = −kL−= k 2 L4 x,∂x ∂p ∂p ∂x
∂f ∂x ∂f ∂x(: f :)5 x = k 2 L4 : f : x = k 2 L4−
= k 2 L5 p,
∂x ∂p
∂p ∂x
···

48

２－ 47

(4.68)

ͱ‫͖Ͱࢉܭ‬Δɻ͢ͳΘͪ n Λ੔਺ͱͯ͠

(: f :)2n x = (−1)n k n L2n x,(: f :)2n+1 x = (−1)n k n L2n+1 p(4.69)

Ͱ͋Δͱ༧૝͞ΕΔɻैͬͯɺ

e:f : x =∞
: f :n

m=0
∞


m!

x=

∞

(: f :)2n x

n=0

2n!

+

∞

(: f :)2n+1 x

n=0
∞


(2n + 1)!

p(−1)n √ 2n(−1)n √ 2n+1
kL
+√
kL
2n!
(2n + 1)!
k
n=0
n=0
√ 
√ 
p
= x coskL + √ sin
kL
k

=x

(4.70)

͜Ε͸࢛‫࣓ిۃ‬ੴʹΑΔࣸ૾Ͱ͋Δɻ࣮ࡍɺLie ม‫ ׵‬e:f : p ΋ಉ༷ʹ‫͖ͯͰࢉܭ‬ɺ
∞
∞

√ 
(−1)n √ 2n+1(−1)n √ 2np = − kx
e p=
(−1)
kL
+p
kL
m!
(2n + 1)!
2n!
m=0
n=0
n=0
√ 
√ 
√
= −x k sin
kL + p cos
kL
:f :

∞


m: f :

n(4.71)

ͱͳΓɺసૹߦྻ (1.58) ʹҰக͢ΔɻLie ม‫׵‬Λ࢖͏৔߹ɺਖ਼४ํఔࣜΛղ͘ૢ࡞͸ӡಈํఔࣜΛ
ੵ෼͢Δ͜ͱͰ͸ͳ͘ɺϋϛϧτχΞϯͰܾ·Δඍ෼ԋࢉΛ‫͢ࢉܭ‬Δ͜ͱͰ͋Δɻ͜͜·Ͱͷྫ͔Βਪ࡯Ͱ͖ΔΑ͏ʹɺ௕͞ L ͷϏʔϜϥΠϯͷߏ੒ཁૉͷϋϛϧτχΞϯΛ Hͱͨ͠৔߹ Lie ม‫׵‬x → e−L:H: x
p → e−L:H: p(4.72)

͸ͦͷϏʔϜϥΠϯ௨ա‫ޙ‬ͷ (x, p) Λ༩͑ΔɻLie ม‫( ׵‬4.72) ͸ਖ਼४ํఔࣜͷղͰ͋Δ͔ΒࣗવʹγϯϓϨΫςΟοΫͳม‫׵‬Λ༩͑Δ͜ͱ͕อূ͞Ε͍ͯΔɻ
͜͜Ͱ࿡‫࣓࣓ిۃ‬ੴͷϋϛϧτχΞϯ

H=

p2x + p2yk 3(x − 3xy 2 ) +6
2(4.73)

Λߟ͑Δɻ࢛‫࣓ిۃ‬ੴͷ࣌ͱಉ༷ʹ Lie ม‫׵‬Λ‫ٻ‬ΊΔͨΊʹ (: H :)m x Λ‫ͯ͠ࢉܭ‬ΈΔͱ(: H :)0 x = x,(: H :)1 x = −∂H= −px ,∂pxk
∂H ∂px= − (x2 − y 2 ),∂x ∂px2kk∂H ∂x2∂H ∂y 23
2
2−+
= k(xpx − ypy ),(: H :) x = − : H : (x − y ) = −22∂px ∂x∂py ∂y(: H :)2 x = − : H : px = −(: H :)4 x = k : H : (xpx − ypy )∂H ∂(xpx ) ∂H ∂(ypy ) ∂H ∂(ypy )∂(xpx ) ∂H−−+=k∂px ∂x∂px ∂x∂y ∂py∂py ∂yk= k −p2x + p2y + (x2 + y 2 ) ,2(: H :)5 x = O(k 2 )
···
49

２－ 48

(4.74)

ͱͳΔɻैͬͯ௕͞ L ͚ͩ࡞༻ͤ͞Δͱ

1
1
e−L:H: x = x + px L − kL2 (x2 − y 2 ) − kL3 (xpx − ypy )
4
6

1
k 2
4
2
2
2
+ kL −px + py + (x + y ) + O(k 2 L5 )24
2

(4.75)

ͱ͍͏ແ‫ͳͱ਺ڃݶ‬Δ͕ɺ࢛‫࣓ిۃ‬ੴͷΑ͏ʹดͨ͡‫Ͱܗ‬ද‫͍ͳʹ͏͖ͦͰݱ‬ɻैͬͯɺཧ࿦ղ
ੳΛߦ͏ʹ͠Ζ਺஋γϛϡϨʔγϣϯΛߦ͏ʹ͠Ζզʑ͸Ͳ͔͜ͷ࣍਺Ͱల։Λଧͪ੾Δඞཁ͕
͋Δɻۙࣅਫ਼౓Λ্͛ΔͨΊʹ͸ΑΓߴ࣍ͷ߲·Ͱ‫͢ࢉܭ‬Δඞཁ͕͋Δ͕ɺͦͷ‫ࢉܭ‬͸Ұൠʹ൥ࡶͰ͋ΔɻMathematica ͳͲͷ਺ࣜॲཧγεςϜΛ࢖ͬͯ‫͢ࢉܭ‬Δͷ͕ಘࡦͰ͋Δɻ
͜͜Ͱॏཁͳ͜ͱ͸ Lie ม‫ࣗ׵‬ମ͸γϯϓϨΫςΟοΫͳม‫׵‬Λ༩͑Δ͕ɺల։Λ్தͰଧͪ੾
Δͱͦͷม‫͕׵‬γϯϓϨΫςΟοΫͰ͋Δ͜ͱ͕Ұൠʹอূ͞Εͳ͍ɺͱ͍͏͜ͱͰ͋Δɻͦ͜Ͱͦͷ໰୊Λղܾ͢ΔͨΊʹγϯϓϨΫςΟοΫ਺஋ੵ෼๏͕ߟҊ͞Εͨɻզʑ͸‫ʹط‬અ 4.2 Ͱ࢛‫ۃ‬ి࣓ੴͷۙࣅϞσϧΛ୊ࡐʹͯ͠γϯϓϨΫςΟοΫ਺஋ੵ෼๏Λ‫ͨݟ‬ɻγϯϓϨΫςΟοΫ਺஋ੵ෼ํͷཧ࿦͸ Lie ୅਺Λ༻͍ͨཧ࿦ʹΑͬͯఆࣜԽ͞Ε͍ͯΔɻҎ߱Ͱ͘͝؆ܿʹͦͷ֓ཁΛड़΂Δɻ

Baker-Campbell-Hausdorﬀ ͷެࣜೋͭͷࢦ਺ԋࢉࢠ e:f : , e:g: ʹରͯ͠ f ͱ g ͕ަ‫׵‬Մೳ [f, g] = 0 Ͱ͋Ε͹ී௨ͷࢦ਺ؔ਺ͱಉ
༷ʹ

e:f : e:g: = e:f +g:(4.76)

͕੒ཱ͢Δɻͱ͜Ζ͕ [f, g] ̸= 0 ͷ৔߹ɺࢦ਺ԋࢉࢠͷੵͷৼΔ෣͍͸ී௨ͷࢦ਺ؔ਺ͱҟͳΓɺࣜ (4.76) ͸੒ཱ͠ͳ͍ɻ͜͜ͰɺBaker-Campbell-Hausdorﬀ(BCH) ͷެࣜͱ‫ݺ‬͹ΕΔॏཁͳؔ܎
͕ࣜ͋Δɻ

e:f : e:g: = e:h:h=A+B+

1
1
1[f, g] +[f [f, g]] + [g [g, f ]] +[f, [g, [g, [f, ]]]] · · ·21224(4.77)

͜ͷެࣜ͸γϯϓϨΫςΟοΫ਺஋ੵ෼๏ʹ‫ݶ‬Βͣඇઢ‫ܗ‬୯ཻࢠྗֶͷଟ͘ͷཧ࿦Ͱ࢖ΘΕΔɻ
γϯϓϨΫςΟοΫ਺஋ੵ෼๏
γϯϓϨΫςΟοΫ਺஋ੵ෼ͷ۩ମྫͱͯ͠ɺn Λ੔਺ͱ͢Δඇઢ‫ܗ‬ͷϋϛϧτχΞϯ H
ɹ H = kxn +p2
2(4.78)

Λߟ͑Δɻྫ͑͹ n = 3 ͕࿡‫࣓ిۃ‬ੴʹରԠ͢ΔɻγϯϓϨΫςΟοΫͳۙࣅࣸ૾Λ‫ٻ‬ΊΔͨΊ
ʹH Λ

H = Hk + H d ,Hk = kxn ,

Hd =

p2
2

(4.79)

ͱ෼ղ͢Δɻ͜ͷͱ͖ Hk , Hd ΛͦΕͧΕ௕͞ L ͚ͩ࡞༻ͤ͞Δ Lie ม‫׵‬͸

e−L:Hd : x = x + pL,
e−L:Hk : x = x,e−L:Hd : p = p

e−L:Hk : p = p − nkxn−1
50

２－ 49

(4.80)

ͱ‫͖Ͱࢉܭʹີݫ‬Δɻ͜͜Ͱ BCH ͷެࣜ (4.77) Λ࢖͏ͱ13

2

e−L:Hd : e−L:Hk : = e−L:H− 2 L[Hd ,Hk ]+O(L ): = e−L:H+O(L ):(4.81)

ͱॻ͚Δɻࣜ (4.80) ͸ബ೑ΩοΫ‫ʹޙ‬υϦϑτεϖʔεΛ௨ࣸ͢૾Ͱ͋Γɺզʑ͕ࣜ (4.15) Ͱߟ͑ͨख๏ʹରԠ͢ΔɻLie ม‫ࣗ׵‬ମ͸γϯϓϨΫςΟοΫͳม‫׵‬Λ༩͑ΔͷͰͦͷੵ΋γϯϓϨΫ
ςΟοΫͰ͋Δɻ޾͍ʹ΋ඇઢ‫࣓ܗ‬৔ͷ৔߹Ͱ΋ബ೑ΩοΫͷ Lie ม‫׵‬͸‫͖Ͱࢉܭʹີݫ‬Δɻैͬ
ͯɺബ೑ΩοΫͱυϦϑτεϖʔεͱ͍͏ม‫׵‬͸ৗʹγϯϓϨΫςΟοΫͳม‫͋Ͱ׵‬ΓγϛϡϨʔγϣϯख๏ͱͯ͠΋ൺֱత༰қʹ࢖͑Δɻ
અ 4.2 Ͱ࢛‫࣓ిۃ‬ੴͷ৔߹ʹബ೑ΩοΫͱυϦϑτεϖʔεͷ਺΍ͦͷύϥϝʔλʔΛ࠷దԽ͢Δ͜ͱͰۙࣅਫ਼౓Λ޲্Ͱ͖Δ͜ͱΛ‫ͨݟ‬ɻ͜ͷૢ࡞͸ɺྫ͑͹ α, β Λ࣮਺ͱͯࣜ͠ (4.81) ʹ͓͍ͯࠨลΛe−αL:Hd : e−L:Hk : e−βL:Hd :(4.82)

ͱมߋ͠ BCH ͷެࣜΛ༻͍ͯϋϛϧτχΞϯͷۙࣅਫ਼౓͕࠷΋ߴ͘ͳΔ α, β Λ‫͚ͭݟ‬Δ͜ͱͰ
͋Δɻ۩ମతʹॻ͖Լ͢ͱ
13e−αL:Hd : e−L:Hk : e−βL:Hd : = e−αL:Hd : e−L:Hk +βHd − 2 βL[Hk ,Hd ]+O(L ):
=e
=e

[
]
2
−L:(α+β)Hd +Hk − L[Hk ,βHd ]+ 12 Hd ,−LHk + L2 β[Hk ,Hd ] +O(L3 ):2(4.83)(α−β)[Hk ,Hd ]+O(L3 ):−L:(α+β)Hd +Hk − L2ͰΔ͜ͱ͕෼͔Δɻैͬͯɺα = β = 1/2 ͷͱ͖ʹӈลͷ L2 ͷ߲͕ফ͑ɺϋϛϧτχΞϯͷۙࣅਫ਼౓͕࠷΋ߴ͘ͳΔɻ͜Ε͸࢛‫࣓ిۃ‬ੴͷྫʹ͓͚Δࣜ (4.17) ʹରԠ͢Δɻͭ·Γɺ௕͞ L ͷి࣓ੴΛബ೑ΩοΫ̍ͭͱυϦϑτεϖʔε̎ͭͰۙࣅ͢Δ৔߹ɺ࣓৔ͷ‫ʹܗ‬ΑΒͣബ೑ΩοΫΛதԝʹஔ͘ͷ͕࠷΋ۙࣅਫ਼౓͕ߴ͍ɻ͜Ε͸௚‫ײ‬తʹ࠷΋Β͍ۙ͠ࣅํ๏Ͱ͋Δ͕ɺLie ม‫׵‬Λ࢖͏͜ͱͰͦΕ͕਺ֶతʹอূ͞ΕΔ͜ͱ͕෼͔Δɻ͞Βʹۙࣅਫ਼౓Λ্͛ΔͨΊʹ͸൥ࡶͰ͋Δ͕5e−L:αHd : e−L:γHk : e−L:βHd : e−L:δHk : e−L:βHd : e−L:γHk : e−L:αHd : = e−L:H+O(L ):(4.84)

Λຬ࣮ͨ͢਺ α, β, γ, δ Λܾఆ͢Ε͹ྑ͍ɻ݁Ռ͸ࣜ (4.30) ʹҰக͢Δɻ
ඇઢ‫࣓ిܗ‬ੴΛಉ͡௕͞ͷυϦϑτͱಉ͡‫͞ڧ‬ͷബ೑ΩοΫͷ‫܁‬Γฦ͠Ͱۙࣅ͢Δख๏͸γϯϓϨΫςΟοΫͳख๏Ͱ͋Δɻ਺஋ԋࢉͷ‫܁‬Γฦ͠ʹ൐͏ͷ‫ࠩޡ‬ͷ֦େΛແࢹͰ͖ΔͷͰ͋Ε͹ɺεςοϓ਺Λ૿΍͢͜ͱͰۙࣅਫ਼౓ͷ޲্͕‫ظ‬଴Ͱ͖ΔɻγϯϓϨΫςΟοΫ਺஋ੵ෼๏Ͱ͸υϦϑτͷ௕͞ͱബ೑ΩοΫͷ‫͞ڧ‬Λ޻෉͢Δ͜ͱͰΑΓগͳ͍εςοϓ਺Ͱ໨ඪͱ͢Δۙࣅਫ਼౓Λ
ୡ੒͢Δ͜ͱ͕Ͱ͖Δɻ3
ਤ 4.6 ʹपճߦྻ (2.22) ͱϋϛϧτχΞϯ

p2(4.85)
2
ʹΑΔཻࢠͷӡಈΛγϯϓϨΫςΟοΫ਺஋ੵ෼๏ʹΑΓ‫݁ͨ͠ࢉܭ‬ՌΛࣔ͢ɻۙࣅਫ਼౓ͷҟͳΔ
ࣜ (4.80)(̍࣍)ɺࣜ (4.83)(̎࣍) ‫( ࣜͼٴ‬4.84)(̐࣍) ʹΑΓ‫ٻ‬ΊͨϙΞϯΧϨஅ໘4 Ͱ͋Δɻߴ࣍ͷख๏͸ΑΓৄࡉͳߏ଄Λද‫͖Ͱݱ‬Δ͜ͱ͕෼͔ΔɻҰൠʹඇઢ‫࣓ܗ‬৔͸ཻࢠͷӡಈΛෆ҆ఆԽͤ͞ΔཁҼͱͳΔɻਤ 4.6 ͔Β΋෼͔ΔΑ͏ʹ‫෇఺ݪ‬ۙʹ͓͍ͯ͸ԿΒ͔ͷۙࣅతͳอଘྔ͕ଘࡏ͠ɺཻࢠͷ‫੻ح‬͸ดͨ͡‫ۂ‬ઢΛඳ͘ɻৼ෯͕૿͑ͯ
͍͘ͱཻࢠͷӡಈ͸ΧΦεঢ়ଶͱͳΔ (ΧΦεͷւ)ɻ͞Βʹେ͖ͳৼ෯Λཻ࣋ͭࢠ͸ແ‫ݶ‬ԕʹӡಈ͕ൃࢄ͢Δɻ҆ఆʹཻࢠ͕पճͰ͖ΔҐ૬ۭؒ಺ͷྖҬΛྗֶ‫Ϳݺͱܘޱ‬ɻ5 ྗֶ‫ܘޱ‬ͷ࠷దԽ͸Ճ଎‫ث‬ઃ‫ͼٴܭ‬ӡసʹඇৗʹॏཁͳ՝୊Ͱ͋Δɻɹ H = kx4 +3͞Βʹߴ࣍ͷࣸ૾Λ‫ٻ‬ΊΔͨΊͷҰൠ࿦ʹؔͯ͠͸ྫ͑͹จ‫[ ݙ‬5] Λࢀর͞Ε͍ͨɻҐ૬ฏ໘্ʹपճ͝ͱͷਖ਼४ม਺Λϓϩοτͨ͠΋ͷɻ5ྗֶ‫ʹܘޱ‬ରͯ͠ɺϏʔϜύΠϓͷ‫͔Ͳͳܘ‬Βܾ·͍ͬͯΔ‫ܘޱ‬Λ෺ཧ‫Ϳݺͱܘޱ‬ɻ
4

51

２－ 50

(a) 1st order map(b) 2nd order map(c) 3rd order map

ਤ 4.6: τϥοΩϯάγϛϡϨʔγϣϯʹΑΓಘΒΕͨपճߦྻ (2.22) ͱϋϛϧτχΞϯ (4.85) ͔ΒͳΔ‫ܥ‬ͷϙΞϯΧϨஅ໘ɻͦΕͧΕɺ(a) ࣜ (4.80)ɺ(b) ࣜ (4.83) ‫( ͼٴ‬c) ࣜ (4.84) Λ༻͍ͯ‫ܭ‬
ࢉͨ͠ɻ

52

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ࢀߟจ‫ݙ‬[1] S. Y. Lee, Accelerator Physics, 2nd ed. (World Scientiﬁc, Singapore, 2005).
[2] S. I. Tzenov, Contemporary Accelerator Physics, (World Scientiﬁc, Singapore 2004).
[3] େ੢޾‫ت‬ɺʡ୯ཻࢠྗֶೖ໳ ʡɺOHO ʟ00 (2000).
[4] ੜग़উએɺϏʔϜྗֶೖ໳ɺJLC-FFIR92 ‫ڀݚ‬ձߨٛ࿥ɾ‫ڀݚ‬ใࠂूɺా಺ར໌ɺࢁຊঢฤूɺKEK Proceedings 93-6 (1993)ɻ
[5] H. Yoshida, Phys. Lett. A 150, 262 (1990)[6] A. Chao, ʠ Lecture Notes on Special Topics in Accelerator Physics ʡ, SLAC-PUB 9574
[7] T. Suzuki, Technical Report No. KEK-78-14, KEK, Tsukuba (unpublished).
[8] T. Suzuki, KEK Report 96-10 (1996)[9] ᚙ஌ӳ໌ɺ”େ‫౓ڧ‬ཅࢠϦϯάͷϏʔϜྗֶ”ɺOHO ʟ10 (2010).
[10] K. Jimbo, Phys. Rev. ST Accel. Beams 19, 010102 (2016).
[11] খү੖୅ɺ”ϏʔϜஷଂϦϯάͷ‫”ૅج‬ɺOHO ʟ91 (1991).
[12] A. Wolski, ”Linear Dynamics in Particle Accelerators”
http://pcwww.liv.ac.uk/ awolski/[13] A. Wolski, ”NonLinear Dynamics in Particle Accelerators”
http://pcwww.liv.ac.uk/ awolski/[14] T. Satogata, USPAS Accelerator Physics 2017 University of California, Davis
http://www.toddsatogata.net/2017-USPAS/[15] W. Herr, ”Mathematical and numerical methods for non-linear beam dynamics”, CERN
Accelerator School (2013).
[16] W. MacKay and T. Satogata, USPAS Accelerator Physics 2011
http://toddsatogata.net/2011-USPAS/

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෇࿥
ਖ਼४ม‫ͱ׵‬฼ؔ਺ਖ਼४࠲ඪ (q1 , q2 , · · · , qi , · · · , qN )ɺਖ਼४ӡಈྔ (p1 , p2 , · · · , pi , · · · , pN )ɺಠཱม਺ t ͔ΒͳΔϋϛϧτχΞϯ H(q, p, t) ͷਖ਼४ํఔࣜ

q̇i =

∂H
,
∂pi

ṗi = −

∂H∂qi

(4.86)

˙ = dx/dt Ͱ͋Δɻ͜͜Ͱɺ(q, p) ͔Β৽͍͠ม਺ (Q(q, p, t), P (q, p, t)) ΁ͷมΛߟ͑Δɻୠ͠ɺ(x)‫׵‬Λߟ͑Δɻ͜ͷ৽͍͠ม਺͕ɺ৽ͨͳϋϛϧτχΞϯ K ͱ‫ʹڞ‬ਖ਼४ํఔࣜ
Q̇i =

∂K
,
∂Pi

Ṗi = −

∂K
∂Qi

(4.87)

Λຬͨ͢ͱ͖ɺ͜ͷ (q, p) ͔Β (Q, P ) ΁ͷม਺ม‫׵‬Λਖ਼४ม‫Ϳݺͱ׵‬ɻྫ͑͹࣮਺ A ΛθϩͰͳ͍ఆ਺ͱͯ͠ q Λ Q = Aq ͱ͢Δม‫׵‬Λߟ͑Δɻਖ਼४ӡಈྔ P ͱϋϛϧτχΞϯ H(Q, P, t) ͕ͦͷ··Ͱ͸໌Β͔ʹਖ਼४ํఔࣜΛຬ଍Ͱ͖ͳ͍ɻਖ਼४ํఔࣜͷ‫ܗ‬ΛอͭͨΊʹ͸ྫ͑͹ Q = Aq ͱಉ࣌ʹ P = p/A ͱ͢Ε͹ Q, P, K = H(Q, P, t) ͸ਖ਼४ํఔࣜͷ‫ܗ‬Λอͭ͜ͱ͕ग़དྷΔɻ͜ͷΑ͏ʹɺϋϛϧτχΞϯ H(q, p) ʹม਺ม‫׵‬Λࢪ͢ࡍʹ͸ͦΕ͕ਖ਼४ม‫׵‬ͱͳ͍ͬͯΔඞཁ͕͋Δɻ͞΋ͳ͚Ε͹ɺਖ਼४ํఔ͔ࣜΒਖ਼͍͠ӡಈํఔࣜΛಘΒΕͳ͍ɻ฼ؔ਺ͱ͍͏֓೦Λಋೖ͢Δ͜ͱͰ‫ࣜܗ‬తʹਖ਼४ม‫׵‬Λຬͨ͢ม਺ม‫׵‬Λߦ͏͜ͱ͕Ͱ͖Δɻৄ͘͠͸ղੳྗֶͷ‫ڭ‬ՊॻʹҕͶΔ͜ͱʹͯ͠ɺ͜͜Ͱ͸ຊςΩετͰ࢖͏݁ՌͷΈΛ·ͱΊΔɻ฼
ؔ਺ G ʹ͸̐ͭͷλΠϓ͕͋Δɻ

Type 1 :G1 (q, Q, t)(4.88)

Type 2 :G2 (q, P, t)(4.89)

Type 3 :G3 (p, Q, t)(4.90)

Type 4 :G4 (p, P, t)(4.91)‫چ‬ม਺ͱ৽ม਺ɺ͓Αͼ‫چ‬ϋϛϧτχΞϯ H ͱ৽ϋϛϧτχΞϯ K ͷؔ܎Λදʹࣔ͢ɻ฼ؔ਺

G1 (q, Q, t)
G2 (q, P, t)
G3 (p, Q, t)
G4 (p, P, t)qipi∂G1 /∂qi∂G2 /∂qi

Qi∂G2 /∂Pi

Pi−∂G1 /∂Qi−∂G3 /∂pi∂G3 /∂Qi−∂G4 /∂pi∂G4 /∂Piද 4.1: ฼ؔ਺ͱ৽‫چ‬ม਺ͷରԠؔ܎ྫ͑͹ Type 1 ͷ฼ؔ਺G(q, Q, t) =


i

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２－ 53

qi Q iK −H∂G1 /∂t∂G2 /∂t∂G3 /∂t∂G4 /∂t(4.92)

͸ਖ਼४ม‫׵‬

∂G
= Qi
∂qi
∂G
= −qi
Pi = −∂Qi∂G=H
K = H+∂tpi =(4.93)(4.94)(4.95)

Λ༩͑Δɻ͜ͷਖ਼४ม‫׵‬͸ਖ਼४࠲ඪͱਖ਼४ӡಈྔΛަ‫͢׵‬Δɻ͜ͷΑ͏ʹϋϛϧτϯ‫͍͓ʹࣜܗ‬ͯ͸࠲ඪͱӡಈྔ͸౳ՁʹѻΘΕΔɻ
͜͜Ͱɺ΋͏ͻͱͭॏཁͳࣄ࣮Λࣔ͢ɻࠓɺ࣮਺ ϵ Λඍখྔͱͯ͠ɺ(q, P ) ͱϋϛϧτχΞϯ H
ʹΑΔ Type 1 ͷ฼ؔ਺
G(q, P, t) =qi Pi + ϵH(q, P )(4.96)iΛߟ͑Δɻ͜Ε͸ඍখਖ਼४ม‫ݺͱ׵‬͹ΕΔਖ਼४ม‫׵‬ͷҰछͰ͋Δɻ͜ͷඍখਖ਼४ม‫ʹ׵‬Αͬͯ q, p
͸ q + δq, p + δp ʹม‫͞׵‬ΕΔͱ͢Δͱ‫چ‬ม਺ͱ৽ม਺ؒͷؔ܎

∂H
∂pi
∂H
= Pi + ϵ∂qi

Qi = qi + ϵ(4.97)

pi(4.98)

͸

δqi
δpi

∂H
∂pi
∂H
=− ϵ
∂qi
=

ϵ

(4.99)(4.100)ͱॻ͚Δɻ͜Ε͸ਖ਼४ํఔࣜͱಉ͡‫ܗ‬Λ͍ͯ͠Δ͜ͱ͕෼͔Δɻ͢ͳΘͪɺϋϛϧτϯ‫ܥ‬ͷ (q, p)ͷ࣌ؒൃల͸ਖ਼४ม‫׵‬ͷҰछͰ͋ΓɺϋϛϧτχΞϯʹΑΔඍখਖ਼४ม‫׵‬ͷ‫܁‬Γฦ͠ͱΈͳͤΔɻ

55

２－ 54