Synchrotron radiation

ERL ͷϏʔϜྗֶ
Electron Gun

͸͡Ίʹ

1

Main LinacbeamΤωϧΪʔճऩ‫ܕ‬ઢ‫ܗ‬Ճ଎‫( ث‬Energy Recovery Li-can, ERL) ͸‫ج‬ຊతʹઢ‫ܗ‬Ճ଎‫͋Ͱث‬Δ͕ɺ௒఻ಋ
Ճ଎ۭಎͰՃ଎ͨ͠ిࢠϏʔϜΛ࠶ͼಉۭ͡ಎΛ௨ͯ͠‫ݮ‬଎͠ɺ‫ޙ‬ଓͷϏʔϜʹΤωϧΪʔΛड͚౉͢͜ͱ
͕େ͖ͳಛ௃ͱͳ͍ͬͯΔɻ͜ͷ͜ͱʹΑͬͯɺERLͰ͸ߴग़ྗʢߴ‫܁‬Γฦ͠, ߴฏ‫ྲྀిۉ‬ʣͷߴً౓ిࢠ
ϏʔϜͷར༻͕ՄೳͱͳΔɻຊߨٛͰ͸, ER ͷͳ͔ͷҰ࿈ͷྲྀΕɺߴग़ྗͷߴً౓ిࢠϏʔϜͷੜ੒ɺՃ

Undulator
Beam Dumpਤ 1: ઢ‫ܗ‬Ճ଎‫( ث‬linac) ͷུ֓ਤɻిࢠ‫ݯ‬ɺߴप೾Ճ଎ۭಎɺར༻෦෼ (์ࣹޫੜ੒ͷͨΊͷૠೖޫ‫)ݯ‬ɺϏʔϜμϯϓʹΑͬͯߏ੒͞ΕΔɻ‫ج‬ຊతʹϏʔϜ͸ɺిࢠ‫Ͱݯ‬ੜ੒͞Εͨ͋ͱɺՃ଎ɾར༻͞ΕɺϏʔϜμϯϓʹࣺͯΒΕΔɻ

Synchrotron radiation଎ɺ༌ૹɺ‫ݮ‬଎ɺഇ‫ؔʹغ‬ΘΔϏʔϜྗֶʹ͍ͭͯɺࢼ‫ݧ‬Ճ଎‫͋Ͱث‬Δ compact ERL (cERL) ͷӡసྫΛަ͑ͳ͕Β঺հ͢Δɻ

1.1

Booster
Synchrotronઢ‫ܗ‬Ճ଎‫ͱث‬Ϧϯά‫ܕ‬Ճ଎‫ث‬ߴΤωϧΪʔిࢠՃ଎‫ث‬͸ɺେ·͔ʹઢ‫ܗ‬Ճ଎‫ͱث‬LinacϦϯά‫ܕ‬Ճ଎‫ʹث‬෼͚ΒΕΔɻઢ‫ܗ‬Ճ଎‫( ث‬ਤ 1) ͸ɺbeamిࢠϏʔϜΛੜ੒͢ΔͨΊͷిࢠ‫ݯ‬ɺͦΕΛߴΤωϧ

Storage ringΪʔ·ͰՃ଎͢ΔͨΊͷߴप೾Ճ଎ۭಎ (RF ۭಎ)ɺՃ଎͞ΕͨిࢠΛར༻͢Δ෦෼ʢ์ࣹޫੜ੒΍ଞͷՙਤ 2: Ϧϯά‫ܕ‬Ճ଎‫༻ޫࣹ์( ث‬ஷଂϦϯά‫ܕ‬Ճ଎‫)ث‬ిཻࢠͱͷিಥͳͲʣɺͦͯ͠ϏʔϜΛࣺͯΔͨΊͷͷུ֓ਤɻઢ‫ܗ‬Ճ଎‫Ͱث‬ϏʔϜΛੜ੒ͨ͋͠ͱɺϒʔϏʔϜμϯϓ͔Βߏ੒͞ΕΔɻઢ‫ܗ‬Ճ଎‫Ͱث‬͸ɺ‫ج‬ຊελʔγϯΫϩτϩϯͰՃ଎͞ΕɺஷଂϦϯάʹೖࣹతʹిࢠϏʔϜ͸Ұ౓͚ͩར༻͞ΕͯɺࣺͯΒΕΔ͜

͞ΕΔɻஷଂϦϯά಺ʹઃஔ͞Εͨภ޲ి࣓ੴɺૠೖ

1ͱʹͳΔ ɻϦϯά‫ܕ‬Ճ଎‫( ث‬ਤ 2) Ͱ͸ɺ͋Δఔ౓ޫ‫ʹݯ‬Αͬͯ์ࣹޫ͕ੜ੒͞ΕΔɻஷଂϦϯάʹೖࣹՃ଎͞ΕͨిࢠϏʔϜΛઢ‫ܗ‬Ճ଎‫ث‬౳͔Βೖࣹ͠ɺϦ͞ΕͨిࢠϏʔϜ͸ɺ࢒ཹΨε΍ిࢠಉ࢜ͷࢄཚ౳ʹϯά‫ܕ‬ͷ‫ي‬ಓ্Λपճͤ͞ଓ͚Δ͜ͱʹͳΔɻϦϯάΑࣦͬͯΘΕΔ·Ͱด‫ي‬ಓ্Λपճ͠ଓ͚Δɻ‫ܕ‬Ճ଎‫಺ث‬ͷిࢠ͸ɺ࢒ཹΨε΍ిࢠಉ࢜ͷࢄཚ౳ʹΑͬͯपճ‫ي‬ಓ্͔Β֎Εͳ͍‫ݶ‬Γ͸ɺͦͷ‫ي‬ಓ্Λ΋ͦ΋҆ఆԽ‫ߏػ‬Λ΋͍ͬͯΔ͜ͱʹͳΓɺ·ͨɺಉपճ͠ଓ͚Δ͜ͱʹͳΔɻϦϯά‫ܕ‬Ճ଎‫Ͱث‬ॏཁͱͳ͡ϏʔϜ͕पճ͠ଓ͚ΔͨΊʹϑΟʔυόοΫʹΑͬΔͷ͸ɺ҆ఆʹిࢠΛपճͤ͞ଓ͚ΒΕΔ͔ͱ͍͏͜

ͯ͞Βʹ҆ఆԽ͢Δ͜ͱ͕ՄೳͰ͋ΔɻՃ଎‫ث‬ͷར༻ͱͰ͋Δɻ͜ͷΑ͏ʹɺઢ‫ܗ‬Ճ଎‫ͱث‬Ϧϯά‫ܕ‬Ճ଎‫ث‬ʹ͓͍ͯ͸ɺϏʔϜͷ҆ఆੑͱ͍͏ͷ͸ॏཁͳ߲໨Ͱ

Ͱ͸େ͖͘ಛ௃͕ҟͳΔɻ͋ΓɺϦϯά‫ܕ‬Ճ଎‫ث‬͸͜ͷ఺Ͱ༗རͳ‫ࣜܗ‬ͷՃ଎‫ث‬ిࢠϏʔϜͷμΠφϛΫε͔Β‫͖ͱͨݟ‬ͷ྆ऀͷҧͰ͋ΔɻҰํɺઢ‫ܗ‬Ճ଎‫Ͱث‬͸Ϧϯά‫ܕ‬Ճ଎‫ث‬ͷΑ͏͍͸ɺઢ‫ܗ‬Ճ଎‫ث‬͸‫ج‬ຊతʹిࢠϏʔϜͷੑ্࣭͕ྲྀ

ʹपճӡಈͷ҆ఆੑΛߟ͑Δඞཁ͸ͳ͍͕ɺ্ྲྀ͔Β͔ΒԼྲྀʹ޲͔ͬͯ఻ൖ͢Δͷʹରͯ͠ɺϦϯά‫ܕ‬ՃԼྲྀ΁Ө‫఻͕ڹ‬ൖ͢Δੑ࣭Λߟྀ͢Δඞཁ͕͋ΔɻԾ଎‫Ͱث‬͸पճ‫ي‬ಓ͕ด͍ͯ͡Δͱ͍͏ੑ࣭͔Βप‫ڥظ‬ʹ্ྲྀʹԿΒ͔ͷిࢠϏʔϜʹର͢Δ֎ཚ͕͋Δ৔߹քͷԼͰͷӡಈͱͳΔɻ‫͑׵͍ݴ‬Δͱɺઢ‫ܗ‬Ճ଎‫ث‬ͷʹ͸ͦͷӨ‫͕ڹ‬Լྲྀʹ఻೻͍ͯ͘͜͠ͱʹͳΓɺిࢠిࢠͷӡಈ͸ॳ‫ظ‬஋໰୊ͱ͍͏͜ͱʹͳΓɺϦϯά‫ܕ‬ϏʔϜͷੑೳΛ֬อ͢Δʹ͸্ྲྀ͔Βઃ‫ܭ‬௨ΓʹϏʔՃ଎‫Ͱث‬͸प‫ڥظ‬ք৚݅Լͷ໰୊ͱ͍͏͜ͱʹͳΔɻ

ϜΛ༌ૹ͢Δ͜ͱ͕ཁ‫͞ٻ‬ΕΔɻ·ͨɺ্ྲྀͰ֎ཚ͕

पճͤ͞ଓ͚Δͱ͍͏ੑ࣭͔ΒɺϦϯά‫ܕ‬Ճ଎‫ث‬͸ͦੜͨ͡৔߹ʹ΋ɺϏʔϜ͸ҰճͰ௨Γա͗ͯ͠·͏ͨΊϑΟʔυόοΫʹΑΔ੍‫͕ޚ‬೉͘͠ɺՃ଎‫ث‬ͷ҆ఆ

1 Ϧϯά‫ܕ‬Ճ଎‫ʹث‬ϏʔϜΛ‫͢څڙ‬ΔͨΊͷՃ଎‫͞༻࢖ث‬ΕΔ

৔߹͸ɺిࢠϏʔϜ͸ઢ‫ܗ‬Ճ଎‫͔ث‬Βग़ࣹ͞ΕɺϦϯά‫ܕ‬Ճ଎‫ʹث‬஝ੵ͞ΕΔ͜ͱʹͳΔɻԽͷͨΊʹ͸ͦ΋ͦ΋֎ཚͷ‫ݪ‬ҼΛઈͭͱ͍͏ରࡦ͕ඞཁͱͳΔɻ㸲㸫 

1.2ಈঢ়ଶΛͳΔ΂͘ଗ͑Δ͜ͱ͕Ճ଎‫ث‬ͷϏʔϜྗֶͰϏʔϜੑೳઢ‫ܗ‬Ճ଎‫ͱث‬Ϧϯά‫ܕ‬Ճ଎‫ث‬ͷಛ௃Λ౿·͑ͯɺՃ଎‫ث‬தͷϏʔϜੑೳʹ͍ͭͯߟ͑ΔɻՃ଎‫ث‬தͰ͸Ұͭͷిࢠ͕ӡಈ͍ͯ͠ΔΘ͚Ͱ͸ͳ͘ɺଟ਺ͷిࢠ͕ूஂͱͳͬͯӡಈ͍ͯ͠ΔɻҰͭͷిࢠͷΈ͕ӡಈ͍ͯ͠Δ৔߹ʹ͸ɺ3 ࣍‫࠲ۭؒݩ‬ඪ (x, y, z) ͱӡಈྔ(px , py , pz ) ͷ 6 ͭͷม਺ (6 ࣍‫ݩ‬Ґ૬ۭؒ࠲ඪ) ͕ܾ·Ε͹ͦͷӡಈঢ়ଶ͕ఆ·ͬͨ͜ͱʹͳΔɻ͔͠͠ͳ͕ΒɺҰͭͷిࢠͷΈΛՃ଎ͯ͠ར༻͢Δͱ͍͏ͷ͸ɺಘΒΕΔग़ྗʢ์ࣹޫͷ‫౓ڧ‬ɺিಥ‫ܕ‬Ճ଎‫ث‬ͷ৔߹ʹ͸িಥස౓ʣ͕‫ۃ‬Ίͯখ͘͞ɺ࣮ࡍͷՃ଎‫ث‬ͷར༻ʹ͸ద͞ͳ͍ɻग़ྗΛ্͛ΔͨΊʹɺ࣮ࡍͷՃ଎‫ث‬Ͱ͸ۭؒతɾ࣌ؒతʹۙ઀ͨ͠ଟ਺ͷిࢠूஂΛಉ࣌ʹՃ଎͢Δ͜ͱʹͳΔɻ͜ͷ৔߹ɺಘΒΕΔग़ྗ͸ిࢠͷ‫ʹ਺ݸ‬ൺྫ͢Δ͜ͱʹͳΔɻ͜ͷΑ͏ͳిࢠूஂ
Λిࢠόϯν (bunchɺմɺ๪) ͱ‫Ϳݺ‬ɻՃ଎‫Ͱث‬͸ిࢠϏʔϜͷੑೳɺ͋Δ͍͸඼࣭ͱ͍͏
‫ݴ‬༿Λ࢖͏͜ͱ͕ଟ͍͕ɺ͜Ε͸ిࢠूஂͷӡಈঢ়ଶ͕ͲΕ͘Β͍ଗ͍ͬͯΔ͔Λද͍ͯ͠Δ͜ͱʹରԠ͢
ΔɻిࢠूஂΛߏ੒͢Δશͯͷిࢠ͕ಉ͡ӡಈঢ়ଶΛ΋ͭͱ͍͏͜ͱ͸Ͱ͖ͳ͍͕ɺ͜Ε͕ͳΔ΂͘ଗ͍ͬͯΔͷ͕඼࣭ͷߴ͍ϏʔϜͱ͍͏͜ͱʹͳΔɻྫ͑͹ਫฏํ޲ͷ࠲ඪ x ͱͦͷӡಈྔ px ͷ 2 ͭͷม਺Ͱߏ੒͞ΕͨҐ૬ۭؒ x-px Λߟ͑ͨͱ͖ɺ͜ͷҐ૬্ۭؒͰిࢠूஂͷ෼෍͕࡞Δ໘ੵΛਫฏํ޲ͷΤϛολ
ϯεͱ‫ Ϳݺ‬2 ɻՃ଎‫Ͱث‬͸ɺ͜ͷిࢠϏʔϜͷΤϛο
λϯεΛখ͘͢͞Δͱ͍͏͜ͱ͕ҰͭͷॏཁͳςʔϚͱͳ͍ͬͯΔɻӡಈঢ়ଶ͕ͳΔ΂͘ଗ͍ͬͯΔ͜ͱͷར఺͸Կ͔ͱ͍͏͜ͱΛߟ͑Δɻͦͷར఺͸ɺྫ͑͹ిࢠϏʔϜͷۭؒతɾ࣌ؒతେ͖͞Λখ͘͞Ͱ͖Δͱ͍͏͜ͱ͕͋
Δɻूஂͷ֦͕Γ͕খ͘͞ͳΔͱ͍͏͜ͱ͸ɺిࢠີ

ॏཁͱͳΔɻಛʹॏཁͱͳΔͷ͕ɺϏʔϜਐߦํ޲Ͱ
͋Δ z ํ޲ʹରͯ͠௚ަ͢Δ໘಺ͷӡಈ (͜ͷํ޲Λԣํ޲ͱ‫)Ϳݺ‬ɺਫฏํ޲ x ͱਨ௚ํ޲ y ͷΤϛολϯεͰ͋Δɻ
ͦΕͰ͸ɺਫฏํ޲ͷిࢠूஂͷӡಈঢ়ଶ͸ԿʹΑͬͯɺͲͷΑ͏ʹܾ·Δ͔ͱ͍͏͜ͱΛՃ଎‫ث‬ͷ‫ࣜܗ‬ຖʹߟ࡯͢Δɻ·ͣɺϦϯά‫ܕ‬Ճ଎‫ث‬ͷ৔߹͸ɺपճதͷ์ࣹޫͷੜ੒ʹ‫ى‬Ҽͯ͠ӡಈঢ়ଶ͕มԽ͢
Δɻ์ࣹޫੜ੒ʹΑͬͯɺ์ࣹ‫ݮ‬ਰͱ์ࣹྭ‫͍ͱى‬͏ 2 ͭ‫ݱ‬৅͕ൃੜ͠ɺిࢠͷӡಈঢ়ଶ (ిࢠͷΤωϧΪʔɺ‫ي‬ಓ) ͕มԽ͢Δ͜ͱʹͳΔɻϦϯά‫ܕ‬Ճ଎‫ث‬ͷΤϛολϯε͸ɺ͜ΕΒ 2 ͭͷ‫ݱ‬৅ͷฏߧঢ়ଶͰ
ܾ·Δɻਤ 3 ʹϦϯά‫ܕ‬Ճ଎‫͚͓ʹث‬Δ์ࣹ‫ݮ‬ਰͱ์ࣹྭ‫ى‬ͷ໛ࣜਤΛࣔ͢ɻҰํઢ‫ܗ‬Ճ଎‫Ͱث‬͸ɺిࢠॐ͔Βੜ੒͞ΕͨϏʔϜ
͸ 1 ճ (͋Δ͍͸਺ճ) ͷΈར༻ (์ࣹޫੜ੒΍িಥͳͲ) ͞Εɺͦͷ͋ͱ͙͢ϏʔϜμϯϓʹࣺͯΒΕΔͨΊɺϏʔϜ͸์ࣹྭ‫ى‬ɾ‫ݮ‬ਰͷฏߧঢ়ଶʹ౸ୡͤͣɺ
ిࢠ‫Ͱݯ‬ੜ੒͞ΕͨϏʔϜͷ࣭͕ɺར༻ՕॴͰͷϏʔϜͷ඼࣭Λܾఆ෇͚Δ͜ͱʹͳΔɻઢ‫ܗ‬Ճ଎‫ث‬ͷར఺͸ɺ͜ͷిࢠ‫Ͱݯ‬ੜ੒͞ΕͨϏʔϜ඼࣭͕์ࣹ‫ݮ‬ਰɾྭ‫ʹى‬ґΒͣɺอ࣋͞ΕΔͱ͍͏͜ͱͰ͋Δɻ͜Ε͕ͲͷΑ͏ͳར఺ʹͳΔ͔ͱ͍͏ͱɺ‫ز‬ԿΤϛολϯε
͕Ճ଎ʹΑͬͯ‫ݮ‬গ͢Δͱ͍͏͜ͱͰ͋Δɻਤ 4 ʹՃ଎ʹΑͬͯɺԣํ޲ͷ܏͖ x ͕‫ݮ‬গ͢Δ༷ࢠΛࣔ͢ɻՃ଎͞Εͨ͋ͱͷిࢠϏʔϜͰॏཁͳͷ͸‫ز‬ԿΤϛολϯεͰ͋ΓɺՃ଎ͯ͠ΤωϧΪʔΛ্͛Δ͜ͱʹΑͬͯ͜ΕΛԼ͛Δ͜ͱ͕Ͱ͖Δɻిࢠ‫͔ݯ‬Βੜ੒͞ΕͨిࢠϏʔϜ͕΋ͭॳ‫ظ‬ͷ‫֨ن‬ԽΤϛολϯεΛ

εn ͱ͢Δͱɺӡಈྔ pz = mcγβ Λ΋ͭిࢠϏʔϜͷ‫ز‬ԿΤϛολϯε͸ɺ౓্͕ঢ͢Δ͜ͱʹରԠ͠ɺిࢠूஂͷϐʔΫ‫౓ڧ‬ε=্͕ঢ͢Δͱ͍͏͜ͱʹରԠ͢Δɻ·ͨɺ࣌ؒతͳ֦
͕ΓΛখ͘͢͞Δ͜ͱʹΑͬͯɺ਺ 10 fs ͱ͍͏‫ۃ‬Ίͯ୹͍࣌ؒߏ଄Λ࡞Δ͜ͱ΋ՄೳͱͳΔɻిࢠՃ଎‫ث‬ͷ์ࣹޫར༻ͷͱ͖ʹ͸ɺ͜ͷ୹͍֦͕࣌ؒΓͷిࢠϏʔϜ͔Βɺ‫ۃ‬Ίͯ୹͍࣌ؒߏ଄Λ࣋ͬͨ์ࣹޫ͕ੜ

εn
,
γβ

(1.1)

ͱͳΓɺిࢠϏʔϜΛՃ଎͢Δ΄Ͳ‫ز‬ԿΤϛολϯ
ε͕Լ͕Δ͜ͱʹͳΔɻ͜͜Ͱɺγ = 1/ 1 − β 2 ɺ

β = vz /cɺc ͸ޫͷ଎͞Ͱ͋Δɻ͜Ε͕ɺઢ‫ܗ‬Ճ଎‫ث‬Λ࢖͏ࡍͷར఺ͱͳΔɻ੒͞ΕΔ͜ͱʹͳΓɺ͜ΕΛར༻ͯ͠ΑΓߴ͍࣌ؒ෼

ERL ΋ઢ‫ܗ‬Ճ଎‫ث‬ͷҰͭͰ͋ΔͨΊɺిࢠϏʔϜղೳͰͷ෺࣭΍ੜ෺ͷμΠφϛΫεͷଌఆ͕Մೳͱͳ

ͷ඼࣭Λ޲্ͤ͞Δʹ͸ɺిࢠ‫Ͱݯ‬௿Τϛολϯε

ΔɻҎ্ͷΑ͏ʹେ·͔ʹͰ͸͋Δ͕ɺిࢠूஂͷӡϏʔϜΛੜ੒͠ɺ೗ԿʹͦΕΛѱԽͤͣ͞ʹ༌ૹͤ͞

2 Ճ଎‫Ͱث‬͸ɺp ͷ୅ΘΓʹ p Λਐߦํ޲ӡಈྔ p Ͱ‫֨ن‬xxzԽͨ͠෺ཧྔ x = px /pz ͰҐ૬ۭؒΛ࡞ΓΤϛολϯεΛ‫ٻ‬ΊΔɻ͜ΕΛ‫ز‬ԿΤϛολϯεͱ͍͍ɺ͜Ε͸ిࢠϏʔϜͷΤωϧ
Ϊʔʹґଘ͢ΔɻຊςΩετͰ͸ɺ୯ʹ‫ز‬ԿΤϛολϯεΛΤϛο
λϯεͱද‫͠ݱ‬ɺ͋ͱͰग़ͯ͘ΔΤωϧΪʔʹґଘ͠ͳ͍ΤϛολϯεΛ‫֨ن‬ԽΤϛολϯεͱ‫Ϳݺ‬ɻΔ͔ͱ͍͏͜ͱ͕‫ͳͱݤ‬Δɻ㸲㸫 ਤ 4: Ճ଎ʹΑΔԣํ޲ӡಈྔͷมԽɻՃ଎ʹΑΔӡ
ಈྔมԽ͸‫ج‬ຊతʹɺਐߦํ޲ (z ํ޲) ͷΈʹੜ͡ΔͨΊɺՃ଎ʹΑͬͯԣํ޲ͷ܏͖ x = vx /vz ͸খ͘͞ͳΔɻx-x Ґ૬্ۭؒͰͷΤϛολϯε (‫ز‬ԿΤϛολϯε) ͸ɺՃ଎ʹΑͬͯ‫ݮ‬গ͢Δ͜ͱʹͳΔɻޫ‫༻ݯ‬Ϧϯά‫ܕ‬Ճ଎‫͋Ͱث‬Δ 2.5 GeV PF-ring 5 Ͱ͸ɺฏ‫ۉ‬ϏʔϜిྲྀ 450 mA Ͱӡస͕ߦΘΕ͍ͯΔɻਤ 3: ์ࣹ‫ݮ‬ਰͱ์ࣹྭ‫ى‬ͷུ֓ਤɻภ޲ి࣓ੴ಺Ͱ

PR-ring Ͱ͸ߴप೾Ճ଎ۭಎͷप೾਺͸ 500.1 MHzͰ͋ΓɺՃ଎‫ث‬ͷதʹ 312 ‫ݸ‬ͷిࢠόϯνΛஷΊΔిࢠͷ‫ي‬ಓ͕‫͛ۂ‬ΒΕͨͱ͖ʹ์ࣹޫΛ઀ઢํ޲ʹ์͜ͱ͕Ͱ͖ΔͨΊɺपճप೾਺͸ 500.1 MHz / 312

ࣹ͢Δɻ(a) ์ࣹ‫ݮ‬ਰ: ์ࣹޫੜ੒ʹΑͬͯిࢠͷ઀ઢํ޲ͷӡಈྔ͕‫ݮ‬গ͢Δɻిࢠ͕ x ํ޲ͷӡಈྔ

= 1.6029 MHz ͱ‫·ٻ‬ΔɻՃ଎‫ʹ಺ث‬ஷଂ͞Ε͍ͯ
Δ૯ిՙྔ͸ 280.7 nC ͱͳΔɻϦϯά‫ܕ‬Ճ଎‫Ͱث‬ͷΛ΋͍ͬͯΔ৔߹ɺ͜ͷํ޲ͷӡಈྔ΋‫ݮ‬ਰ͢ΔɻՃ

ΤωϧΪʔऩࢧΛߟ͑ΔͱɺϏʔϜଛࣦΛߟ͑ͳ͚Ε଎ۭಎʹΑͬͯిࢠ͸Ճ଎͞Ε z ํ޲ͷӡಈྔ͕૿͹ɺ์ࣹޫੜ੒ʹΑࣦͬͯΘΕͨ෼Λपճ͍ͯ͠Δి

Ճ͢Δɻ͜ͷͱ͖ɺ܏͖ x = px /pz ͸์ࣹલͷঢ়ଶࢠϏʔϜʹ‫͢څڙ‬Δ͚ͩͱͳΔɻʹൺ΂ͯ‫ݮ‬গ͢Δ͜ͱʹͳΔɻ(b) ์ࣹྭ‫ى‬: ์ࣹޫҰํɺઢ‫ܗ‬Ճ଎‫ث‬ͷ৔߹͸ɺϦϯά‫ܕ‬Ճ଎‫ͱث‬͸ҟੜ੒ʹΑͬͯిࢠͷӡಈྔ͕‫ݮ‬ਰ͢ΔͨΊɺภ޲ి࣓

ͳΓৗʹిࢠ‫͔ݯ‬Βిࢠ͕‫͞څڙ‬Εଓ͚Δ͜ͱʹͳΔ

ੴͷ࣓৔ͷ‫ʹ͞ڧ‬Αܾͬͯ·Δपճ‫ي‬ಓ͕มԽ͢Δɻ

ͱ͍͏େ͖ͳҧ͍͕͋Δɻઢ‫ܗ‬Ճ଎‫Ͱث‬͸ɺՃ଎ͯ͠

͜ͷͨΊɺӡಈྔ͕‫ݮ‬গͨ͋͠ͱͷઃ‫يܭ‬ಓ͔Β‫ݟ‬ΔҰ౓ར༻ͨ͠ϏʔϜΛͦͷ··ࣺͯͯ͠·͏ͨΊɺిͱɺ์ࣹޫΛ์ࣹͨ͋͠ͱͷిࢠ͸‫ي‬ಓ͕ͣΕ͍ͯΔ

ࢠϏʔϜΛՃ଎͢Δͷʹඞཁͳిྗ͸ɺ(ϏʔϜిྲྀ)͜ͱʹͳΓɺͦͷઃ‫يܭ‬ಓʹରͯ͠ৼಈ͠ͳ͕Βपճ× (Ճ଎ిѹ) ͱ͍͏୯७ͳؔ܎ͱͳΔɻERL Λ์ࣹޫ‫ʹݯ‬ར༻ͨ͠ͱ͖ͷ࠷ऴ໨ඪ͸ɺϏʔϜΤωϧΪʔ

͢Δ͜ͱʹͳΔɻ

1.3

3 GeV Ͱฏ‫ۉ‬ϏʔϜిྲྀ 100 mA ͱͳ͍ͬͯΔ͕ɺ͜ͷͱ͖ϏʔϜΛՃ଎͢Δͷʹඞཁͳग़ྗΛ୯७ʹߟϏʔϜిྲྀͱग़ྗ

Ճ଎‫ث‬ͷग़ྗΛߟ͑Δͱ͖ʹॏཁͱͳΔͷ͕ɺฏ‫ۉ‬ϏʔϜిྲྀͰ͋Δ 3 ɻిྲྀͷఆٛ͸ɺ͋Δ໘Λ୯Ґ࣌ؒ͋ͨΓʹ௨աͨ͠ిՙྔͰ͋Δɻઢ‫ܗ‬Ճ଎‫ث‬ͷ৔߹͸୯७Ͱ͋Γɺ‫ج‬ຊతʹిࢠ‫͔ݯ‬Βੜ੒͞Εͨ୯Ґ࣌ؒ͋ͨΓͷిՙྔΛߟ͑Ε͹ྑ͍ɻҰํϦϯά‫ܕ‬Ճ଎‫ث‬ͷ৔߹ʹ͸ɺిࢠϏʔϜ͸पճ͠ଓ͚Δ͜ͱʹͳΔͨΊɺϏʔϜిྲྀ͸‫ج‬ຊతʹपճप೾਺ 4 ͱஷଂిՙ
ྔͷੵͱͯ͠‫ٻ‬ΊΒΕΔɻ͜ͷͨΊɺϦϯά‫ܕ‬Ճ଎‫ث‬ͷ৔߹ʹ͸ɺಉ͡ిՙྔΛஷଂ͍ͯ͠Δ৔߹Ͱ΋पճप೾਺͕ߴ͍΄Ͳిྲྀ͸ߴ͘ͳΔɻKEK ʹ͋Δ์ࣹ

͑Δͱɺ3 GV × 100 mA = 300 MW ͱ͍͏ඇৗʹେ͖ͳ஋ͱͳΔɻ·ͨɺ΋͏Ұͭ໰୊ͱͳΔͷ͕ɺࣺ
ͯΔग़ྗ΋‫ۃ‬Ίͯେ͖͍ͱ͍͏͜ͱͰ͋ΔɻߴΤωϧΪʔϏʔϜΛࣺͯΔʹ͸ɺԿΒ͔ͷ෺࣭ʹ౰ͯͯిࢠ
ΛࢭΊΔ͜ͱʹͳΔ͕ɺࣺͯΔϏʔϜͷΤωϧΪʔ͕
೤͋Δ͍͸์ࣹઢʹม‫͞׵‬ΕΔ͜ͱʹͳΓɺൃ೤ྔɺ
์ࣹઢൃੜྔͱ΋ʹ‫ۃ‬Ίͯେ͖ͳ΋ͷͱͳΔɻ͜ͷΑ͏ʹɺઢ‫ܗ‬Ճ଎‫Ͱث‬୯७ʹฏ‫ۉ‬ϏʔϜిྲྀΛ૿‫͢ڧ‬Δ
ͱ͍͏ͷ͸‫࣮ݱ‬తͰ͸ͳ͍ɻ

ERL ͱ͍͏ͷ͸ɺਤ 5 ʹࣔ͢Α͏ʹɺՃ଎͠ར༻
͞ΕͨిࢠϏʔϜΛ࠶ͼओՃ଎ۭಎΛ௨ͯ͠‫ݮ‬଎͠ɺΤωϧΪʔͷԼ͕ͬͨϏʔϜΛϏʔϜμϯϓʹࣺͯΔ

3 ϏʔϜిྲྀͱͯ͠ɺ͜͜Ͱ͸·ͣฏ‫ۉ‬ϏʔϜిྲྀΛߟ͑Δɻ͜Ε͸ిࢠϏʔϜ͕͋Δ࣌ؒߏ଄Λ΋͍ͬͯͨͱ͖ʹɺͦͷߏ଄Λߟྀ͠ͳ͍Ͱɺฏ‫ۉ‬తͳిྲྀΛߟ͑Δ΋ͷͰ͋ΔɻҰํిࢠϏʔϜͷ࣌ؒߏ଄Λߟྀͯ͠ɺҰͭͷ࣌ؒߏ଄ʹ஫໨ͯ͠ϐʔΫిྲྀͱ͍͏΋ͷΛߟ͑Δ৔߹΋͋Δɻ
4 ୯Ґ࣌ؒ͋ͨΓԿճपճ͢Δ͔Λࣔ͢ྔɻ

5 Ճ଎‫Ͱث‬͸ΤωϧΪʔͷ୯Ґͱͯ͠ eV (ిࢠϘϧτ) Λ༻͍Δɻ·ͨΤωϧΪʔͱ͍ͬͯ΋ӡಈΤωϧΪʔͱશΤωϧΪʔͷ
2 ͕ͭ͋Γɺిࢠͷ৔߹ʹ͸੩ࢭΤωϧΪʔ me c2 = 0.511 MeV
෼͕ࠩͰΔɻ௿ΤωϧΪʔྖҬͰ͸͜ͷ͕ࠩແࢹͰ͖ͳ͍ͷͰɺඞཁͳͱ͖ʹ͸ӡಈΤωϧΪʔ͔શΤωϧΪʔ͔Λ໌ࣔ͢Δɻ㸲㸫 

(a) Burst mode

ERL

...

...

(b) CW mode
...

...

ਤ 6: ERL ͷೖࣹ‫Ͱث‬ॏཁͱͳΔ෺ཧ‫ݱ‬৅ɻͨͩ͠ɺ
ޫӄ‫ࢠిۃ‬ॐͰͷϨʔβʔ੔‫Ͳͳܗ‬͸আ͘ɻ
ERL
10 MeV)௃ͱ͍͏ઢ‫ܗ‬Ճ଎‫ث‬ͷಛ௃Λ΋ͭɻERL Λར༻ͨ͠ਤ 5: ERL ͷུ֓ਤɻిࢠॐɺՃ଎ۭಎ͔Βߏ੒͞Εޫ‫Ͱݯ‬͸͜ΕʹՃ͑ͯɺฏ‫ۉ‬ϏʔϜిྲྀͷ૿‫ʹڧ‬ΑͬΔೖࣹ‫ث‬ɺओՃ଎ۭಎɺपճ෦ɺϏʔϜμϯϓʹΑͬͯ์ࣹޫͷฏ‫ۉ‬ग़ྗΛߴΊΑ͏ͱ͍͏΋ͷͰ͋Δɻ

ͯߏ੒͞ΕΔɻೖࣹ‫Ͱث‬ͷΤωϧΪʔ͸ɺ·ͨ΄΅ઢ‫ܗ‬Ճ଎‫ݯޫܕث‬ͷར༻ͱͯ͠͸ɺX ઢࣗ༝ిࢠ

ϏʔϜμϯϓʹࣺͯΔϏʔϜͷΤωϧΪʔͱͳΔɻ͜Ϩʔβʔ (X-ray Free Electron Laser, X-FEL) ͕͢ͷͨΊɺೖࣹ‫ث‬ͷΤωϧΪʔ͕ɺࣺͯΔϏʔϜͷશग़ͰʹՔಇ͓ͯ͠Γɺ໨֮·͍͠੒ՌΛ্͍͛ͯΔ [2]ɻྗΛܾΊΔ͜ͱʹͳΔͨΊɺ૿େͤ͞Δͱ์ࣹઢःณ

͜Ε·Ͱͷઢ‫ܗ‬Ճ଎‫Ͱݯޫܕث‬͸ɺϏʔϜͷ࣌ؒߏͷ໰୊͕‫ͳ͘͠ݫ‬Δɻ଄͸ਤ 6 (a) ʹࣔ͢Α͏ͳϚΫϩύϧεߏ଄ (όʔετӡస) ͱͳ͍ͬͯΔɻϚΫϩύϧε͸ෳ਺ͷύϧεͱ͍͏΋ͷͰ͋Δɻ‫ݮ‬଎͞ΕͨϏʔϜ͕ࣦͬͨΤωϧ͔Βߏ੒͞Εɺ͜ͷؒ͸ϏʔϜ͕Ճ଎͞ΕΔ͜ͱʹΪʔ͸ి࣓৔ͱͯ͠ओՃ଎ۭಎ಺ʹ஝͑ΒΕɺ࣍ʹདྷͳΔɻͦͷ͋ͱ࢑͘ϏʔϜ͕ͳ͍ঢ়ଶͱͳΓɺ·ͨ࣍ΔೖࣹϏʔϜΛՃ଎͢Δ͜ͱʹͳΓɺΤωϧΪʔͷ΍ͷϚΫϩύϧε͕ଓ͘͜ͱʹͳΔɻ͜ͷ৔߹͸ɺϏʔΓऔΓ (ΤωϧΪʔճऩ) ͕ߦΘΕΔ͜ͱʹͳΔɻओ

Ϝ͸ؒܽӡసͱͳΓɺ࣌ؒߏ଄શମͰ‫͖ͱͨݟ‬ͷฏՃ଎ۭಎ಺Ͱ͸࠷ॳʹి࣓৔Λ࡞ΔͨΊͷిྗ͕ඞཁ

‫ۉ‬ϏʔϜిྲྀ͸௿͍΋ͷͱͳΔɻ‫ࡏݱ‬Քಇ͍ͯ͠Δ

ͱͳΔ͕ɺҰ౓ΤωϧΪʔճऩ͕੒ཱͨ͋͠ͱ͸‫ج‬ຊ

X-FEL ޫ‫ݯ‬͸ɺ਺ 10 Hz ∼ 120 Hz ఔ౓ͷ‫܁‬Γฦ͠తʹେ͖ͳిྗΛඞཁͱ͠ͳ͍ɻERL Ͱ͸ೖࣹ‫Ͱث‬प೾਺Ͱӡస͞Ε͍ͯΔɻ͜Εʹରͯ͠ɺύϧε͕࿈ిࢠϏʔϜΛՃ଎͢ΔͨΊͷిྗ͕ඞཁͱͳΔ͕ɺྫ͑͹ͦͷΤωϧΪʔΛ 10 MeV ͱ͢ΔͱɺిࢠϏʔଓͯ͠ଓ͘ӡసϞʔυ (CW ӡస, Continuous Wave
Operation) ͕͋Δɻ͜ͷͱ͖ͷ࣌ؒߏ଄Λਤ 6 (b) ʹϜͷग़ྗ͸ 10 MeV × 100 mA = 1 MW ͱͳΓɺ300

ࣔ͢ɻX-FEL ͷ࣍ͷൃలͱͯ͠ɺถࠃͷ SLAC Λத

MW ʹൺ΂Δͱඇৗʹখ͘͞ͳΔɻ·ͨɺϏʔϜμϯϓʹࣺͯΔग़ྗ΋ 1 MW ͱͳΓɺμϯϓͷൃ೤ྔɺ৺ͱͨ͠‫܁͍ߴͰؔػڀݚ‬Γฦ͠प೾਺ͰͷӡసΛ໨์ࣹઢൃੜྔ΋େ෯ʹ‫ݮ‬গ͞ΕΔɻ͜ͷΑ͏ʹɺERL

͠ԽʹΑͬͯɺ์ࣹޫͷग़ྗ૿‫͕ͳͭʹڧ‬Δ͕ɺ୯७͸ిࢠ‫Ͱݯ‬ੜ੒͞Εͨখ͍͞ॳ‫ظ‬ΤϛολϯεͷϏʔʹߦ͏ͱઌ΄Ͳड़΂ͨΑ͏ʹϏʔϜͷग़ྗ͕‫ۃ‬ΊͯߴϜΛར༻͢Δͱ͍͏ઢ‫ܗ‬Ճ଎‫͍͓ͯʹث‬ɺେฏ‫ۉ‬Ϗʔ͍΋ͷʹͳͬͯ͠·͏ɻ࠷΋଎͍‫܁‬Γฦ͠प೾਺͸ɺϜిྲྀΛѻ͏ͨΊͷҰͭͷํ๏Ͱ͋Δɻ

ߴप೾Ճ଎ۭಎͷप೾਺ͱಉ͡प೾਺ͰϏʔϜΛՃ଎ࢦͨ͠ LCLS-II ‫ܭ‬ը͕ਐΊΒΕ͍ͯΔ [3]ɻߴ‫܁‬Γฦ͢Δ͜ͱͰ͋Δ͕ɺ͜ΕΛߦ͏ʹ͸ฏ‫ۉ‬ϏʔϜిྲྀ͕

1.4্ঢ͗͢͠ΔͨΊɺLCLS-II ‫ܭ‬ըͰ͸࠷େ 1 MHz ͷ

ERL ͷར༻

‫܁‬Γฦ͠प೾਺͕໨ඪͱͳ͍ͬͯΔɻERL ͸ઢ‫ܗ‬Ճ଎‫Ͱث‬େฏ‫ۉ‬ϏʔϜిྲྀӡసΛՄೳͱ͢Δ͜ͱ͕େ͖ͳಛ௃Ͱ͋Γɺ͜ΕΛ‫͔ͨ͠׆‬ར༻๏͕‫ݕ‬౼͞Ε͍ͯΔɻERL ͷར༻ઌ͸ओʹɺ์ࣹޫ

ERL Ͱ͸ΤωϧΪʔճऩʹΑͬͯ‫׬‬શͳ CW ӡస͕ՄೳͰ͋ΔɻKEK ͷ ERL Ͱ͸ 1.3 GHz ͷप೾਺Λ΋ͭ௒఻ಋՃ଎ۭಎ͕։ൃ͞Ε͓ͯΓɺ͜ͷ৔

Δɻͱ΋ʹ௿Τϛολϯε͔ͭେిྲྀΛѻ͏ͱ͍͏͜

߹ϏʔϜͷ‫܁‬Γฦ͠प೾਺͸࠷େ 1.3 GHz ͱͳΔɻERL ޫ‫ݯ‬ͷ໨ඪ͸ɺ௒௿Τϛολϯε΍௒୹ύϧεͱ͕‫͍ͯͬͳͱݤ‬Δɻ

௕Λ΋ͭిࢠϏʔϜʹΑͬͯɺඇৗʹߴ͍ฏ‫౓ًۉ‬ɺ

ERL Λ༻͍ͨ์ࣹޫ‫ݯ‬ར༻Ͱ͸ɺిࢠॐ͔Βੜ੒͞ΕͨϏʔϜ͸ 1 ճʢ͋Δ͍͸਺ճʣͷΈૠೖޫ‫ݯ‬Λۭؒ‫ׯ‬বੑͷྑ͍์ࣹޫͷੜ੒ͱ͍͏ઢ‫ܗ‬Ճ଎‫ث‬ͷ௨ա͠ϏʔϜμϯϓʹࣺͯΒΕΔͨΊɺϏʔϜ͸์ࣹ͑ͯɺϏʔϜΛ࠶ͼओۭಎ·Ͱ༌ૹ͢ΔͨΊʹઃஔ͞‫ݮ‬ਰɾྭ‫ʹى‬ΑΔฏߧঢ়ଶʹ౸ୡͤͣɺೖࣹ‫Ͱث‬ੜ੒

ΕΔपճ෦ʹɺଟ਺ͷ์ࣹޫར༻ͷϏʔϜϥΠϯΛ࡞͞ΕͨϏʔϜͷ࣭͕์ࣹޫͷ࣭Λܾఆ෇͚Δͱ͍͏ಛ

Δ͜ͱ͕Մೳͱ͍͏ಛ௃͕͋Δɻ͜ͷΑ͏ʹɺઢ‫ܗ‬Ճ‫ݯ‬ར༻ [1] ͱૉཻࢠ‫༻ݧ࣮֩ࢠݪ‬ͷར༻ͷ 2 ͭͱͳେฏ‫ྲྀిۉ‬Խͷಛ௃Λ‫͔ͨ͠׆‬΋ͷͰ͋Δɻ͜ΕʹՃ㸲㸫 ଎‫ث‬ͷಛ௃Λ‫͔ͨ͠׆‬ઌਐੑɺෳ਺ͷϏʔϜϥΠϯΛ
‫ݐ‬ઃՄೳͰ͋Δͱ͍͏൚༻ੑ͕ ERL ޫ‫ݯ‬ͷେ͖ͳಛ
௃Ͱ͋Δɻ‫ࡏݱ‬೔ຊͰ͸ɺKEKɺJAEA Λத৺ͱͯ͠೔ຊͷ‫ͯ͠ྗڠ͕ؔػڀݚ‬ɺERL ‫ݯޫܕ‬ͷ࣮ূ‫ػ‬RFcERL (compact ERL) [1] Λ༻͍ͨϏʔϜӡస͕ਐΊ

RF kickΒΕ͍ͯΔ [4]ɻGun focusing

ERL ͷૉཻࢠ‫֩ࢠݪ‬ར༻ʹ͓͍ͯ΋ɺ์ࣹޫ‫ݯ‬ͷ৔߹ͱಉ༷ʹɺߴ඼࣭͔ͭେฏ‫ۉ‬ϏʔϜిྲྀͷಛ௃Λphoto emission‫͔ͨ͠׆‬ར༻͕‫ݕ‬౼͞Ε͍ͯΔ [5]ɻ

1.5

CSR

ਤ 7: ERL ͷೖࣹ‫Ͱث‬ॏཁͱͳΔ෺ཧ‫ݱ‬৅ɻͨͩ͠ɺ

ERL Ͱॏཁͳ߲໨

ޫӄ‫ࢠిۃ‬ॐͰͷϨʔβʔ੔‫Ͳͳܗ‬͸আ͘ɻ

ERL ͷಛ௃͸ɺిࢠ‫ݯ‬ɾೖࣹ‫Ͱث‬ͷϏʔϜͷ࣭Λ޲্ͤ͞Ε͹์ࣹޫͷ࣭Λ޲্ͤ͞ΒΕΔͱ͍͏͜ͱͰ
͋Δɻ‫͑ݴʹٯ‬͹ɺೖࣹ‫Ͱث‬೗Կʹ௒௿Τϛολϯε
ϏʔϜΛੜ੒͠ɺ૿େͤͣ͞ʹૠೖޫ‫Ͱ·ݯ‬༌ૹ͢Δ
͔ͱ͍͏͜ͱ͕ɺERL Λར༻͢Δ্Ͱͷ࠷େͷ՝୊
Ͱ͋Δͱ͍͏͜ͱΛ͍ࣔͯ͠Δɻಛʹɺ100 mA(‫܁‬Γ
ฦ͠ 1.3 GHz Ͱ 1 όϯν͋ͨΓ 77 pCʣͷେిྲྀͰ‫֨ن‬ԽΤϛολϯε 0.1 mm mrad ͱ͍͏௒௿ΤϛολϯεϏʔϜΛೖࣹ‫Ͱث‬ੜ੒ɾ༌ૹ͢Δͱ͍͏ͷ͸௅ઓతͳ՝୊Ͱ͋Γɺେ‫ܕ‬ͷ GeV Ϋϥεͷ ERL ޫ‫ݯ‬Λ࣮‫͢ݱ‬ΔͨΊʹ͸ɺ࣮ূ͠ͳ͚Ε͹ͳΒͳ͍ςʔϚΛѻ͏৔߹ʹ͸ɺϏʔϜ͕Ճ଎ۭಎ಺ʹྭ‫͢ى‬Δి࣓
৔ʹΑΔϏʔϜෆ҆ੑ‫ݱ‬৅Λߟྀ͢Δඞཁ͕͋Δɻ
·ͨɺERL ͸͜Ε·Ͱʹͳ͍େిྲྀΛઢ‫ܗ‬Ճ଎‫ث‬Ͱѻ͏ͱ͍͏͜ͱ͔ΒɺՃ଎‫ث‬γεςϜશମͱͯ͠
ͷ૯߹։ൃ͕ॏཁͱͳΔɻERL Λޫ‫ݯ‬Ճ଎‫ͯ͠ͱث‬ར༻͢Δ৔߹Λߟ͑Δͱɺͦ͜Ͱཁ‫͞ٻ‬ΕΔੑೳ͸ɺ҆ఆͳޫ (มಈ͠ͳ͍ɺதஅ͠ͳ͍)ɺ௿͍ίετ (‫ݐ‬ઃɾӡస) ͱ͍͏͜ͱʹͳΔɻޫ‫ݯ‬ར༻ʹ޲͚ͨ ERLՃ଎‫ݱ࣮ث‬ͷ՝୊ (ະ஌਺͸Կ͔ʁ) ʹ͍ͭͯ·ͱΊΔͱ࣍ͷΑ͏ʹͳΔɻͰ͋Δɻ

1. ߴً౓ɾେిྲྀిࢠϏʔϜͷੜ੒: 100 mA Λ‫ڙ‬௒௿ΤϛολϯεϏʔϜΛੜ੒͢Δ্Ͱࠀ෰͠ͳ͚
Ε͹ͳΒͳ͍෺ཧ‫ݱ‬৅͸ɺۭؒిՙޮՌͱίώʔϨϯτ์ࣹʢCSRʣʹΑΔϏʔϜ඼࣭ͷ௿ԼͰ͋Δɻਤ 7ʹೖࣹ‫ۙث‬๣ͰͷॏཁͱͳΔ෺ཧ‫ݱ‬৅Λࣔ͢ɻۭؒి
ՙޮՌ͸ϏʔϜ಺ిࢠؒͰͷ Coulomb ྗʹΑΔ΋ͷ
Ͱ͋Γɺ10 MeV ఔ౓ͷ௿͍ϏʔϜΤωϧΪʔΛ΋ͭೖࣹ‫Ͱ಺ث‬͸ɺશମʹ౉ͬͯͦͷӨ‫ڹ‬Λߟྀ͠ͳ͚Ε‫͠څ‬ଓ͚Δిࢠ‫ݯ‬͸͜Ε·Ͱʹͳ͍։ൃཁૉͰ
͋Δɻ

2. େిྲྀిࢠϏʔϜͷՃ଎: 100 mA ΛՃ଎͠ଓ͚
Δ௒఻ಋՃ଎ۭಎ͸͜Ε·Ͱʹͳ͍։ൃཁૉͰ
͋Δɻ͹ͳΒͳ͍ɻిࢠϏʔϜ͕Ճ଎͞Εͨ͋ͱ͸ɺۭؒి

3. ϏʔϜੑೳ: ೖࣹ‫( ث‬௿͍ΤωϧΪʔྖҬ) Ͱͷࢼ‫ݧ‬ɺपճ෦Ͱͷੑೳ‫ূݕ‬Λߦ͍ͬͯ͘ඞཁ͕͋

ՙޮՌ͸ऑ͘ͳ͍ͬͯ͘ɻपճ෦·ͰϏʔϜΛ༌ૹ͢ΔɻಛʹɺେిྲྀՃ଎ͷӨ‫ڹ‬ɺ୹όϯν௕ͷӨ‫ڹ‬Δؒʹɺόϯν௕ΛΑΓ୹͘͢ΔͨΊͷόϯνѹॖͱ(CSR) ͷ‫ূݕ‬΍ɺ௿ΤϛολϯεϏʔϜΛ࣮ূ
͢Δ͜ͱ͕ඞཁͰ͋Δɻ

͍͏ૢ࡞Λߦ͏͕ɺ͜ͷͱ͖ʹ΋ CSR ͷӨ‫ڹ‬Λ೗Կʹখ͘͞཈͑Δ͔ͱ͍͏͜ͱ͕ॏཁͱͳΔɻCSR ͷӨ‫ڹ‬Λେ·͔ʹ‫ͱ͏ݴ‬ɺిࢠ͕ภ޲ి࣓ੴͰ‫͛ۂ‬ΒΕͨ‫ي‬ಓΛ௨Δͱ͖ɺిࢠϏʔϜ͕์ग़ͨ͠์ࣹޫ͕‫ي‬ಓ௕ࠩʢޫ͸௚ਐɺిࢠ͸ԕճΓʣʹΑͬͯ‫ݩ‬ͷϏʔϜʹ௥͍͖ͭɺ‫ݩ‬ͷϏʔϜΛՃ଎ɾ‫ݮ‬଎ͤ͞ΤωϧΪʔΛมԽͤ͞Δͱ͍͏΋ͷͰ͋ΔɻۭؒిՙޮՌ΍

CSR ͸ɺ୯७Խ͞ΕͨϞσϧͰ͸ղੳతʹऔΓѻ͑Δ͕ɺ‫࣮ݱ‬ͷόϯνԽ͞ΕͨϏʔϜͰ͸ूஂతͳޮՌ
ͱͯ͠‫ݱ‬ΕΔͨΊɺղੳతͳѻ͍͕ࠔ೉Ͱ͋Δɻ͜ͷΑ͏ͳࣄ৘͔Βɺओʹ਺஋‫ʹࢉܭ‬ΑΔํ๏͕༻͍ΒΕ͍ͯΔɻ·ͨɺERL ͷಛ௃Ͱ͋Δେฏ‫ۉ‬ϏʔϜిྲྀ㸲㸫 

4. ௕࣌ؒɾ҆ఆʹ͔ͭҰ༷ʹӡస͢ΔͨΊʹඞཁͳ҆ఆੑ: ERL ͸‫ج‬ຊతʹઢ‫ܗ‬Ճ଎‫ͳݯޫܕث‬ͷͰɺஷଂϦϯάͷΑ͏ͳ҆ఆԽ‫͍ͳ͕ߏػ‬ͷͰɺมಈ‫ݯ‬Λஅͭ͜ͱ͕ඞཁͰ͋Δɻ·ͨɺϢʔβʔར༻Λதஅ͢Δ‫ݪ‬ҼʹͳΓͦ͏ͳͷ͸ɺΧιʔυަ‫( ׵‬ण໋͕ͲΕ͘Β͍͔ʁ)ɺ௒఻ಋۭಎͷఀࢭස౓Ͱ͋Γɺ࣮ࡍͷϏʔϜӡసΛ௨ͯ͠‫͠ূݕ‬͍ͯ͘ඞཁ͕͋Δɻ

5. ӡసίετ: ΤωϧΪʔճऩ͸Ұ‫ݟ‬ΤίͬΆ͘‫ݟ‬͑Δ͕ɺϏʔϜग़ྗͱ͸ผʹɺ௒఻ಋۭಎΛ 2 ∼4 K ʹྫྷ٫͢ΔͨΊͷྫྷౚ‫ػ‬ͷӡసʹίετ͕͜Ͱ͸ɺ·ͣओۭಎʹΑΔՃ଎ɾ‫ݮ‬଎ͷϏʔϜ༌ૹ৚ֻ͔Δɻ

݅ 7 ʹ͍ͭͯ঺հ͢ΔɻओۭಎΛग़ͨϏʔϜ͸ɺपճ

6. ์ࣹઢःณ: ΤωϧΪʔճऩʹΑͬͯμϯϓʹࣺ
ͯΔग़ྗ͸͔ͳΓԼ͕Δ͜ͱʹͳΔ͕ɺ༌ૹதʹͲ͜ͰɺͲΕ͘Β͍ϏʔϜଛࣦ͕‫͖ى‬Δͷ͔Λɺࢼ‫͢ূݕͰػݧ‬Δඞཁ͕͋Δɻ෦ (पճϧʔϓ) Λ௨ͬͯిࢠϏʔϜΛར༻͢Δ۠ؒ
·Ͱ༌ૹ͞ΕΔɻ͜ͷࡍʹɺར༻৚݅ʹ߹ΘͤͯϏʔϜ৚݅Λ੍‫͢ޚ‬Δඞཁ͕͋Δɻྫ͑͹ɺERL ͷޫ‫ݯ‬ར༻Ͱ͸ӡసϞʔυʹΑͬͯ 100 fs ҎԼͷ୹͍ిࢠόϯν௕Λ࣮‫͢ݱ‬Δ͜ͱ͕ཁ‫͞ٻ‬ΕΔ͕ɺ͜ͷૢ࡞͸͜ͷΑ͏ʹɺERL Λར༻ͨ͠ޫ‫ݯ‬Λ࣮‫͢ݱ‬Δʹ͸ɺओۭಎͱपճ෦Λར༻ͯ͠ߦΘΕΔɻόϯν௕͕୹͘ཁૉٕज़։ൃͱͱ΋ʹɺޫ‫ݯ‬Ճ଎‫ͯ͠ͱث‬ͷ૯߹ੑೳ

ͳͬͯ͘ΔͱɺCSR ʹΑΔӨ‫ݦ͕ڹ‬ஶʹ‫ݱ‬Εͯ͘Δ

Λ‫͢ূݕ‬Δඞཁ͕͋Δɻ͜ΕΛߦ͏ͨΊʹ։ൃ͞ΕͨͨΊɺ͜ΕΛ೗Կʹ੍‫͢ޚ‬Δ͔ͱ͍͏͜ͱ͕όϯνѹͷ͕ɺcERL Ͱ͋Γɺ2014 ೥͔Β։࢝͞Εͨ૯߹ӡॖʹ͓͍ͯॏཁͱͳΔɻ

సͰɺ্‫ه‬ͷ՝୊ʹର͢Δճ౴Λ༩͑ͭͭ͋Δͱ͜Ζ

ຊςΩετͰ͸ɺిࢠϏʔϜͷ෺ཧΛͰ͖Δ͚ͩΘͰ͋Δɻ͜͜Ͱ͸ɺcERL ࣮ূ‫Ͱػ‬ͷ੒ՌΛ঺հ͠ͳ͔Γ΍͘͢঺հ͠Α͏ͱࢼΈͨɻERL ೖࣹ‫Ͱث‬͸ɺ͕ΒɺERL ͰॏཁͱͳΔϏʔϜྗֶʹ͍ͭͯ·ͱΊ୯ཻࢠʹ͍ͭͯͷ૬ର࿦తͳӡಈํఔ͔ࣜΒ࢝Ίͯɺ͍ͯ͘ɻͦΕΛϏʔϜΤϯϕϩʔϓํఔࣜʹ֦ு͠ɺۭؒిՙ
ޮՌΛऔΓೖΕΔͱ͍͏ΞϓϩʔνΛͱΔɻԁ‫ܗ‬Ճ଎

1.6‫ث‬தͰͷϏʔϜμΠφϛΫεͷ෺ཧͰ͸ɺ‫ۂ‬ઢ࠲ඪ‫ܥ‬ຊςΩετͷྲྀΕͰͷ୯ཻࢠͷӡಈ͔Β࢝ΊͯɺCourant-Synder ύϥຊςΩετͰ͸ 6 ɺERL ͷதͰߦΘΕΔҰ࿈ͷྲྀϝλͱ‫ݺ‬͹ΕΔϏʔϜޫֶؔ਺Λߟ͑Δํ๏͕ҰൠతΕɺੜ੒ɺՃ଎ɺ༌ૹɺ‫ݮ‬଎ɺഇ‫঺͍ͯͭʹغ‬հͯ͠ͳྲྀΕͰ͋Δ͕ɺ͜͜Ͱ͸ͦͷղઆ͸͠ͳ͍Ͱɺೖࣹ͍͘ɻERL ೖࣹ‫Ͱث‬͸ϏʔϜͷΤωϧΪʔ͕௿͍ͨ

‫Ͱث‬ͷϏʔϜ෺ཧΛߟ͑Δ্ͰศརͳΞϓϩʔνΛͱΊʹɺۭؒిՙޮՌʹΑΔूஂޮՌ͕‫ݦ‬ஶʹ‫ݱ‬ΕΔɻ

Δ͜ͱʹͨ͠ɻຊςΩετͷதͰ΋ɺϏʔϜޫֶؔ਺์ࣹޫ‫༻ݯ‬ͷ GeV ΫϥεͷϦϯά‫ܕ‬Ճ଎‫Ͱث‬͸ɺి͕ग़ͯ͘Δ͕ɺͦΕΒʹ͍ͭͯ͸ɺOHO’2008 ͷ‫ాݪ‬ࢠϏʔϜͷӡಈ͸ۭؒిՙޮՌʹΑΔޮՌΛແࢹͯ͠ࢯͷߨٛ΍ଞͷࢀߟจ‫[ ݙ‬6, 7] Λࢀরͯ͠ཉ͍͠ɻ

୯ཻࢠతʹѻ͍ɺඞཁʹԠͯ͡ϏʔϜෆ҆ఆੑͳͲͷ͜͜Ͱ͸ɺ·ͣର৅ͱ͢Δ ERL Ճ଎‫ث‬Λߏ੒͢Δ

ूஂޮՌΛऔΓೖΕΔͱ͍͏ํ๏͕औΒΕΔɻҰํɺཁૉʹ͍ͭͯηΫγϣϯ 2 Ͱ঺հ͠ɺిࢠϏʔϜத

ERL ೖࣹ‫Ͱث‬͸ϏʔϜͷΤωϧΪʔ͕௿͘ɺ·ͨిՙີ౓΋ߴ͍ͨΊʹɺۭؒిՙޮՌͷӨ‫ࢧ͕ڹ‬഑తʹͰॏཁͱͳΔ෺ཧ‫ݱ‬৅ʹ͍ͭͯηΫγϣϯ 3 Ͱ঺հͳΓɺͦ΋ͦ΋୯ཻࢠతͳऔΓѻ͍͕Ͱ͖ͳ͍ɻ͜ͷ

Ͱ͋ΔɻηΫγϣϯ 4 Ͱ͸ɺۭؒిՙޮՌʹΑΔӨ‫ڹ‬ͨΊʹɺೖࣹ‫ث‬தͰͷిࢠϏʔϜͷ෺ཧ͸ෳࡶʹͳΓΛධՁ͢Δ४උͱͯ͠ɺۙ࣠ޫઢۙࣅΛ༻͍ͨϏʔϜ͕ͪͰ͋Δɻ‫࣮ݱ‬ͷೖࣹ‫ث‬ͷઃ‫Ͱܭ‬͸ɺ্ྲྀͷཁૉͷΤϯϕϩʔϓํఔࣜΛಋग़͠ɺۭؒిՙޮՌ͕ͳ͍৔

Ө‫͕ڹ‬Լྲྀʹ఻೻͍ͯͨ͘͠ΊɺϏʔϜμΠφϛΫε

߹ͷ֎෦͔ΒՃ͑ͨి࣓৔ʹର͢ΔӨ‫ڹ‬Λ঺հ͢ΔɻΛߟ͑Δࡍʹ͸ೖࣹ‫ث‬શମΛऔΓѻ͏ඞཁ͕͋ΓɺଟηΫγϣϯ 5 Ͱ͸ɺԁ౵ରশੑΛ΋ͭϏʔϜʹର͘͠ͷ৔߹͸਺஋‫ʹࢉܭ‬ΑΔํ๏͕࠾༻͞Ε͍ͯΔɻ͠

ۭͯؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜΤϯϕϩʔϓํఔ͔ࣜ͠ɺೖࣹ‫ث‬Λߏ੒͢Δ֤ཁૉຖʹిࢠϏʔϜͷ෺ཧΛಋग़͢Δɻ͜͜Ͱ঺հ͢ΔۭؒిՙޮՌʹ͍ͭͯͷΛߟ͑ɺͦͷ‫ج‬ຊΛ཈͓͑ͯ͘͜ͱ͸ɺ਺஋‫ࢉܭ‬Λߦٞ࿦ͷଟ͘͸ɺࢀߟจ‫[ ݙ‬8, 9] Λࢀߟʹ͍ͯ͠Δɻη͏্Ͱ΋ॏཁͰ͋Δɻ

Ϋγϣϯ 6 Ͱ͸ϏʔϜΤϯϕϩʔϓํఔࣜΛ֦ு͠ిࢠॐɾೖࣹ‫Ͱث‬ੜ੒͞ΕͨిࢠϏʔϜ͸ɺ௒఻ಋ

͢Δɻ͜ͷͳ͔ͰɺಛʹॏཁͱͳΔͷ͕ۭؒిՙޮՌ

ͯɺۭؒిՙޮՌʹΑࣹͬͯӨΤϛολϯε͕૿େۭ͢ಎ (ओۭಎ) ʹΑͬͯ௿ΤωϧΪʔྖҬ͔ΒߴΤωΔ༷ࢠΛௐ΂ΔɻηΫγϣϯ 7 Ͱ͸ɺۭؒిՙޮՌ

ϧΪʔྖҬ·ͰՃ଎͞ΕΔɻे෼ʹߴ͍ΤωϧΪʔ·Λ‫ؚ‬ΜͩϏʔϜγϛϡϨʔγϣϯʹ͍ͭͯ঺հ͢Δɻ

ͰՃ଎͞Εͨ৔߹ʹ͸ɺۭؒిՙޮՌΛແࢹͯ͠औΓηΫγϣϯ 8 Ͱ͸ిࢠॐۙ๣ͰͷϏʔϜ෺ཧʹ͍ͭѻ͏͜ͱ͕Ͱ͖Δɻे෼Ճ଎͞Εͨ͋ͱͷϏʔϜʹͭͯ঺հ͠ɺ࣮ࡍͷ cERL ిࢠॐͷղੳϞσϧͷิਖ਼͍ͯ͸‫ج‬ຊతʹ୯ཻࢠͱͯ͠ղੳ͢Δ͜ͱʹͳΔɻ͜

ʹ͍ͭͯ঺հ͢ΔɻηΫγϣϯ 9 Ͱ͸ೖࣹ‫ث‬ͷ༌ૹ

6 2015 ೥ 8 ݄ 17 ೔൛ɻςΩετ͸ਵ࣌ߋ৽͍ͯ͘͠༧ఆͳͷͰɺ࠷৽൛ʹ͍ͭͯ͸ɺOHO ηϛφʔͷ web ϖʔδΛࢀরͯ͠ཉ͍͠ɻ৚݅ͷ࠷దԽʹ͍ͭͯ঺հ͢ΔɻηΫγϣϯ 10 Ͱ͸
7 ϏʔϜ༌ૹ৚݅͸ɺϏʔϜޫֶؔ਺ɺϏʔϜΦϓςΟΫεʹ
Αͬͯ‫ه‬ड़͞ΕΔɻ㸲㸫 ߹ྲྀ෦Ͱ໰୊ͱͳΔ෺ཧʹ͍ͭͯ֓‫͢؍‬ΔɻηΫγϣϯ 11 Ͱ͸ೖࣹ‫͔ث‬Βओۭಎ΁ͷϏʔϜ༌ૹʹ͍ͭͯ
঺հ͢ΔɻηΫγϣϯ 12 Ͱ͸ओۭಎʹΑΔՃ଎ɾ‫ݮ‬଎༌ૹ৚݅ͷઃ‫͍ͯͭʹܭ‬ɺηΫγϣϯ 13 Ͱ͸पճ෦Ͱͷ༌ૹ৚݅ʹ͍ͭͯɺηΫγϣϯ 14 Ͱ͸ϏʔϜͷ‫ݮ‬଎ͱϏʔϜμϯϓʹ͍ͭͯ঺հ͢ΔɻηΫγϣϯ

15 Ͱ͸पճ෦Ͱͷόϯνѹॖʹ͍ͭͯ঺հ͢ΔɻηΫγϣϯ 12 ∼ 15 ͸ౡాඒൕࢯʹΑͬͯ࡞੒͞Εͨɻ·ͨɺຊςΩετ͸ OHO’2008 ͷςΩετΛϕʔεʹͯ͠ɺՃචमਖ਼͞Εͨ΋ͷͰ͋Δɻ
ຊςΩετͰ͸ɺERL ೖࣹ‫ث‬Λྫʹͱͬͯɺઢ‫ܗ‬Ճ଎‫ث‬தͰͷۭؒిՙޮՌͷӨ‫ڹ‬ΛऔΓѻ͍ͬͯΔɻԁ‫ܗ‬Ճ଎‫Ͱث‬͸ઢ‫ܗ‬Ճ଎‫ͱث‬ҟͳΓɺӡಈ͕प‫ظ‬৚݅ԼͰߦΘΕΔͱ͍͏େ͖ͳҧ͍͕͋Δɻԁ‫ܗ‬Ճ଎‫ث‬தͰ͸ɺۭؒిՙޮՌʹΑΔॏཁͳӨ‫ͯ͠ͱڹ‬νϡʔϯγϑτͱ‫ݺ‬͹ΕΔ‫ݱ‬৅͕͋Δ͕ɺ͜͜Ͱ͸ͦΕʹ͍ͭͯ͸औΓѻΘͳ͍ɻա‫ڈ‬ͷ OHO’2000 ͷொా৻ೋࢯ
ʹΑΔςΩετ [10] Λࢀরͯ͠ཉ͍͠ɻ·ͨɺଟ͘ͷ༗༻ͳจ‫͍ͯͭʹݙ‬͸ɺ‫ר‬຤ʹ·ͱΊͨͷͰࢀর͍͖͍ͯͨͩͨ͠ɻ㸲㸫 

r

ERL Ճ଎‫ث‬Λߏ੒͢Δཁૉ2

ERL ͸ߴً౓ిࢠϏʔϜΛੜ੒͠Ճ଎͢Δిࢠॐɾೖࣹ‫ث‬ɺΤωϧΪʔճऩΛߦ͏ओՃ଎ۭಎɺϏʔϜΛ

Anode
Ez
Electron bunch

पճͤ͞Δपճ෦ɺ‫ݮ‬଎‫ޙ‬ͷϏʔϜΛࣺͯΔϏʔϜμzϯϓΑΓߏ੒͞ΕΔɻ͜ΕΒͷதͰ ERL ͷ‫ݤ‬ΛѲΔ

Laser

ཁૉ͸ɺߴً౓͔ͭେిྲྀిࢠϏʔϜΛੜ੒͢ΔͨΊͷిࢠॐͱͦΕΒΛՃ଎͢ΔͨΊͷ௒఻ಋՃ଎ۭಎͰ
͋Δɻ͜͜Ͱ͸ɺ͜ΕΒͷཁૉʹ͍ͭͯͷ঺հ͢Δɻ

2.1

Photo emission
Cathode(Semiconductor)ޫӄ‫ ۃ‬DC ిࢠॐ

ERL ೖࣹ‫ث‬ͷੑೳΛܾఆ͢Δ্Ͱɺ࠷΋ॏཁͳཁૉ͸ిࢠॐͰ͋ΔɻిࢠॐͰ඼࣭ͷྑ͍ϏʔϜΛੜ੒Ͱ͖ͳ͚Ε͹ɺͦΕҎ߱ͷཁૉͰ͍͘Β඼࣭Λҡ࣋͢ΔΑ͏ʹͯ͠΋ɺॳ‫ظ‬ͷ඼࣭Ҏ্ʹ͸Ͱ͖ͳ͍͔Β
Ͱ͋Δɻిࢠॐͷ։ൃͰ͸ɺޫӄ‫ۃ‬ද໘Ͱͷిࢠͷੜ੒ͷ෺ཧͷཧղɺߴ͍Ճ଎ిѹͷ࣮‫ݱ‬ɺߴ͍ྔࢠޮ཰
ͷ࣮‫ݱ‬ɺΧιʔυද໘ͷ௕ण໋ԽͳͲͷ։ൃ՝୊͕͋
Γɺ‫ࡏݱ‬ண࣮ͳ։ൃ͕ਐΊΒΕ͍ͯΔɻ

ERL ༻ͷిࢠॐͱͯ͠‫͍༻ࡏݱ‬ΒΕ͍ͯΔͷ͕ɺޫӄ‫( ۃ‬Χιʔυ, cathode) Λ༻͍ͨ DC ిࢠॐͰ͋ΔɻcERL ༻ʹ࠷େՃ଎ిѹ 500 kV ͷޫӄ‫ ۃ‬DCిࢠॐ͕։ൃ͞ΕɺcERL ʹ͓͚ΔϏʔϜӡసʹ༻͍ΒΕ͍ͯΔɻޫӄ‫ ۃ‬DC ిࢠॐͷ໛ࣜਤΛਤ ‘8 ʹࣔ͢ɻΧιʔυ໘ʹ GaAs ͷΑ͏ͳޫӄ‫࣭ࡐۃ‬Λઃஔ͠ɺΞϊʔυి‫ిʹؒۃ‬ѹΛҹՃ͓ͯ͘͠ɻిࢠϏʔϜΛऔΓग़͍ͨ͠λΠϛϯάͰྭ‫ى‬ϨʔβʔΛরࣹ͠ɺిࢠΛऔΓग़͢ɻද໘͔ΒऔΓग़͞Εͨిࢠ͸ɺਤ 8: ޫӄ‫ۃ‬Λ༻͍ͨ DC ిࢠॐͷ໛ࣜਤɻΧιʔυ্ʹޫӄ‫ޮిޫ( ࣭ࡐۃ‬ՌʹΑͬͯిࢠΛੜ੒) Λ഑ஔ͠ɺΞϊʔυి‫ͱؒۃ‬ͷؒʹ DC ిѹΛҹՃ͢
Δɻྭ‫ى‬Ϩʔβʔޫ͕রࣹ͞ΕΔͱిࢠ͕औΓग़͞ΕɺΧιʔυͱΞϊʔυؒͷిѹʹΑͬͯిࢠ͕Ճ଎͞ΕΔɻ

ERL Ճ଎‫ث‬։ൃʹ͓͍ͯɺిࢠॐͷ։ൃ͸ඇৗʹॏཁͳ໾ׂΛ୲͍ͬͯΔ͕ɺOHO’2015 Ͱ͸ʮిࢠॐʯʹ͍ͭͯͷߨٛ΋͋ΔͨΊɺ͜͜Ͱ͸ిࢠॐʹ͍ͭͯ͸ϏʔϜྗֶʹؔ͢Δུ֓Λ঺հ͢ΔͷΈͱ͢Δɻిࢠॐʹ͍ͭͯ͸ɺOHO’2008 ͷ੢৿৴ߦࢯͷߨٛ΍ɺOHO’2015 ͷࢁຊকതࢯͷߨٛͰৄ͘͠঺հ͞
ΕΔͷͰɺͦͪΒΛࢀরͯ͠ཉ͍͠ɻ·ͨଟ͘ͷ༗༻ͳจ‫[ ݙ‬11, 12] ΋͋ΔͷͰɺͦͪΒ΋ࢀর͍ͯͨͩ͠
͖͍ͨɻΧιʔυͱΞϊʔυؒͷిѹʹΑͬͯՃ଎͞ΕɺϏʔ

2.2

ϜϥΠϯԼྲྀ΁༌ૹ͞ΕΔɻ

ERL ʹར༻͢ΔిࢠॐͰ͸ɺߴً౓͔ͭେฏ‫ిۉ‬ྲྀͷిࢠϏʔϜΛੜ੒͠ଓ͚Δ͜ͱ͕ཁ‫͞ٻ‬ΕΔɻిࢠॐͷ‫ͯ͠ͱࣜܗ‬ɺޫӄ‫ ۃ‬DC ిࢠॐͷ΄͔ʹɺDC೤ిࢠॐɺߴप೾ (RF) ೤ిࢠॐɺৗ఻ಋޫӄ‫ ۃ‬RFిࢠॐɺ௒఻ಋޫӄ‫ ۃ‬RF ిࢠॐ͕͋Δɻ೤ిࢠॐ
ͷ৔߹ʹ͸೤ిࢠ์ग़Λར༻͢ΔͨΊॳ‫ظ‬Τϛολϯ
ε͕େ͖͘ͳΔͱ͍͏͜ͱ͕͋Γɺ·ͨৗ఻ಋͷ RFిࢠॐͰ͸େฏ‫ۉ‬ϏʔϜిྲྀΛग़͠ଓ͚Δ৔߹ʹ͸ɺߴप೾ి࣓৔Λྭ‫͠ى‬ଓ͚Δ͜ͱʹΑΔ೤ෛՙͷ໰୊
͕ੜ͡Δɻ௒఻ಋޫӄ‫ ۃ‬RF ిࢠॐ͸ɺERL ిࢠॐʹཁ‫͞ٻ‬ΕΔੑೳΛຬͨ͢Մೳੑ͕ߴ͍͕ɺ·ͩൃల్্ͷٕज़Ͱ͋Γɺ҆ఆͨ͠ϏʔϜӡసʹ͸·ͩ։ൃ͕ඞཁͰ͋Δɻޫӄ‫ ۃ‬DC ిࢠॐ͜Ε·Ͱʹ։ൃ͞Εٕͨज़Λ‫͍ͯ͠ૅج‬ΔͨΊɺERL ͷཁ‫ٻ‬Λຬͨ͢࠷΋‫࣮ݱ‬తͳిࢠॐͱͳ͍ͬͯΔɻޫӄ‫ʹۃ‬ΑΔిࢠϏʔϜͷੜ੒cERL Ͱ͸ޫӄ‫ۃ‬ͷࡐ࣭ͱͯ͠ɺ൒ಋମͷ GaAs ͕༻͍ΒΕ͍ͯΔɻిࢠॐ͔ΒͷϏʔϜͷ඼࣭͸ɺ·ͣୈҰʹޫӄ‫͔ۃ‬Βੜ੒͞ΕΔిࢠϏʔϜͷ඼࣭ʹΑܾͬͯ·ΔͨΊɺGaAs ͷଞʹɺGaAsP ΍ GaInP [13]ɺ
௒֨ࢠߏ଄ͳͲଟ͘ͷࡐ࣭ͷ‫ߦ͕ূݕ‬ΘΕ͍ͯΔɻΧιʔυࡐ࣭ͷ෺ཧͰॏཁͱͳΔͷ͕ɺॳ‫ظ‬Τϛολϯεͱ࣌ؒԠ౴Ͱ͋Δɻ·ͣɺॳ‫ظ‬Τϛολϯεʹ
͍ͭͯߟ͑ΔɻGaAs Χιʔυͷ৔߹ʹ͸ɺՁిࢠଳ
ʹ͍ΔిࢠΛྭ‫ى‬ϨʔβʔʹΑͬͯ఻ಋଳ·Ͱྭ‫ى‬͠ɺ͜ΕΛΧιʔυද໘Խ͔Βਅۭ४ҐʹऔΓग़͢͜ͱʹͳΔ (ਤ 9)ɻ͜ͷͱ͖ɺిࢠΛऔΓग़͠қ͘͢ΔͨΊʹɺ఻ಋଳʹରͯ͠ਅۭ४ҐΛԼ͛ΔΑ͏ͳද໘ (Negative Electron Aﬁnity, NEA ද໘) ͕࡞੒͞ΕΔɻ͜ͷ NEA ද໘ʹΑͬͯɺরࣹͨ͠Ϩʔβʔग़㸲㸫 

kB T Λਪଌ͢Δ͜ͱ͕Ͱ͖ΔɻERL ‫ܭ‬ը͕ਐΊΒΕ͍ͯΔίʔωϧେֶͰ͸ɺޫӄ‫ ۃ‬DC ిࢠॐͷςετϏʔϜϥΠϯͰɺGaA s ΧιʔυɺGaAsP Χιʔυʹ͍ͭͯͷ೤Τϛολϯεͷଌఆ͕ߦΘΕɺΧιʔυͰͷ kB T ͷ஋͕‫ܭ‬ଌ͞Ε͍ͯΔ [14, 15]ɻ·ͨ KEKͰ΋ޫӄ‫ ۃ‬DC ిࢠॐͷςετϏʔϜϥΠϯΛ༻͍ͯ GaAs ͷ༨৒ΤωϧΪʔͱྭ‫ى‬Ϩʔβʔ೾௕ͷؔ܎͕ଌఆ͞Ε͍ͯΔ [16]ɻ
·ͨɺΧιʔυࡐ࣭ͰॏཁͱͳΔ΋͏Ұͭͷੑೳ͕ਤ 9: ޫӄ‫͔ۃ‬ΒͷిࢠͷҾ͖ग़͠ɻྭ‫ى‬ϨʔβʔʹΑͬͯిࢠ͕Ձిࢠଳ (VB) ͔Β఻ಋଳ (CB) ʹྭ‫ى‬͞ΕΔɻྭ‫͞ى‬Εͨిࢠ͸൒ಋମதΛද໘·ͰҠಈ͢Δɻਅۭ४Ґ͸ NEA ද໘ʹΑͬͯԼ͛ΒΕ͍ͯΔͨ
Ίɺిࢠ͕ਅۭʹҾ͖ग़͞ΕΔɻ

ྔɺྔࢠޮ཰ (Quantum Eﬃciency) ্͕ঢ͢Δɻྭ‫ى‬Ϩʔβʔͷ೾௕͸ɺ఻ಋଳͱՁిࢠଳͷΤωϧΪʔࠩʹ߹Θͤͨ΋ͷ͔ɺͦΕΑΓ΋େ͖͍ΤωϧΪʔΛ΋ͭ΋ͷ͕࢖༻͞ΕΔɻ೾௕ͷ୹͍ϨʔβʔΛ࢖ͬͨ৔߹ɺਅۭ४ҐΑΓ΋ߴ͍ΤωϧΪʔΛ΋ͬͨిࢠ͕
૿͑ΔͨΊɺΑΓଟ͘ͷిࢠ͕ਅۭʹऔΓग़͞ΕΔ͜ͱʹͳΓɺྔࢠޮ཰্͕ঢ͢Δͱ͍͏ޮՌ͕͋ΔɻҰํͰɺਅۭʹऔΓग़͞Εͨͱ͖ʹిࢠ͕༨෼ʹ΋ͭΤωϧΪʔ͕ɺऔΓग़͞ΕͨిࢠϏʔϜͷॳ‫ظ‬ΤϛολϯεΛܾΊΔ͜ͱʹͳΓɺΤϛολϯεΛԼ͛Δͱ͍͏໘Ͱ͸ྭ‫ى‬Ϩʔβʔ೾௕Λ୹͘͢Δͱ͍͏ͷ͸ಘࡦͰ͸ͳ͍ɻޫӄ‫͔ۃ‬ΒऔΓग़͞ΕͨిࢠϏʔϜͷॳ‫ظ‬Τϛολϯε͔Βɺޫӄ‫ۃ‬ද໘Λग़ͨ͋ͱʹͲΕ͘Β͍ͷ༨৒ͳΤωϧΪʔΛ͔࣋ͭΛ‫ੵݟ‬΋Δ͜ͱ͕Ͱ͖Δɻ͜ͷ༨৒ͳΤωϧΪʔ͕ԣํ޲ (xɺy ํ޲) ͷॳ‫ظ‬Τϛολϯεͷ‫ͳͱݯى‬Δɻ༨৒ΤωϧΪʔΛ kB T 8 ͱ͢
Δͱɺޫӄ‫͔ۃ‬Βੜ੒͞Εͨిࢠͷ‫֨ن‬ԽΤϛολϯ
ε͸ɺ

ε x = σxରͯ͠ిࢠϏʔϜ͕ͲΕ͘Β͍ͷ࣌ؒԠ౴ͰऔΓग़͞ΕΔ͔ͱ͍͏͜ͱʹରԠ͢Δɻ΋͠ɺরࣹͨ͠Ϩʔβʔύϧεʹରͯ͠ɺऔΓग़͞ΕͨిࢠϏʔϜ͕௕͍ςʔϧΛ΋ͭͱ͢Δͱɺςʔϧ෦෼͸Լྲྀʹ͋Δߴप
೾Ճ଎ۭಎͷ࠷దͳҐ૬͔ΒͣΕͨঢ়ଶͰՃ଎͞Εྗʹରͯ͠ͲΕ͘Β͍ͷిࢠ͕औΓग़͞ΕΔ͔Λࣔ͢࣌ؒԠ౴Ͱ͋Δɻ͜Ε͸ɺরࣹͨ͠ϨʔβʔύϧεʹΔ͜ͱʹͳΓɺઃ‫ܭ‬৚͔݅ΒΤωϧΪʔͣΕͨΤωϧΪʔΛ΋ͭ͜ͱʹͳΔɻΤωϧΪʔͷͣΕͨిࢠ͸ภ޲ి࣓ੴͰ‫ي‬ಓ͕‫͛ۂ‬ΒΕΔ۠ؒͰɺઃ‫يܭ‬ಓ͔ΒͣΕͨྖҬΛ௨ա͢Δ͜ͱʹͳΓɺ๬·ͳ͍ϏʔϜଛࣦͷ‫ݪ‬Ҽͱ΋ͳΔɻ·ͨɺ͋ͱͷηΫγϣϯͰ঺հ͢Δ
Τϛολϯεิঈʹରͯ͠΋ѱ͍Ө‫ڹ‬Λ༩͑Δ͜ͱʹͳΔɻΧιʔυͷ࣌ؒԠ౴͸ɺྭ‫ى‬Ϩʔβʔ೾௕ͱΧιʔυࡐ࣭ͷް͞ʹґଘ͢Δɻ࣌ؒԠ౴͕஗ΕΔ‫ݪ‬Ҽ͸ɺΧιʔυද໘͔Βԕ͘Ͱ (ΑΓԞͰ) ྭ‫͞ى‬Εͨిࢠ͕ɺ֦ࢄʹΑͬͯද໘·ͰͨͲΓண͘ͷʹ༗‫ݶ‬ͷֻ͕͔࣌ؒΔͨΊͰ͋ΔɻԾʹΧιʔυࡐ࣭͕े෼ʹബ͚Ε͹ɺද໘ʹ୧Γண͘·Ͱͷ͕࣌ؒ୹͘ͳΓɺ࣌ؒԠ౴͸଎͘ͳΔɻ·ͨɺྭ‫ى‬Ϩʔβʔޫ͕Χιʔυ಺ʹͲͷఔ౓·Ͱ৵ೖ͢Δ͔ͱ͍͏͜ͱ΋ॏཁͱͳΔɻ͜ΕΛࣔ͢෺ཧྔ͕৵ೖ௕ α Ͱ͋Γɺޫ͕Χιʔυ໘ʹ৵ೖ͢Δ‫౓ڧ‬͸ exp (−αz) ʹൺྫ͢Δɻ೾௕

785 nm ͷ৔߹ʹ͸ α ∼ 1000 nm ͱͳΓɺ͔ͳΓԞ·Ͱ৵ೖ͢ΔͨΊ࣌ؒԠ౴͕஗͘ͳΔɻ೾௕ 544 nmͷ৔߹ʹ͸ α ∼ 100 nm ͱͳΓɺද໘ۙ๣ͰͷΈిࢠ͕ੜ੒͞ΕΔͨΊ଎͍࣌ؒԠ౴Λ΋ͭ͜ͱʹͳΔɻྭ‫ى‬Ϩʔβʔ೾௕͕୹͍৔߹ɺ࣌ؒԠ౴ͱྔࢠޮ཰

kB T
mc2

(2.1)

Ͱ͸ಘʹͳΔ͕ɺԣํ޲ͷΤϛολϯεΛߟ͑Δͱ༨
৒ΤωϧΪʔ͕૿͑Δ͜ͱʹͳΓಘͰ͸ͳ͍ɻ͜ͷΑͱ‫͞ࢉܭ‬ΕΔ [8,11]ɻ͜͜ͰɺkB ͸ Bolzmann ఆ਺ɺ͏ʹɺྭ‫ى‬Ϩʔβʔ೾௕ɺྔࢠޮ཰ɺΧιʔυࡐ࣭ɺm ͸ిࢠͷ࣭ྔɺc ͸ਅۭதͰͷޫ଎Ͱ͋Δɻσx ͸Χιʔυද໘Ͱͷ x ํ޲ͷ rms ϏʔϜαΠζͰ͋Δɻ͜ΕΑΓɺॳ‫ظ‬Τϛολϯε͸ kB T ͱॳ‫ظ‬ϏʔϜαΧιʔυް͞ɺ࣌ؒԠ౴ɺॳ‫ظ‬ΤϛολϯεΛߟྀ͠Πζʹґଘ͢Δ͜ͱ͕Θ͔Δɻࣜ (2.1) ͔Βɺిࢠॐӄ‫ۃ‬ද໘Ͱͷిࢠͷੜ੒͚ͩͰͳ͘ɺۭؒిՙޮՌɺ

ͷ‫ޙ‬ΖͰΤϛολϯε͕ଌఆͰ͖Ε͹ɺΧιʔυʹΧιʔυද໘Ͱͷ‫ి૾ڸ‬ՙͷޮՌͳͲͷॏཁͳ෺ཧ͕

౰ͯͨϨʔβʔεϙοταΠζ͔Βɺ༨৒ΤωϧΪʔ

͋ΔɻిࢠΛੜ੒͢Δࡍʹ͸ɺύϧεঢ়ͷϨʔβʔΛ

8 MTE, Mean Transverse Energy ͱදࣔ͢Δ৔߹΋͋ΔɻΧιʔυද໘ʹ౰ͯͯిࢠΛੜ੒͢Δ͕ɺͦͷύϧεͯɺӡస৚݅ΛܾΊ͍ͯ͘ඞཁ͕͋Δɻ
ిࢠॐͰͷॏཁͳ෺ཧ‫ݱ‬৅ͱͯ͠ɺઌʹ঺հͨ͠ޫ

㸲㸫 

x

z

4σx

1/2

(12) σzਤ 10: ϏΞ؈‫ܕ‬ͷཻࢠ෼෍ɻԁ౵಺Ͱ͸ిՙີ౓͸Ұఆɻσx ͸ x ํ޲ͷ rms ϏʔϜαΠζɺσz ͸ z ํ޲ͷ rms όϯν௕ɻਤ 11: cERL ͷޫӄ‫ ۃ‬DC ిࢠॐͷి‫ܗۃ‬ঢ়ɻΞϊʔυͱΧιʔυؒͷ‫཭ڑ‬͸ɺ160 mm ͱͳ͍ͬͯ
Δɻ2013 ೥͔ΒͷӡసͰ͸ɺ390 kV ΛҹՃͯ͠ӡస‫ܗ‬ঢ়ͱԣํ޲ͷ෼෍‫ܗ‬ঢ়͕ɺۭؒిՙޮՌͷӨ‫ڹ‬Λখ͘͢͞Δ্ͰॏཁͰ͋Δ͜ͱ͕Θ͔͍ͬͯΔɻ‫׵͍ݴ‬
͑Δͱɺۭؒɾ࣌ؒ྆ํ޲ʹ͍ͭͯͷϨʔβʔ෼෍ͷ੔‫͕ܗ‬௿ΤϛολϯεΛಘΔ্Ͱ͸ॏཁͱͳΔɻ࣮ࡍ
ʹ࣌ؒํ޲ͱԣํ޲ͷϨʔβʔ෼෍ͷ‫ܗ‬ঢ়Λ੔‫͢ܗ‬Δٕज़։ൃ͕ਐΊΒΕ͓ͯΓɺΑΓ௿ΤϛολϯεϏʔϜΛ࣮‫͢ݱ‬Δʹ͸ɺ͜ΕΒͷ੍‫ʹޚ‬Αͬͯ೗Կʹઃ‫ܭ‬৚݅ʹ͚͍͔ۙͮͯ͘ͱ͍͏͜ͱ͕‫ͳͱݤ‬Δɻ

ERL ೖࣹ‫ث‬ͷγϛϡϨʔγϣϯΛߦ͏ࡍʹ͸ɺిࢠॐͷΧιʔυද໘Ͱిࢠ͕͋Δ෼෍Λ΋ͬͯੜ੒͞
ΕΔͱԾఆͯ͠ɺཻࢠτϥοΩϯά͕։࢝͞ΕΔɻ‫ݱ‬࣮ͷిࢠॐͰ͸ͲͷΑ͏ͳ෼෍Ͱిࢠ͕ੜ੒͞ΕΔ͔
Λ஌Δ͜ͱ͸ɺ‫ػࢉܭ‬γϛϡϨʔγϣϯͷਫ਼౓Λ্͛ΔͨΊʹ΋ॏཁͰ͋Δɻಛʹɺޫӄ‫ࢠిۃ‬ॐͰ͸ɺ೤ిࢠॐͱ͸ҟͳΓɺϨʔβʔͷ೾௕͕औΓग़͢ిՙྔ
΍ϏʔϜͷॳ‫ظ‬Τϛολϯεɺ࣌ؒߏ଄ʹӨ‫ڹ‬Λ༩͑
ΔͨΊෳࡶͰ͋Δɻిࢠॐͷ‫Ͱڀݚ‬͸ɺ͜ΕΒͷଌఆ΋ਫ਼ྗతʹߦΘΕ͍ͯΔ [14, 15]ɻ·ͨɺERL ೖࣹ‫ݱ࣮Ͱث‬Մೳͳ࠷খΤϛολϯε
Λௐ΂ΔͨΊʹɺॳ‫ظ‬ͷిࢠ෼෍ͷ‫ܗ‬ঢ়ΛมԽͤ͞࠷
దͳ෼෍Λ୳͢‫ڀݚ‬΋ߦΘΕ͍ͯΔ [17]ɻଟ͘ͷ৔߹ʹ͸ɺϏΞ؈‫ܕ‬ͷ෼෍ (ਤ 10) ͕ॳ‫ظ‬෼෍ͱͯ͠༻͍
ΒΕ͍ͯΔ͕ɺԣํ޲ͷ෼෍Ͱத৺෼෍ͷີ౓ΛԼ͛ͨ৔߹ͷํ͕ɺΑΓۭؒిՙޮՌͷӨ‫ڹ‬Λখ͘͞཈͑
ΒΕΔ͜ͱ͕Θ͔͍ͬͯΔ [12]ɻ

Λߦ͍ͬͯΔɻԣ࣠͸ z ࣠ɺॎ࣠͸ r ࣠Λද͢ɻਤ͸ɺz ࣠पΓʹԁ౵ରশੑ͕͋Δ৔߹ͷஅ໘ͷ൒෼Λࣔ͢ɻΧιʔυʢࠨଆͷԁ౵ʣ͔ΒΞϊʔυʢ݀ͷ։͍ͨฏ൘ʣʹ޲͔ͬͯɺՃ଎ి৔͕‫ܗ‬੒͞ΕΔɻؒʹ͸੩ి৔͕ੜ͡Δ͜ͱʹͳΔɻిࢠॐ͔ΒԼྲྀʹ޲͚ͯϏʔϜΛ༌ૹ͢Δࡍʹɺ͜ͷ۠ؒͷి৔෼෍͕
ిࢠϏʔϜʹରͯ͠ऩଋ͋Δ͍͸ൃࢄྗΛ༩͑Δɻి৔෼෍͸ి‫ܗۃ‬ঢ়ʹΑܾͬͯ·ΔͨΊɺϏʔϜ༌ૹ΋ߟྀͨ͠ి‫ܗۃ‬ঢ়ઃ‫͕ܭ‬ඞཁͱͳΔɻͨͩ͠ɺి‫ܗۃ‬ঢ়͸ϏʔϜͷ༌ૹ৚͚݅ͩͰͳ͘ɺి‫ۃ‬ද໘ͷ࠷େి৔ʹ΋Ө‫͢ڹ‬Δɻి‫ۃ‬ද໘ͷ࠷େి৔͕͋·Γʹߴ͍ͱ์ిΛҾ͖‫ͳʹͱ͜͢͜ى‬Γɺ҆ఆͳిѹҹՃΛ્֐͢Δ͜ͱʹͳΔͨΊɺ͜ΕΒͷෳ਺ͷ৚݅Λߟྀͨ͠ઃ‫͕ܭ‬ॏཁͱͳΔɻ
ਤ 11 ʹ cERL ࣮ূ‫࢖Ͱػ‬ΘΕ͍ͯΔޫӄ‫ ۃ‬DC
ిࢠॐ [18] ͷஅ໘‫ܗ‬ঢ়Λࣔ͢ɻΧιʔυɾΞϊʔυؒͷΪϟοϓ͸ 160 mm Ͱ͋Δɻ͜͜Ͱɺిࢠॐ͸ z࣠ͷपΓʹԁ౵ରশੑΛ࣋ͭͱͯ͠ɺ൒෼ͷஅ໘Λࣔͨ͠ɻਤ 12 ʹ z ্࣠ͰͷՃ଎ి৔ Ez0 (z) Λࣔ͢ɻి৔͸ POISSON [19] ʹΑͬͯ‫͞ࢉܭ‬Εͨɻରশੑ
͔Βɺz ্࣠Ͱ͸൒‫޲ํܘ‬ͷ੒෼ Er ɺԁपํ޲ͷ੒෼ Eθ ͱ΋ʹθϩͱͳΔɻ͔͠͠ɺz ্͔࣠Β཭Εͨ৔߹ (r = 0)ɺEr ͸θϩʹ͸ͳΒͳ͍ɻΑΓৄࡉͳ‫Ͱࢉܭ‬͸ɺ2 ࣍‫ݩ‬ͷి৔ͷϚοϓ͕ඞཁʹͳΔ͕ɺz࣠ۙ๣Ͱ͸ Ez0 (z) ͔Β Er Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻz ্࣠ͷి৔ Ez0 (z) Λ༻͍Δͱɺz ࣠ۙ๣Ͱɺr = 02.3

ͷ৔߹ͷి৔ͷ੒෼͸ɺిࢠॐͷి‫ܗۃ‬ঢ়ޫӄ‫͔ۃ‬Βੜ੒͞ΕͨిࢠϏʔϜ͸ɺΧιʔυͱΞ

Ez (r, z)=

Er (r, z)=ϊʔυి‫ʹؒۃ‬ҹՃ͞ΕͨిѹʹΑͬͯՃ଎͞ΕΔɻ

DC ిѹ͕ҹՃ͞ΕΔͨΊɺΧιʔυͱΞϊʔυి‫ۃ‬㸲㸫 

1 ∂ 2 Ez0 (z),Ez0 (z) − r2
4∂z 2
1 ∂Ez0 (z)
− r
2
∂z

(2.2)
(2.3)

0
−5000
−10000−15000Ez

Ez (V/cm)−20000−25000−30000−35000−40000−45000−50000ਤ 13: ిࢠॐ (ਤ 11) ͷి‫ܗۃ‬ঢ়ʹΑΔऩଋɾൃࢄ

−55000ྗͷ‫ࢉܭ‬ɻΧιʔυද໘ʹ z ࣠ͷ‫͔఺ݪ‬Β r ํ޲ʹ−60000
0

50

100

150

200

250

300

Z (mm)

ਤ 12: ిࢠॐ (ਤ 11) ͕ z ্࣠ʹ࡞ΔՃ଎ి৔ɻz

1 mm ͣͭͷִؒͰిࢠΛ഑ஔ͠ɺτϥοΩϯά‫ࢉܭ‬Λߦͬͨɻిࢠॐશମͱͯ͠͸ൃࢄྗ͕ಇ͘ɻ
r࣠पΓͷԁ౵ରশੑ͕͋ΔͨΊɺz ্࣠Ͱ͸ Ez ੒෼

ironͷΈͱͳΓɺEr ͓Αͼ Eθ ͸θϩͱͳΔɻz = 0 (m)coil͕Χιʔυද໘ɺz = 0.16 (m) ͕Ξϊʔυද໘ɻelectron bunch
Bzͱද͢͜ͱ͕Ͱ͖Δɻ͜Ε͸ɺԣํ޲ͷి৔ Er ͸z

∂Ez0 (z)/∂z ʹґଘ͢Δ͜ͱΛ͍ࣔͯ͠Δɻਤ 12 ʹࣔͨ͠Α͏ʹɺ∂Ez0 (z)/∂z > 0 ͱͳΔͨΊɺΧιʔυۙ๣Ҏ֎͸ෛͷిՙΛ࣋ͭిࢠϏʔϜʹରͯ͠ൃࢄ

Bz(z)ྗ͕ಇ͘͜ͱʹͳΔɻ͜ΕΛࣔͨ͢Ίʹɺਤ 11 ͷి‫ܗۃ‬ঢ়͕࡞Δి৔ͷதͰͷిࢠͷτϥοΩϯά (ཻࢠ௥੻) ‫ࢉܭ‬Λߦͬͨɻ͜ͷ݁ՌΛਤ 13 ʹࣔ͢ɻτϥοzΩϯά‫Ͱࢉܭ‬͸ɺిࢠؒͷۭؒిՙޮՌ͸ແࢹ͍ͯ͠
Δɻਤ 12 ͷ 1 ࣍‫ిݩ‬৔෼෍͔Β༧૝ͨ͠Α͏ʹɺΧιʔυ෇ۙͰ͸ऩଋྗ͕ಇ͕͘ɺిࢠॐશମͱͯ͠͸ਤ 14: ιϨϊΠυి࣓ੴͷ໛ࣜਤɻϏʔϜ࣠ (z ࣠)ൃࢄྗ͕ಇ͘͜ͱ͕Θ͔Δɻ͜ͷΑ͏ʹɺి‫ܗۃ‬ঢ়͸্Ͱ͸ɺ࣓৔ Bz ͷΈ͕ੜ͡Δɻ

ԣํ޲ͷϏʔϜྗֶʹ΋Ө‫ڹ‬Λ༩͑ΔͨΊɺͦΕΒΛߟྀͨ͠ઃ‫͕ܭ‬ඞཁͱͳΔɻ·ͨϏʔϜͷ (x-y) அ໘Ͱͷ෼෍Λճసͤ͞Δͱ͍͏ERL ೖࣹ‫ث‬ͷγϛϡϨʔγϣϯͰ͸ɺۭؒిՙޮ
ՌΛ‫ؚ‬Μͩίʔυ͕࢖ΘΕΔ͕ɺऔΓೖΕΔ෺ཧ‫ݱ‬৅ͱཁ‫͢ٻ‬Δਫ਼౓ʹΑͬͯɺ֎෦ి࣓৔ͷ༩͑ํ͕ҟ͢ɻιϨϊΠυͷ໾ׂ͸ɺిࢠॐͷి৔ͱۭؒిՙޮͳͬͯ͘Δɻߴ଎ʹ‫͍ͨ͠ࢉܭ‬৔߹ʹ͸ɺԁ౵ରশੑ

͘͢Δͱ͍͏͜ͱͷଞʹɺηΫγϣϯ 6 Ͱࣔ͢Α͏ͳΛ࣋ͭϏʔϜΛԾఆ͠ɺϏʔϜਐߦํ޲ (z ࣠) ͷࣹۙӨΤϛολϯεΛิঈ͢Δͱ͍͏͜ͱͰ͋ΔɻERL๣ͰͷӡಈΛߟ͑ɺ্‫ه‬ͷΑ͏ͳ 1 ࣍‫ݩ‬ͷ֎෦ి࣓ೖࣹ‫Ͱث‬͸ɺଟ͘ͷ৔߹ɺిࢠॐԼྲྀͱόϯνϟʔۭ৔Λ࢖༻͢Δɻ͔͠͠ɺz ࣠ۙ๣Ͱͷి࣓৔ͷల։͕ಎԼྲྀʹιϨϊΠυΛઃஔ͢ΔҊ͕࠾༻͞Ε͍ͯΔɻ

ಇ͖͕͋Δɻਤ 14 ʹιϨϊΠυి࣓ੴͷ໛ࣜਤΛࣔՌʹΑΓൃࢄͨ͠ϏʔϜαΠζΛऩଋޮՌʹΑΓখ͞ਖ਼͘͠ͳ͍ (ϏʔϜαΠζ͕େ͖͍৔߹)ɺ͋Δ͍͸Α

JAEA ͷ 250 kV ޫӄ‫ ۃ‬DC ిࢠॐͰ༻͍ΒΕ͍ͯΓਖ਼֬ʹిࢠॐͷ࡞Δి৔ͷӨ‫ڹ‬ΛऔΓೖΕΔʹ͸ɺΔιϨϊΠυి࣓ੴͷஅ໘Λਤ 15 ʹࣔ͢ɻి࣓ੴ͸2 ࣍‫ݩ‬ɺ͋Δ͍͸ 3 ࣍‫ݩ‬ͷి৔෼෍Λ༩͑Δඞཁ͕

ԁ౵ରশੑΛ࣋ͭͨΊɺஅ໘ͷ൒෼Λࣔͨ͠ɻਤ 16

͋Δɻʹ z ্࣠ͷ࣓৔෼෍ Bz0 (z) Λࣔ͢ɻిࢠॐͱͷҐஔؔ܎͸ɺz = 0 ͕ిࢠॐͷΧιʔυද໘ʹରԠ͢

2.4

Δɻ͜ͷιϨϊΠυి࣓ੴ͸ɺओίΠϧͷଞʹɺΧιϨϊΠυιʔυද໘Ͱͷ࣓৔Λθϩʹ͢ΔͨΊͷิਖ਼ίΠϧΛ

ERL ೖࣹ‫Ͱث‬͸ɺిࢠॐͷ௚‫ʹޙ‬ιϨϊΠυి࣓ੴ͕ઃஔ͞ΕΔɻιϨϊΠυి࣓ੴ͕࡞Δ࣓৔ʹ͸ɺηΫγϣϯ 4.3 Ͱࣔ͢Α͏ʹϏʔϜʹऩଋྗΛ༩͑ɺ͍࣋ͬͯΔɻ΋͠ɺΧιʔυද໘Ͱ༗‫ݶ‬ͷ࣓৔͕͋Δ

Bz0 (z = 0) = 0 ͱ͢ΔͱɺΧιʔυͰද໘Ͱੜ੒͞㸲㸫 ΕͨిࢠϏʔϜ͸ɺηΫγϣϯ 4.2 ͷࣜ (4.20) Ͱࣔ͢Α͏ʹɺ༗‫ظॳͳݶ‬ͷԣํ޲ӡಈྔ pθ Λ࣋ͭ͜ͱʹͳΓɺॳ‫ظ‬Τϛολϯεͷ૿େΛট͘ɻ͜ΕΛආ͚
ΔͨΊʹɺิਖ਼ίΠϧΛ࢖༻ͯ͠ɺΧιʔυද໘Ͱͷ0࣓৔Λθϩʹௐ੔͍ͯ͠Δɻ

5

10

15

20

25

30

35

40

45

50

55

60

60

60

55

55

ిࢠॐͷి৔ͷٞ࿦ͷͱ͖ͱಉ༷ʹɺιϨϊΠυ࣓50

50

৔ͷ 1 ࣍‫ݩ‬෼෍ Bz0 (z) ͔Βɺ2 ࣍‫࣓ݩ‬৔෼෍Λ‫ࢉܭ‬

45

45

͢Δ͜ͱ͕Ͱ͖Δɻz ࣠ۙ๣Ͱ͸ɺBz0 (z) ͔Β r = 04040ͷ৔߹ͷ࣓৔੒෼͸ɺ

35

35

30

30

25

25

20

20

15

15

10

10

ਤ 15 ͷιϨϊΠυి࣓ੴΛిࢠ͕௨աͨ͠ͱ

55͖ͷ‫੻ي‬Λਤ 17 ʹࣔ͢ɻιϨϊΠυೖ‫ Ͱޱ‬x =0

Br (r, z)=

Bz (r, z)=

1 ∂Bz0 (z),2 ∂z
1 ∂ 2 Bz0 (z)Bz0 (z) − r2
4
∂z 2
−

(2.4)
(2.5)

ͱ‫͞ࢉܭ‬ΕΔɻ

0
0

5

10

15

20

25

30

35

40

45

50

55

60

−0.4, −0.2, 0, 0.2, 0.4 mm ͷҐஔͷిࢠΛ഑ஔ͠ (y= 0)ɺཻࢠτϥοΩϯάΛ࣮ࢪͨ͠ɻιϨϊΠυͷதਤ 15: ιϨϊΠυ (JAEA-250 kV ޫӄ‫ ۃ‬DC ిࢠॐ৺࣓৔ͷ‫ ͞ڧ‬0.02, 0.04 T ͷೋͭʹ͍ͭͯ‫ͨ͠ࢉܭ‬ɻ༻) ͷஅ໘‫ܗ‬ঢ়ɻԣ࣠͸ r ࣠ɺॎ࣠͸ z ࣠Λද͢ɻਤਤ 17 ʹࣔ͢Α͏ʹɺத৺࣓৔͕ 0.04 T ͷ৔߹ʹ͸͸ɺz ࣠पΓʹԁ౵ରশੑ͕͋Δ৔߹ͷஅ໘ͷ൒෼Λ

x, y ํ޲ͱ΋ʹ͋Δ఺Ͱऩଋ͞ΕΔ͜ͱ͕Θ͔Δɻ·ࣔ͢ɻz = 0 ͷҐஔ͕ిࢠॐͷΧιʔυද໘ͱͳΔɻ

ͨɺऩଋྗ͸த৺࣓৔ͷ‫ʹ͞ڧ‬ΑͬͯมԽ͢Δɻਤ 18Χιʔυද໘Ͱͷ࣓৔Λθϩʹ͢ΔͨΊʹɺओίΠϧʹ x-y ্ۭؒͷ֤ిࢠͷ‫੻ي‬Λࣔ͢ɻೖ‫ Ͱޱ‬x ্࣠Ҏ֎ʹิਖ਼ίΠϧ͕༻͍ΒΕΔɻʹฒΜͰ͍ͨిࢠ͕ɺऩଋ͞ΕΔͱͱ΋ʹճస͠ɺyํ޲੒෼Λ΋ͭɻਤ 19ɺਤ 20 ʹ x ํ޲ͱ y ํ޲ͷҐ૬্ۭؒͷ֤ిࢠͷ‫੻ي‬Λࣔ͢ɻೖ‫Ͱޱ‬͸ y = 0 ֤ͩͬͨిࢠͷ࠲ඪ͕ɺιϨϊΠυʹΑΔճసʹΑͬͯग़‫Ͱޱ‬͸༗‫ͳݶ‬஋Λ࣋ͭ͜ͱ͕Θ͔Δɻ͜ͷΑ͏ʹɺιϨϊΠυ͸ऩଋ࡞༻ͱͱ΋ʹ x-y ۭؒͰͷճసΛ༩͑Δ͜ͱʹͳΔɻ
0.03

Bz (T)

0.02

0.01

0
0

0.2

0.4

z (m)

0.6

0.8

1

ਤ 16: ιϨϊΠυ (ਤ 15) ͕ z ্࣠ʹ࡞Δ࣓৔ Bz ɻz ࣠पΓͷԁ౵ରশੑ͕͋ΔͨΊɺz ্࣠Ͱ͸ Bz ੒෼ͷΈͱͳΓɺBr ͓Αͼ Bθ ͸θϩͱͳΔɻ

㸲㸫 

−4

[1×10 ]

Bz = 0.02 T

0.2

5
0
−0.2

Horizontal velocity, βx

Offset of horizontal position, x (mm)Offset of vertical position, y (mm)

0.4

−0.4
Center of solenoid
0.4

Bz = 0.02 T
Bz = 0.04 T

0.2

x0 = 0.4 mm

Exit of solenoid
0
Entrance of solenoid

0

−5

−0.2
x0 = −0.4 mm

−0.4
0

0.2

0.4

0.6

Longitudinal position, z (m)

−0.4
0.8

−0.2
0
0.2
Horizontal position, x (mm)0.4

ਤ 19: ιϨϊΠυͷத৺࣓৔ 0.02 T ͷͱ͖ͷɺx-βx
ਤ 17: ιϨϊΠυʹΑΔऩଋ࡞༻ͷ‫ࢉܭ‬ɻιϨϊΠυೖ‫ Ͱޱ‬x = −0.4, −0.2, 0, 0.2, 0.4 mm ͷҐஔͷిۭؒ࠲ඪͷมԽɻ͜͜Ͱɺβx = vx /cɻࢠΛ഑ஔ͠ (y = 0)ɺཻࢠτϥοΩϯάΛ࣮ࢪͨ͠ɻ
ιϨϊΠυͷத৺࣓৔ͷ‫ ͞ڧ‬0.02, 0.04 T ͷೋͭʹ͍ͭͯ‫ࢉܭ‬ɻ

Bz = 0.02 T

0.002

x0 = 0.4 mm

Bz = 0.02 T

0.2

Exit of solenoid
Verticall velocity, βyVertical position, y (mm)0.4

x0 = 0.2 mm

Entrance of solenoid00

Entrance of solenoid
x0 = −0.4 mm−0.2

x0 = −0.2 mm

Exit of solenoid−0.002
−0.2

−0.4
−0.4

−0.2
0
0.2
Horizontal position, x (mm)0Vertical position, y (mm)

0.2

0.4

ਤ 20: ιϨϊΠυͷத৺࣓৔ 0.02 T ͷͱ͖ͷɺx-βyۭؒ࠲ඪͷมԽɻ͜͜Ͱɺβy = vy /cɻೖ‫Ͱޱ‬͸ y = 0

ਤ 18: ιϨϊΠυͷத৺࣓৔ 0.02 T ͷͱ͖ͷɺx-y֤ͩͬͨిࢠͷ࠲ඪ͕ɺιϨϊΠυʹΑΔճసʹΑۭͬؒ࠲ඪͷมԽɻೖ‫ Ͱޱ‬x ্࣠ʹฒΜͰ͍ͨిࢠ͕ɺ

ͯग़‫Ͱޱ‬͸༗‫ͳݶ‬஋Λ࣋ͭɻऩଋ͞ΕΔͱͱ΋ʹճస͠ɺy ํ޲੒෼Λ΋ͭɻ

㸲㸫 

2.5

όϯνϟʔۭಎ

(a) after cathode−FLSCޫӄ‫ࢠిۃ‬ॐͰ͸ɺύϧεঢ়ͷϨʔβʔΛಋೖͯ͠ɺόϯνԽ͞ΕͨిࢠϏʔϜΛੜ੒͢Δɻ͔͠͠ɺ

(b) after drift

FLSC

σz0

Χιʔυ௚‫Ͱޙ‬͸ϏʔϜͷΤωϧΪʔ͕௿͍ͨΊʹɺ

σz1 > σz0ۭؒిՙޮՌͷӨ‫͘ڧ͕ڹ‬ಇ͖ɺϏʔϜ͸ਐߦํ޲ͷ(c) buncher cavityൃࢄྗʹΑͬͯҾ͖৳͹͞ΕΔɻERL Ͱ͸ɺ୹͍όϯν௕ͷిࢠϏʔϜ͕ཁ‫͞ٻ‬ΕΔͨΊɺҾ͖৳͹͞ΕͨdecelerateిࢠόϯνΛ࠶౓ѹॖ͢Δඞཁ͕͋ΔɻͦΕΛߦ͏ͷz͕ɺిࢠॐͷ‫ʹޙ‬ઃஔ͞ΕΔόϯνϟʔۭಎͰ͋Δɻόϯνѹॖͷ໛ࣜਤΛਤ 21 ʹࣔ͢ɻόϯνϟʔͷ໾

accelerateׂ͸ɺిࢠॐͰੜ੒͞ΕɺՃ଎͞ΕͨϏʔϜͷόϯν
௕Λ୹͘͢Δ͜ͱͰ͋Δɻ͜ͷৄࡉʹ͍ͭͯ͸ɺηΫ(d) after drift

γϣϯ 4.5 Ͱ঺հ͢Δɻίʔωϧେֶͷ ERL ‫ܭ‬ըͰ࢖༻͞ΕΔόϯνϟʔσz2༻ۭಎͷ 1 ࣍‫ిݩ‬৔෼෍ Ez0 (z) Λਤ 22 ʹࣔ͢ɻ‫ܥ‬σz2 < σz1ʹԁ౵ରশੑΛԾఆཻͨ͠ࢠτϥοΩϯάίʔυͰ͸ɺి࣓৔ͱͯ͠ɺz ্࣠ͷి৔ Ez0 (z) ͷΈ͕ೖྗਤ 21: όϯνѹॖͷ໛ࣜਤɻిࢠॐΛग़ͨ͋ͱͷిࢠσʔλͱͯ͠༩͑ΒΕΔɻ͜Ε͸ɺz ͷΈґଘ͢ΔؔϏʔϜ͸ۭؒిՙޮՌʹΑͬͯɺόϯν௕͕৳ͼΔɻ਺Ͱɺ࣌ؒґଘ͸‫͍ͳ·ؚ‬ɻ͜ͷ৔߹ɺEz0 (z) ʹ࣌όϯνϟʔۭಎʹΑͬͯɺόϯνઌ୺Λ‫ݮ‬଎͠ɺ‫୺ޙ‬ؒґଘΛ෇͚Ճ͑ͨ Ez (z, t) ͔Βɺz ࣠ۙ๣ͰͷଞͷΛՃ଎͢Δ͜ͱʹΑͬͯɺόϯνѹॖΛ͢Δ͜ͱ͕Ͱి࣓৔ͷ੒෼Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻz ্࣠Ͱ൒‫ํܘ‬

͖Δɻ͜ͷํ๏Λ଎౓มௐʹΑΔόϯνѹॖͱ͍͏޲ r ͓Αͼԁपํ޲ θ ͷ੒෼͸ҰఆͰ͋ΔͱԾఆ͕͠ɺిࢠϏʔϜͷ଎౓͕ޫͷ଎͞ΑΓ΋஗͍௿Τωϧ

ͯɺEz (z, t) Λ Maxwell ํఔࣜʹ୅ೖ͢Δͱɺz ࣠ͷΪʔྖҬͰ͔͠༗ޮͰ͸ͳ͍ɻۙ๣ɺ͢ͳΘͪ r ͕খ͍͞৔߹ʹ͸ɺ൒‫޲ํܘ‬ͷి৔޲ (z ํ޲) ʹ ±6.9 mm ͷҐஔ (ઌ୺ͱ‫ࢠిʹ )୺ޙ‬Er (z) ͓Αͼԁपํ޲ͷ࣓৔ Bθ (z) ͸Er (z, t) =Bθ (z, t) =r ∂Ez0 (z, t)−,2∂zr ∂Ez0 (z)2c2∂tΛ഑ஔ͠ɺτϥοΩϯά‫ࢉܭ‬Λߦͬͨɻਤ 24 ʹτϥο

(2.6)
(2.7)

ͱ༩͑ΒΕΔɻ͜͜Ͱ͸ɺr ʹ͍ͭͯల։ͨ͠‫ʹޙ‬ɺr ʹ͍ͭͯͷ 1 ࣍ͷ߲ͷΈΛ࢒͍ͯ͠ΔɻTM Ϟʔυͷి࣓৔ͷ৔߹ɺz ࣠ۙ๣Ͱ͸ి࣓৔ͷ੒෼͸ɺΩϯά݁ՌΛࣔ͢ɻx ্࣠ʹ 0.2 mm ִؒͰ 5 ͭͷిࢠ (x0 = −0.4, −0.2, 0.0. 0.2, 0.4 mm) Λ഑ஔ͠ɺy ্࣠ͷҐஔ͸͢΂ͯθϩ (y0 = 0) ͱ͍ͯ͠Δɻ·ͨɺԣํ޲ͷॳ‫ظ‬ӡಈྔ͸θϩ (Px = Py = 0ɺ͢ͳΘͪɺβx = βy = 0) ͱͨ͠ɻిࢠ͸ z ํ޲ʹਐߦ͢
Δͱ͠ɺ250 keV ͷॳ‫ظ‬ӡಈΤωϧΪʔΛ࣋ͭͱԾఆ
ͨ͠ɻ‫Ͱࢉܭ‬͸ɺ࠷େՃ଎ి৔ 1 MV/mɺՃ଎Ґ૬Ez (r, z)=

Er (r, z)=Bφ (r, z)=

Ez0 (z) cos(ωt + φ),(2.8)
1 ∂Ez0 (z)− rcos(ωt + φ), (2.9)
2∂zrω
Ez0 (z) sin(ωt + φ)
(2.10)2c2ͱ‫͞ࢉܭ‬ΕΔɻ͜͜Ͱɺω ͸ి࣓৔ͷ֯ৼಈ਺ɺφ ͸ॳ‫ظ‬Ґ૬Ͱ͋Δɻ͜ͷΑ͏ʹɺԁ౵ରশੑ͕͋Δ‫ܥ‬Ͱ͸ి৔ͷ 1 ࣍‫ݩ‬෼෍ Ez0 (z) ͔Βɺଞͷ੒෼΋‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻͨͩ͠ɺ্‫ه‬ͷల։͸ɺr ͕খ͍࣌͞ͷΈ༗ޮͰ͋ΓɺϏʔϜαΠζ͕େ͖͍৔߹ͳͲʹ͸ɺ2 ࣍‫ݩ‬ɺ3 ࣍‫ݩ‬ͷి࣓৔ͷ෼෍͕ඞཁͱͳΔɻ

͸࠷େՃ଎Λ༩͑ΔҐ૬͔Β −90 ౓ 9 ͱͨ͠ɻ·ͨిࢠؒͷۭؒిՙޮՌ͸ແࢹ͍ͯ͠Δɻਤ 24 ʹࣔ͢Α͏ʹ‫Ͱ͔ۇ‬͸͋Δ͕όϯνϟʔۭಎ಺Ͱԣํ޲ͷྗΛड͚Δ͜ͱ͕Θ͔Δɻߴप೾ి࣓৔͸࣌ؒมಈ͢Δ
ͷͰɺόϯνͷਐߦํ޲ͷҐஔؔ܎ʹΑͬͯͰड͚Δ
ྗ͕มԽ͢Δ͕ɺ͜ͷόϯνϟʔۭಎͰ͸ɺόϯνલ‫Ͱޙ‬ͷԣํ޲ͷྗͷมԽ͸খ͍͜͞ͱ͕Θ͔ΔɻόϯνϟʔۭಎͰ͸Ճ଎ి৔͕খ͍ͨ͞Ίɺ࣌ؒมԽʹΑΔӨ‫ڹ‬͸খ͍͕͞ɺ࣍ʹࣔ͢௒఻ಋۭಎͰ͸όϯνͷ
લ‫ʹ܎ؔޙ‬Αͬͯɺऩଋྗ͕มԽ͢Δ͜ͱ͕Θ͔Δɻ

όϯνϟʔۭಎʹΑΔӨ‫ڹ‬Λࢹ֮తࣔͨ͢Ίʹɺ
ਤ 23 ʹࣔ͢Α͏ʹόϯνͷத৺ͱத৺͔Βਐߦํ㸲㸫 

9 όϯνѹॖΛߦ͏Ґ૬ɻ

Offset of horizontal position, x (mm)Ez (MV/m)

2

1

0
−0.1

0
z (m)

0.4
head
centertail
0

Center of buncher
−0.4
0

0.1

0.2
Longitudinal position, z (m)0.4ਤ 22: γϯάϧηϧɾόϯνϟʔۭಎ಺Ͱͷ z ্࣠

ਤ 24: όϯνϟʔۭಎʹΑΔిࢠͷ‫ي‬ಓมԽɻਤ 23ͷՃ଎ి৔ɻz ࣠पΓͷԁ౵ରশੑ͕͋ΔͨΊɺz ࣠ʹࣔͨ͠ॳ‫ࢠཻظ‬ΛτϥοΩϯάͨ͠ɻ্Ͱ͸ Ez ੒෼ͷΈͱͳΓɺEr ͓Αͼ Eθ ͸θϩͱ

• 250 kW ‫ڃ‬ͷେిྗΛ౤ೖՄೳͳೖྗΧοϓϥʔͳΔɻ

• 100 mA ͷ࿈ଓӡస (CW ӡస) ʹରԠͨ͠ HOM

electron bunchtail:s = −6.9 mm(Higher Order Mode, ߴ࣍ߴௐ೾) Χοϓϥʔhead:s = 6.9 mms• Τϛολϯε૿େΛ‫͍ͳ͜͞ى‬Ճ଎‫ث‬ͷઃ‫ͦͱܭ‬ͷ࣮‫ݱ‬center: s = 0ͱ͍͏Α͏ͳ͜ͱ͕͋Δ [1]ɻϦχΞίϥΠμʔ༻ͷਤ 23: όϯνͷத৺ͱɺத৺͔Βਐߦํ޲ (z ํ޲)ۭಎ։ൃͷ‫͔ݧܦ‬Β͸ɺ30 MV/m ͷՃ଎ޯ഑ͷୡ੒

ʹ ±6.9 mm ͷҐஔ (ઌ୺ͱ‫ࢠిʹ )୺ޙ‬Λ഑ஔɻ͸े෼ՄೳͰ͋Δͱใࠂ͞Ε͍ͯΔ͕ɺՃ଎ޯ഑Λ

15 MV/m ఔ౓ʹԼ͛ͯɺҐ૬΍Ճ଎ޯ഑ͳͲͷӡస

2.6ύϥϝλͷࣗ༝౓Λ֬อͨ͠ํ͕ɺ௿ΤϛολϯεΛ

ೖࣹ‫ث‬௒఻ಋՃ଎ۭಎ

࣮‫͢ݱ‬Δ্Ͱ༗རͱߟ͑ΒΕΔɻcERL ͷӡసͰ͸ɺ

ERL Ͱ͸ɺೖࣹ‫ۭ༻ث‬ಎ͓Αͼपճ෦༻ۭಎ (ओۭಎ) ʹ͸ɺ௒఻ಋՃ଎ۭಎ (Super conducting RFՃ଎ޯ഑ 3 MV/m ∼ 7.1 MV/m ఔ౓Ͱೖࣹ‫ث‬௒఻ಋۭಎ͸ӡస͞Ε͍ͯΔɻ

Cavity) ͕࢖༻͞ΕΔɻ௒఻ಋۭಎ͕ඞཁͳཧ༝͸ɺ࿈ଓ೾ (CW) ͰେిྲྀϏʔϜΛՃ଎͠ɺ·ͨߴ͍Ճ͖ɺRF ۭಎ͸ࣹӨΤϛολϯε΁ͷӨ‫ڹ‬Λ༩͑Δཁ

଎ޯ഑ΛಘΔͨΊͰ͋ΔɻRF ి࣓৔ͷप೾਺͸ɺ1.3

ૉͷҰͭͰ͋ΔɻηΫγϣϯ 4.4 Ͱ঺հ͢ΔΑ͏ʹɺ

GHz (L-band) ͕࢖༻͞ΕΔɻcERL ‫ܭ‬ըͰ͸ɺೖࣹRF ి࣓৔ʹΑͬͯόϯνԽ͞ΕͨϏʔϜͷࣹӨΤ‫ ʹ಺ث‬3 ୆ͷ 2 ηϧۭಎΛઃஔ͢Δ͜ͱͰઃ‫͕ܭ‬ਐϛολϯεͷ૿େ͕Ҿ͖‫͜͞ى‬ΕΔՄೳੑ͕͋Δɻ·ΊΒΕ͍ͯΔɻͨɺRF ి࣓৔ͷඇઢ‫ܗ‬ੑʹΑͬͯ΋Τϛολϯεͷ

ೖࣹ‫༻ث‬Ճ଎ۭಎͷ໾ׂ͸ɺిࢠॐͰੜ੒ͨ͠େి

ERL ೖࣹ‫Ͱ಺ث‬ͷϏʔϜμΠφϛΫεΛߟ͑ͨͱ૿େ͕Ҿ͖‫͜͞ى‬ΕΔɻೖࣹ‫ۭث‬ಎͷ RF ి࣓৔ʹྲྀϏʔϜ (100 mA) Λ 10 MeV ఔ౓ 10 ·ͰՃ଎͢Δ

ΑΔ͜ΕΒͷӨ‫ڹ‬Λ‫ྗۃ‬཈͑ΔͨΊʹɺۭಎͷӡసύ͜ͱͰ͋Δɻपճ෦ͷՃ଎ۭಎ͸ΤωϧΪʔճऩΛߦϥϝλͰ͋ΔՃ଎ޯ഑ͱҐ૬ͷௐ੔͕ඞཁͰ͋Δɻ͜

͏͕ɺೖࣹ‫ۭ༻ث‬ಎͰ͸ΤωϧΪʔճऩ͕ߦΘΕͳ͍Ε͕ɺ࠷େՃ଎ޯ഑ΛԼ͛ͯ΋ɺࣗ༝౓Λ֬อͨ͠ํ

ͨΊɺՃ଎ʹඞཁͳύϫʔ͸શͯ֎෦ͷ RF ‫͔ݯ‬Β͕ྑ͍ͱߟ͑Δཧ༝Ͱ͋Δɻ

‫͞څڙ‬Εͳ͚Ε͹ͳΒͳ͍ͱ͍͏େ͖ͳҧ͍͕͋Δɻ‫ʹ࣮ݱ‬͸ɺRF ి࣓৔ʹΑΔӨ‫ڹ‬͸ɺۭಎʹೖͬͯ͜ͷͨΊɺRF ύϫʔΛۭಎʹೖྗ͢Δ݁߹‫ث‬ʢΧο͘ΔϏʔϜͷॳ‫ظ‬৚݅ʢ෼෍΍λΠϛϯάʣʹΑͬͯϓϥʔʣ͸େిྗʹରԠͨ͠΋ͷ͕ඞཁͱͳΔɻೖࣹӨ‫͞ڹ‬Εɺ·ͨ 3 ୆ͷՃ଎ۭಎΛௐ੔͢Δඞཁ͕͋‫ۭ༻ث‬ಎͰͷ։ൃ՝୊ͱͯ͠͸ɺΔͨΊɺ࠷దԽ͢΂͖ύϥϝλͷ਺͕ଟ͘ͳΔɻ͜ͷ

10 cERL Ͱ͸पճ෦Λ௨ͯ͠ΤωϧΪʔճऩӡసΛ࣮ࢪ͢ΔͨΊʹೖࣹ‫ث‬ϏʔϜͷશΤωϧΪʔΛ 2.9 MeV ʹԼ͛ͯӡసΛߦ͍ͬͯΔɻͨΊɺύϥϝλௐ੔͸‫ػࢉܭ‬γϛϡϨʔγϣϯʹΑͬͯɺೖࣹ‫ث‬શମͷύϥϝλʹରͯ͠ߦΘΕΔɻ

㸲㸫 

−20

−15

−10

−5

0

5

10

15

electron bunch

20

24

24

22

22

20

20

18

18

16

16

14

14

12

12

10

10

8

8

6

6

4

4

2

2

0

0
−20

−15

−10

−5

0

5

10

15

tail:
s = −5.2 mm

center: s = 0
ਤ 27: όϯνͷத৺ͱɺத৺͔Βਐߦํ޲ (z ํ޲)ʹ ±5.2 mm ͷҐஔ (ઌ୺ͱ‫ࢠిʹ )୺ޙ‬Λ഑ஔɻ

20

ਤ 25: 2 ηϧۭಎ͔ΒͳΔೖࣹ‫ث‬௒఻ಋՃ଎ۭಎͷhead
centertailஅ໘‫ܗ‬ঢ়ɻԣ࣠͸ z ࣠ɺॎ࣠͸ x ࣠Λද͢ɻਤ͸ɺz
Offset of horizontal position, x (mm)࣠पΓʹԁ౵ରশੑ͕͋Δ৔߹ͷஅ໘ͷ൒෼Λࣔ͢ɻ
4

Ez (MV/n)

2
0
−2
−4
−0.2

0
z (m)

head:
s = 5.2 mm
s

0.5

0

Center of cavity

−0.5
0

0.2
0.4
Longitudinal position, z (m)

0.6

0.2

ਤ 28: 2-cell ೖࣹ‫ث‬௒఻ಋۭಎʹΑΔిࢠͷ‫ي‬ಓม
Խɻਤ 27 ʹࣔͨ͠ॳ‫ࢠཻظ‬ΛτϥοΩϯάͨ͠ɻਤ 26: ೖࣹ‫ث‬௒఻ಋՃ଎ۭಎ (ਤ 25) ͕ z ্࣠ʹ࡞ΔՃ଎ి৔ Ez ɻz ࣠पΓͷԁ౵ରশੑ͕͋ΔͨΊɺzۭؒ) ͷ‫੻ي‬ͷมԽΛࣔ͢ɻೖࣹ‫ث‬௒఻ಋۭಎग़‫ʹޱ‬্࣠Ͱ͸ Ez ੒෼ͷΈͱͳΓɺEr ͓Αͼ Eθ ͸θϩ͓͍ͯɺόϯνઌ୺ɺத৺ɺ‫͔͓ʹ୺ޙ‬Εͨిࢠ͕࡞ͱͳΔɻ

Δ௚ઢΛ‫ݟ‬ΔͱɺͦΕͧΕ܏͖͕ҟͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻ͜Ε͕ࣹӨΤϛολϯεΛมԽͤ͞Δ‫ݪ‬Ҽͱ

ਤ 25 ʹ cERL ༻ͷೖࣹ‫ث‬Ճ଎ۭಎͷஅ໘ਤΛࣔ

ͳΔɻ͜ͷޮՌΛ͏·͘࢖͏͜ͱͰɺۭؒిՙޮՌʹ͢ɻՃ଎ۭಎ͸ɺ2 ͭͷηϧΛ࣋ͭɻਤ 26 ʹɺz ࣠ΑͬͯͣΕͨҐ૬ۭؒ෼෍Λิঈ͢Δ͜ͱ͕Ͱ͖Δɻ্ͰͷՃ଎ి৔ Ez0 (z) Λࣔ͢ɻ͜ͷ Ez0 (z) Λ༻͍Δͱɺόϯνϟʔۭಎͷ৔߹ͱಉ༷ʹɺి࣓৔ͷ෼෍2.7

͸ࣜ (2.8)ɺ(2.9)ɺ(2.10) ʹΑͬͯද͞ΕΔɻ
όϯνϟʔۭಎͷ৔߹ͱಉ༷ʹɺೖࣹ‫ث‬௒఻ಋۭಎ
ͷӨ‫ڹ‬Λࢹ֮తʹ‫ݟ‬ΔͨΊʹɺཻࢠτϥοΩϯά‫ࢉܭ‬Λߦͬͨɻਤ 27 ʹࣔ͢Α͏ʹόϯνͷத৺ͱத৺͔
Βਐߦํ޲ (z ํ޲) ʹ ±5.3 mm ͷҐஔ (ઌ୺ͱ‫ޙ‬୺) ʹిࢠΛ഑ஔ͠ɺτϥοΩϯά‫ࢉܭ‬Λߦͬͨɻి
ࢠ͸ z ํ޲ʹਐߦ͢Δͱ͠ɺ250 keV ͷॳ‫ظ‬ӡಈΤωϧΪʔΛ࣋ͭͱԾఆͨ͠ɻ‫Ͱࢉܭ‬͸ɺ࠷େՃ଎ి৔

10 MV/mɺҐ૬͸࠷େՃ଎Ґ૬ͱͨ͠ɻਤ 28 ʹࣔ͢Α͏ʹɺిࢠ͸Ճ଎ۭಎ಺Ͱԣํ޲ͷྗΛड͚Δɻ
·ͨɺόϯνલ‫Ͱޙ‬ԣํ޲ʹड͚Δྗͷେ͖͕͞ҟ
ͳΔɻόϯνͷઌ୺͸ൃࢄྗΛड͚ɺ‫୺ޙ‬͸ऩଋྗΛ
ड͚Δ͜ͱ͕Θ͔Δɻਤ 29 ʹԣํ޲Ґ૬ۭؒ (x-βx4 ‫࣓ిۃ‬ੴʹΑΔϚονϯά෦

ERL ೖࣹ‫Ͱث‬͸ɺೖࣹ‫༻ث‬Ճ଎ۭಎͷ‫ʹޙ‬ɺϏʔϜޫֶؔ਺ௐ੔༻ʹෳ਺ͷ 4 ‫࣓ిۃ‬ੴ͕ઃஔ͞ΕΔɻ͜ͷ෦෼ΛϚονϯά෦ͱ‫Ϳݺ‬ɻcERL Ͱ͸ɺ5 ୆ͷ

4 ‫࣓ిۃ‬ੴ͕ઃஔ͞Ε͍ͯΔɻ͜ΕΒͷ 4 ‫࣓ిۃ‬ੴͷ໾ׂ͸ɺϏʔϜαΠζΛௐ੔͢ΔͨΊ͚ͩͳ͘ɺ͜
ͷ‫ʹޙ‬ଓ͘߹ྲྀ෦Ͱͷ෼ࢄؔ਺ͱΤωϧΪʔ֦͕Γʹ
ΑΔࣹӨΤϛολϯεͷ૿େΛආ͚ΔͨΊʹɺϏʔϜޫֶؔ਺Λௐ੔͢Δ͜ͱͰ͋Δɻ
ిࢠॐ͔ΒՃ଎ۭಎ·Ͱ͸ɺ‫ج‬ຊతʹԁ౵ରশੑΛ࣋ͭཁૉͰ͕͋ͬͨɺ4 ‫࣓ిۃ‬ੴ͸ԁ౵ରশੑΛ΋͍ͬͯͳ͍ɻैͬͯɺϚονϯά෦Ͱ͸ϏʔϜͷԁ౵ରশੑ่͕ΕΔ͜ͱʹͳΔɻ㸲㸫 

x (m)

0.001
Horizontal velocity, βx

0.5

head
centertail
Exit of cavity

B1
B2

0
Q1

-0.5
-1

0

0

B3

Q2

1 z (m) 2

3

ਤ 30: ίʔωϧେֶ ERL ‫ܭ‬ըͷ߹ྲྀ෦ͷϨΠΞ΢
τɻۣ‫ܗ‬ภ޲ి࣓ੴ 3 ୆ͱɺ4 ‫࣓ిۃ‬ੴ 2 ୆͔Βߏ

Entrance of cavity−0.001

−0.4

−0.2
0
0.2
Horizontal position, x (mm)

੒͞ΕΔɻ
0.4

ηx (m)

0.2

ਤ 29: 2-cell ೖࣹ‫ث‬௒఻ಋۭಎʹΑΔԣํ޲Ґ૬ۭؒ

(x-βx ۭؒ) ͷ‫੻ي‬ͷมԽɻਤ 27 ʹࣔͨ͠ॳ‫ࢠཻظ‬

0.1ΛτϥοΩϯάͨ͠ɻ
0

2.8

0

߹ྲྀ෦

߹ྲྀ෦͸ɺERL ೖࣹ‫͔ث‬Βͷ௿ΤωϧΪʔϏʔϜΛɺपճ෦͔Β͖ͨߴΤωϧΪʔϏʔϜͱ߹ྲྀͤ͞Δ
෦෼Ͱ͋Δɻ߹ྲྀ෦͸ɺҰൠʹ 3 ͭ͋Δ͍͸ 4 ͭͷภ޲ి࣓ੴ͔Βߏ੒͞ΕΔ [20]ɻਤ 30 ʹίʔωϧେֶ ERL ‫ܭ‬ըͰͷ߹ྲྀ෦ͷϨΠΞ΢τΛࣔ͢ɻ3 ͭͷภ޲ి࣓ੴͱ 2 ͭͷ 4 ‫࣓ిۃ‬ੴ͔Βߏ੒͞ΕΔɻ4 ‫ۃ‬ి࣓ੴ͸ɺ߹ྲྀ෦ग़‫Ͱޱ‬෼ࢄؔ਺͕θϩͱͳΔΑ͏ʹௐ੔͞ΕΔɻ·ͨɺ4 ‫࣓ిۃ‬ੴΛ‫ࣜܗ͍ͳ·ؚ‬ͷ߹ྲྀ෦Ͱ͸ɺภ޲ి࣓ੴͷ୺ͷ‫ܗ‬ঢ়Λௐ੔͠ɺͦ͜ͰͷऩଋྗΛར༻ͯ͠෼ࢄؔ਺Λθϩʹ͢ΔΑ͏ʹઃ‫͞ܭ‬

1

s (m)

2

3

ਤ 31: cERL ઃ‫ظॳܭ‬ஈ֊ͷ߹ྲྀ෦Ͱͷ෼ࢄؔ਺ (4‫࣓ిۃ‬ੴΛ‫͍ͳ·ؚ‬৔߹)ɻ߹ྲྀ෦ग़‫Ͱޱ‬෼ࢄؔ਺͕
θϩʹͳΔΑ͏ʹɺภ޲ి࣓ੴͷ୺෦‫ܗ‬ঢ়͕ௐ੔͞Ε͍ͯΔɻͳ͍‫ݶ‬ΓɺҰ౓ઃ‫ࣗͱ͏·ͯ͠͠ܭ‬༝ʹௐ੔Ͱ͖Δύϥϝλ͕ͳ͍ɻ͜ͷͨΊɺ߹ྲྀ෦͔ΒԼྲྀͰͷϏʔϜͷ࣭͸ɺ߹ྲྀ෦खલͷϚονϯά෦ͷύϥϝλʹΑͬͯࠨӈ͞ΕΔɻΕΔ [1]ɻਤ 31 ʹɺcERL ઃ‫ܭ‬ͷॳ‫ݕʹظ‬౼͞Ε͍ͯͨ߹ྲྀ෦Ͱͷ෼ࢄؔ਺Λࣔ͢ɻERL ͷ߹ྲྀ෦Ͱ͸ɺ

2.9

ਤ 30 ͷ߹ྲྀ෦ͱ͸ҧ͍ 4 ‫࣓ిۃ‬ੴ͸‫·ؚ‬Ε͍ͯͳ͍ɻͦͷ୅ΘΓɺภ޲ి࣓ੴͷ୺෦ͷ‫ܗ‬ঢ়Λௐ੔͠ɺग़‫Ͱޱ‬෼ࢄؔ਺Λθϩʹ͍ͯ͠Δɻ
߹ྲྀ෦Ͱ໰୊ͱͳΔ෺ཧ‫ݱ‬৅͸ɺॎํ޲ۭؒిՙ
ྗͱίώʔϨϯτ᫔ࣹ (Coherent Synchrotron Radi-

ation, CSR) ʹΑΔΤωϧΪʔ֦͕Γͷ૿େͰ͋Δɻ
෼ࢄؔ਺͕θϩͰͳ͍෦෼ͰΤωϧΪʔ֦͕Γ͕૿େ
͢Δͱɺ෼ࢄؔ਺Λ௨ࣹͯ͠ӨΤϛολϯεͷ૿େ͕Ҿ͖‫͜͞ى‬ΕΔɻ͔͠͠ɺॎํ޲ۭؒిՙྗɺ͋Δ͍͸ CSR ʹΑΔࣹӨΤϛολϯεͷ૿େ͸ɺ߹ྲྀ෦ख
લͷϚονϯά෦ͰϏʔϜޫֶؔ਺Λௐ੔͢Δ͜ͱʹΑΓɺ࠷খʹ཈͑Δ͜ͱ͕ՄೳͰ͋Δ͜ͱ͕ࣔ͞Εͯ
͍Δ [21, 22]ɻ‫ج‬ຊతʹɺ߹ྲྀ෦͸ͦͷ಺෦ʹ 4 ‫࣓ిۃ‬ੴΛઃஔ͠ΤωϧΪʔճऩΛߦ͏௒఻ಋՃ଎ۭಎ

ೖࣹ‫͔ث‬Β߹ྲྀ෦Λ‫ͯܦ‬༌ૹ͞ΕͨిࢠϏʔϜΛՃ଎͢ΔͨΊͷ௒఻ಋۭಎΛ͜͜Ͱ͸ओۭಎ͋Δ͍͸ओՃ଎ۭಎͱ‫Ϳݺ‬ɻओۭಎͷେ͖ͳ໾ׂ͸ɺೖࣹϏʔϜΛՃ଎ͤ͞Δͱͱ΋ʹɺपճͯ͠࠶ͼ໭͖ͬͯͨϏʔϜΛ‫ݮ‬଎͢Δ͜ͱͰ͋ΔɻΤωϧΪʔऩࢧΛߟ͑Δ
ͱɺೖࣹ‫ث‬௒఻ಋۭಎ͸ϏʔϜΛՃ଎͢ΔͨΊͷిྗΛ౤ೖ͠ଓ͚ͳ͚Ε͹ͳΒͳ͍͕ɺओۭಎͰ͸࠷ॳʹߴप೾ి࣓৔Λྭ‫͢ى‬ΔͨΊͷిྗ͕ඞཁͱͳΔ͕ɺҰ౓ΤωϧΪʔճऩ͕੒ཱ͢Ε͹‫ج‬ຊతʹϏʔϜΛՃ଎͢ΔͨΊͷిྗΛඞཁͱ͠ͳ͍ɻ͜Ε͕ɺೖࣹ‫ث‬௒఻ಋۭಎͱͷେ͖ͳҧ͍ͱͳΔɻ·ͨɺ‫ݱ‬ঢ়ͷ cERLͰ͸ओۭಎ͸ 2 ୆͔͠ઃஔ͞Ε͍ͯͳ͍͕ɺGeV Ϋϥεͷ ERL Ͱ͸ 100 ୆‫ن‬໛ͷओۭಎ͕ઃஔ͞ΕΔ͜
ͱʹͳΔɻϏʔϜ͕ओۭಎͷத৺͔ΒͣΕͨ‫ي‬ಓΛ௨

㸲㸫 

2.10

पճ෦

पճ෦͸ओۭಎͰՃ଎͞ΕͨϏʔϜΛ༌ૹ͢Δͱͱ΋ʹɺΤωϧΪʔճऩ͢ΔͨΊʹॏཁͳप௕ิਖ਼΍ϏʔϜར༻ʹ޲͚ͨόϯνѹॖ౳ͷϏʔϜ৚݅ͷ੍‫ޚ‬Λߦ͏͜ͱ͕ॏཁͱͳΔɻ͜͜Ͱ͸ਤ 34 ʹࣔͨ͠ cERL ͷུ֓ਤΛ༻͍ͯपճ෦ͷ‫ج‬ຊཁૉΛ঺հ͢ΔɻओۭಎͷԼྲྀʹ͸μϯϓγέΠϯ͕ઃஔ͞ΕΔ͕ɺओۭಎͰՃ଎͞ΕͨϏʔϜ͸͜ͷγέΠϯΛ௨աͯ͠ୈҰΞʔΫ෦ʹ༌ૹ͞ΕΔɻୈҰΞʔΫ෦Ͱ͸όϯνѹॖΛߦ͏ͨΊͷ༌ૹ৚݅ͷௐ੔΍ɺ͋Δ͍͸प௕ิਖ਼͕ߦΘΕΔɻGeV Ϋਤ 32: 9 ηϧۭಎ͔ΒͳΔೖࣹ‫ث‬௒఻ಋՃ଎ۭಎͷஅ໘‫ܗ‬ঢ়ɻԣ࣠͸ z ࣠ɺॎ࣠͸ x ࣠Λද͢ɻਤ͸ɺz࣠पΓʹԁ౵ରশੑ͕͋Δ৔߹ͷஅ໘ͷ൒෼Λࣔ͢ɻ

Ez (MV/m)ઃஔͯ͠์ࣹޫར༻͕ߦΘΕΔ͜ͱʹͳΔ͕ɺcERLͰ͸ୈҰΞʔΫ෦Լྲྀͷ௕௚ઢ෦ʹϏʔϜར༻ͷͨΊͷ͕۠ؒઃ͚ΒΕ͍ͯΔɻcERL ͷ௕௚ઢ෦ʹ͸ɺ

4 ͭͷภ޲ి࣓ੴͰߏ੒͞Εͨप௕ิਖ਼γέΠϯ͕ઃஔ͞ΕΔɻ͜ͷ۠ؒͰͷόϯϓͷߴ͞Λௐ੔͢Δ͜ͱ

Electromagnetic field data from file ERL9CELL.AF
Problem title line 1: Sample problem for tuning elliptical cavity
ERL9cell_1D.TBL

ϥεͷ ERL Ͱ͸ΞʔΫ෦ʹ์ࣹޫ༻ϏʔϜϥΠϯΛ

7−09−2008 19:11:50

30

ʹΑͬͯɺϏʔϜ͕ओۭಎʹ໭ΔλΠϛϯάΛௐ੔͢

20

Δ͜ͱ͕Ͱ͖Δ 11 ɻcERL ͷप௕ิਖ਼γέΠϯԼྲྀ10ʹ͸ɺϨʔβʔͱిࢠϏʔϜΛিಥͤͯ͞ X ઢΛൃ

0ੜͤ͞ΔͨΊͷ Laser Compton Scattering (LCS) ۠−10͕ؒઃ͚ΒΕ͍ͯΔ [23]ɻ͜ͷ۠ؒʹ͸ϨʔβʔΛ஝−20ੵ͢ΔͨΊͷϨʔβʔ‫͕ثৼڞ‬ઃஔ͞Ε͍ͯΔɻϨʔ

−30βʔͱিಥͤ͞Δࡍ͸ɺిࢠϏʔϜ͸਺ 10 μm ·Ͱ

Ez

−40

−20

0

20

40

60

80

100

120

ߜΒΕΔɻ௕௚ઢ෦ͷԼྲྀʹ͸ɺୈೋΞʔΫ෦͕ઃஔ

140

Z (cm)

͞ΕΔɻୈೋΞʔΫ෦Ͱ͸ɺѹॖ͞Εͨόϯν௕Λ෮ਤ 33: ओۭಎ (ਤ 25) ͕ z ্࣠ʹ࡞ΔՃ଎ి৔ Ez ɻ‫͢ݩ‬ΔͨΊʹɺόϯνѹॖͱ͸‫ٯ‬ͷաఔͷόϯν৳௕z ࣠पΓͷԁ౵ରশੑ͕͋ΔͨΊɺz ্࣠Ͱ͸ Ez ੒͕ߦΘΕΔɻୈೋΞʔΫ෦ΛऔͬͨϏʔϜ͸ɺ߹ྲྀ෦෼ͷΈͱͳΓɺEr ͓Αͼ Eθ ͸θϩͱͳΔɻ

γέΠϯΛ௨աͯ͠ओۭಎʹ࠶ͼ໭Δ͜ͱʹͳΔɻ
ΤωϧΪʔճऩΛߦ͏ͨΊʹ͸ɺՃ଎Ґ૬͔Β 180

Δͱ͖ɺۭಎ಺ʹ͸ϏʔϜΛԣํ޲ʹΩοΫ͢Δߴ࣍

౓ͣΒͨ͠Ґ૬ʹ‫ݮ‬଎ϏʔϜΛ߹ΘͤΔඞཁ͕͋Δɻߴௐ೾ (HOM) ͕ൃੜ͢Δ͜ͱʹͳΔɻ͜ͷΑ͏ͳߴ͜ͷͨΊʹɺઌ΄Ͳ঺հͨ͠ΞʔΫ෦΍प௕ิਖ਼Λརௐ೾͸ϏʔϜͷΤϛολϯεΛѱԽͤͨ͞ΓɺϏʔϜ༻͢Δ͜ͱʹͳΔɻ·ͨɺ߹ྲྀ෦γέΠϯ͔Βμϯϓෆ҆ఆੑͷ‫ݪ‬ҼͱͳΓɺಛʹओՃ଎ۭಎͷ୆਺ͷଟ͍γέΠϯͷ۠ؒ͸ೖࣹϏʔϜͱ‫ݮ‬଎ϏʔϜͷ 2 ͭͷΤେ‫ن‬໛ͳ ERL Ͱ͸ਂࠁͳ໰୊ͱͳΔɻओۭಎͷઃ‫ܭ‬ωϧΪʔͷҟͳΔϏʔϜ͕௨ա͢Δ͜ͱʹͳΔɻಉ͡Ͱ͸ɺ࠷େฏ‫ۉ‬ϏʔϜిྲྀɺओՃ଎ۭಎͷ୆਺Λߟྀ۠ؒΛΤωϧΪʔͷҟͳΔϏʔϜ͕ಉ࣌ʹ௨ա͢Δ͜

͠ɺHOM ͷӨ‫ڹ‬Λ཈͑ΔΑ͏ͳઃ‫͕ܭ‬ॏཁͱͳΔɻ

ͷ͕۠ؒ ERL ͷಛ௃Λද͍ͯ͠Δͱ͍͑Δɻ͜ͷ۠ਤ 32 ʹ cERL ༻ͷओۭಎͷஅ໘ਤΛࣔ͢ɻओۭಎ

ؒͷ‫ي‬ಓௐ੔͸ɺ‫ج‬ຊతʹΤωϧΪʔͷ௿͍ଆͷϏʔ͸ɺ9 ͭͷηϧΛ࣋ͭɻਤ 33 ʹɺz ্࣠ͰͷՃ଎ి৔Ϝʹ߹Θͤͯߦ͏͜ͱʹͳΔɻ͜Ε͸ɺΤωϧΪʔ͕Ez0 (z) Λࣔ͢ɻ͜ͷ Ez0 (z) Λ༻͍Δͱɺόϯνϟʔۭಎͷ৔߹ͱಉ༷ʹɺి࣓৔ͷ෼෍͸ࣜ (2.8)ɺ(2.9)ɺ

ߴ͍ϏʔϜ͸ಉ࣓͡৔Λ௨աͯ͠΋ͦͷӨ‫͕ڹ‬ऑ͘ͳΔ͔ΒͰ͋Δɻ

(2.10) ʹΑͬͯද͞ΕΔɻ

ओۭಎͰ‫ݮ‬଎͞ΕͨϏʔϜ͸ɺμϯϓγέΠϯͷ࠷ॳͷภ޲ి࣓ੴʹΑͬͯ‫͛ۂ͘ڧ‬ΒΕɺμϯϓϥΠϯ
11 ಉ༷ͷप௕ิਖ਼ΛΞʔΫ෦ͷ‫ي‬ಓΛมԽͤ͞Δ͜ͱͰ΋ߦ͏͜ͱ͕Ͱ͖Δɻ㸲㸫 ਤ 34: cERL ͷ഑ஔਤɻʹಋ͔ΕΔ͜ͱʹͳΔɻ

2.11

ϏʔϜμϯϓ

μϯϓϥΠϯʹ༌ૹ͞Εͨ‫ݮ‬଎ϏʔϜ͸ɺ࠷ऴతʹϏʔϜμϯϓʹࣺͯΒΕΔ͜ͱʹͳΔɻ͜͜Ͱॏཁͱ
ͳΔͷ͕ɺ͋Δ 1 Օॴͷूதͨ͠఺ʹϏʔϜΛࣺͯଓ͚ͳ͍ͱ͍͏͜ͱͰ͋ΔɻϏʔϜμϯϓͰ͸େฏ‫ۉ‬ిྲྀͷϏʔϜ͕ࣺͯΒΕଓ͚Δ͜ͱʹͳΔͨΊɺͦͷ೤ෛՙΛ෼ࢄ͢Δඞཁ͕͋ΔͨΊͰ͋ΔɻେిྲྀϏʔϜΛࣺͯΔࡍʹ͸ɺ༌ૹ࿏Λௐ੔ͯ͠ϏʔϜαΠζΛ
͋Δఔ౓޿͛Δͱͱ΋ʹɺϏʔϜμϯϓ্ྲྀʹઃஔ͞Εͨ‫ي‬ಓิਖ਼ి࣓ੴΛ࣌ؒมԽͤͯ͞ɺϏʔϜ͕౰ͨΔҐஔΛৗʹมԽͤ͞Δ͜ͱ͕ඞཁͰ͋Δɻ㸲㸫 

2. ֎෦͔ΒՃ͑ͨಈి৔ɾಈ࣓৔͔Βͷి࣓ྗిࢠϏʔϜதͷ෺ཧ3

3. ཻࢠؒͷ Coulomb ૬‫༻࡞ޓ‬ʢۭؒిՙޮՌʣ

ՙిཻࢠ͔ΒͳΔϏʔϜ͸ɺి࣓৔ʹΑΔ૬‫࡞ޓ‬༻Λ௨ͯ͠ɺ͞·͟·ͳӨ‫ڹ‬Λड͚Δɻ͜͜Ͱ͸ɺ͜

4. ϏʔϜࣗ਎͕ੜ੒͢Δి࣓৔ʢߤ੻৔ʣ

ΕΒͷ૬‫༻࡞ޓ‬ΛऔΓ্͛ɺࠓճςʔϚͱ͍ͯ͠Δ

5. ภ޲৔ʹΑΔ์ࣹޫͷൃੜʢ์ࣹ‫ݮ‬ਰɺCSRʣERL ͷిࢠϏʔϜΛࢧ഑͢Δ෺ཧΛ঺հ͢Δɻͨͩ͠ɺϏʔϜͷूஂޮՌʹ‫ى‬Ҽ͢ΔϏʔϜෆ҆ఆੑʹͭ

6. ์ࣹޫͷྔࢠԽʢ์ࣹྭ‫ى‬ʣ͍ͯ͸ɺ͜͜Ͱ͸৮Εͳ͍ɻͦΕΒʹ͍ͭͯ͸ɺଞͷ

7. ଞͷϏʔϜ͕࡞Δి࣓৔ʢϏʔϜɾϏʔϜ૬‫ޓ‬࡞༻ʣจ‫[ ݙ‬1, 24] Λࢀরͯ͠ཉ͍͠ɻ

ి࣓৔தͷՙిཻࢠͷӡಈํఔࣜ

8. ࢒ཹΨεͱͷࢄཚɺΠΦϯั֫ϏʔϜͱ͸ɺ୯Ұͷछྨͷཻࢠʢిࢠ΍ཅࢠͳͲʣɺ

9. ޫిࢠʹΑΔిࢠӢͱͷ૬‫༻࡞ޓ‬

3.1

͋Δ͍͸ෳ਺ͷछྨͷཻࢠ͕ɺ΄΅ಉ͡଎౓Λ࣋ͬͯಉҰํ޲ʹਐߦ͢ΔूஂͰ͋Δͱߟ͑Δ͜ͱ͕Ͱ͖
ΔɻՃ଎‫ث‬தͰ͸ɺϏʔϜ͸֎෦͔ΒՃ͑ΒΕͨి࣓৔͔ΒྗΛड͚Δ͜ͱʹͳΔɻి࣓৔தͰͷՙిཻࢠͷӡಈͷ‫ج‬ຊํఔࣜ͸ɺMaxwell ํఔࣜɺ

∇·B

=

∇×H

=

∇·D

=

∇×E

=

0,
∂D
+ J,
∂t
ρ,
∂B
,
−
∂t

(3.1)
(3.2)

ͱͷ૬‫༻࡞ޓ‬Λআ͍ͨ΋ͷͰ͋ΔɻERL ೖࣹ‫ث‬Λઃ‫͢ܭ‬Δࡍʹɺ·ͣॏཁͱͳΔ૬‫༻࡞ޓ‬͸ɺ͜ΕΒͷதۭؒిՙޮՌͷ 3 ͭͰ͋Δɻೖࣹ‫Ͱث‬͸ϏʔϜͷΤωϧΪʔ͕௿͍ͨΊʹɺ·ͣ͜ΕΒͷޮՌ͕ࢧ഑తͱ
ͳΔɻΑΓ‫࣮ݱ‬తͳঢ়‫گ‬Λߟ͑Δ৔߹ʹ͸ɺߤ੻৔΍(3.3)

ΠΦϯั֫ͷӨ‫ڹ‬Λಋೖ͢Δඞཁ͕͋Δ͕ɺERL ೖ(3.4)ࣹ‫ث‬ͷઃ‫ͯͬͨ͋ʹܭ‬͸ɺ‫ج‬ຊతʹ্‫ه‬ͷ 3 ͭͷޮՌΛߟ͑Ε͹ྑ͍ɻ·ͨɺ߹ྲྀ෦ͰͷӨ‫ڹ‬Λߟྀ͢Δ
ͱ͖ʹ͸ɺ্‫ه‬ͷ 3 ͭͷޮՌʹՃ͑ͯ CSR ʹΑΔӨ

D

=

εE,

(3.5)

B

=

μH,

(3.6)

‫ڹ‬Λߟྀ͢Δඞཁ͕͋Δɻ
‫͍༻ͯ͠ͱݯޫࣹ์ࡏݱ‬ΒΕ͍ͯΔɺGeV ఔ౓ͷΤωϧΪʔͷ์ࣹޫετϨʔδϦϯάͰ͸ɺERL ೖࣹ‫ͱث‬͸ҟͳΓɺ௨ৗۭؒిՙޮՌ͸ແࢹ͞Εɺଟ͘ిՙͷ࿈ଓͷํఔࣜɺͷ৔߹ϏʔϜ͸୯ཻࢠతʹऔΓѻΘΕΔɻͰ͸ɺͲ͏

(3.7)ͯ͠ՙిཻࢠͷू·ΓͰ͋ΔϏʔϜͷதͰɺཻࢠؒͷ

Coulomb ૬‫༻࡞ޓ‬Λແࢹ͢Δ͜ͱ͕ՄೳʹͳΔ͔ʁ͓Αͼӡಈํఔࣜɺ

d(γmβc) = e(E + v × B),dt͔ΒɺϏʔϜɾϏʔϜ૬‫ʹࢠిޫͱ༻࡞ޓ‬ΑΔిࢠӢͰ΋ɺ੩ి࣓৔ʹΑΔྗɺಈి࣓৔ʹΑΔྗɺͦͯ͠

D ͱE ɺB ͱH ͷؔ܎ࣜɺ

∂ρ
∇·J +
= 0,
∂t

ERL ೖࣹ‫͖΂ྀ͢ߟͰث‬૬‫༻࡞ޓ‬͸ɺ্‫ه‬ͷ෺ཧ·ͣ͸ɺԁ౵ରশੑΛ΋ͭ࿈ଓϏʔϜΛྫʹͯ͜͠ͷ

(3.8)

Ͱ͋ΔɻՃ଎‫ث‬தͰͷՙిཻࢠͷӡಈΛߟ͑Δͱ͖ʹ͸ɺ͜ΕΒͷӡಈํఔࣜʹ֎෦͔Β༩͑ΒΕͨి࣓͜ͱΛߟ͑ͯΈΔɻ

3.3

৔ɺ͋Δ͍͸ՙిཻࢠؒͷ૬‫Ͳͳ༻࡞ޓ‬Ճ͑ͯɺ‫ه‬ड़͢Δ͜ͱ͕Ͱ͖Δɻ

ۭؒిՙޮՌΛແࢹͰ͖ΔϏʔϜΤωϧΪʔཻࢠؒͷ Coulomb ૬‫͕༻࡞ޓ‬ϏʔϜதͰ͸ͲͷΑ͏ʹͳΔ͔Λɺ࣍ͷΑ͏ͳ؆୯ͳϞσϧʹରͯ͠ߟ͑

3.2

Δɻz ํ޲ʹҰ༷͔ͭ࿈ଓͳϏʔϜ͕ɺz ্࣠Λ଎౓ϏʔϜΛࢧ഑͢Δ෺ཧϏʔϜΛࢧ഑͢Δ෺ཧͱͯ͠ɺҎԼͷΑ͏ͳ͜ͱ͕
ߟ͑ΕΕΔ [25]ɻ

1. ֎෦͔ΒՃ͑ͨ੩ి৔ɾ੩࣓৔͔Βͷి࣓ྗvz ͰҠಈ͍ͯ͠Δͱ͢Δ (ਤ 35)ɻ͜͜ͰɺϏʔϜͷஅ໘͸൒‫ ܘ‬r ͷԁͰ͋ΔͱԾఆ͢Δɻ·ͨɺిՙີ౓
͸ϏʔϜ಺ͰҰ༷Ͱ͋Δͱ͢Δ (ρ0 = const.)ɻ͜ͷͱ͖ɺϏʔϜ͕࡞Δి৔ɺ࣓৔΋ԁ౵ରশੑΛ΋ͭɻ㸲㸫 

Er(r)Bθ(r)ͱද͞ΕΔɻ·ͨϏʔϜશΤωϧΪʔ͸ɺ

E = γmc2 ,

r0

v

z

(3.15)ͱද͞ΕΔɻྫ͑͹ T = 2.5 GeV ͷ৔߹ɺ1/γ 2 =4.2 × 10−8 ͱे෼খ͍͞஋ͱͳΓɺϏʔϜ಺Ͱͷ
Coulomb ૬‫༻࡞ޓ‬΋খ͘͞ͳΔ͜ͱ͕Θ͔ΔɻҰํɺ
ERL ೖࣹ‫Ͱث‬͸ɺE0 ∼ 10 MeV ఔ౓Ͱ͋ΓɺϏʔϜਤ 35: z ্࣠Λ଎౓ vz ͰҠಈ͢Δ z ํ޲ʹҰ༷͔ͭ࿈ଓͳϏʔϜɻϏʔϜͷஅ໘͸൒‫ ܘ‬r ͷԁͰ͋Δ
͢Δɻ

಺Ͱͷ Coulomb ૬‫༻࡞ޓ‬Λແࢹ͢Δ͜ͱ͕Ͱ͖ͳ͍ɻ
Ҏ্ͷΑ͏ʹ؆୯ͳϞσϧ͔ΒɺۭؒిՙޮՌͷΤωϧΪʔґଘੑΛௐ΂ɺߴΤωϧΪʔϏʔϜͷ৔߹ʹ͸ɺͦͷӨ‫े͕ڹ‬෼খ͘͞ͳΔ͜ͱΛࣔͨ͠ɻ͜ͷ͜͞Βʹɺz ࣠ํ޲ͷ࿈ଓϏʔϜͱͨͨ͠Ίɺి৔ɺ࣓

ͱ͔ΒɺGeV Ϋϥεͷ์ࣹޫஷଂϦϯά౳Ͱ͸ɺۭ৔ͷ z ํ޲੒෼͸ zero ͱͳΓɺEr ɺBφ ͷΈ͕࢒Δɻ

ؒޮՌిՙͷӨ‫ڹ‬Λແࢹ͢Δ͜ͱ͕Ͱ͖Δɻಉ༷ͷ݁

Gauss ͷ๏ଇΑΓɺ൒‫޲ํܘ‬ͷి৔͸࣍ͷΑ͏ʹ༩͑ΒΕΔɻ
1
Er (r) =
ρ0 r.
(3.9)
2ε0

Ռ͸ɺϏʔϜͷ੩ࢭ‫ Ͱܥ‬Coulomb ྗʹΑΔి৔Λ‫ܭ‬ࢉ͠ɺϏʔϜ͕ಈ͍ͯ‫͑ݟ‬Δ࣮‫ ʹܥࣨݧ‬Lorentz ม‫ͯ͠׵‬΋ಘΒΕΔɻ
ۭؒిՙޮՌͷӨ‫Ͳ͕ڹ‬ͷఔ౓Ͱ͋Δ͔Λ‫ݟ‬Δ্Ͱ

࣍ʹɺAmpere ͷ๏ଇΛ༻͍ͯɺԁपํ޲ͷ࣓৔੒෼͸ɺηΫγϣϯ 5.1 Ͱ঺հ͢ΔσόΠ௕Λ༻͍ͯɺΑΛ‫ٻ‬ΊΔɻిྲྀີ౓͸Ұ༷ͱԾఆ͍ͯ͠ΔͷͰɺిྲྀΓৄࡉʹٞ࿦͢Δɻີ౓͸࣍ͷΑ͏ʹ༩͑ΒΕΔɻ

j = ρ0 v.

(3.10)

3.4

͜ΕΛ༻͍Δͱ࣓৔͸ɺόϯνԽ͞ΕͨϏʔϜϏʔϜ (beam) ʹ͸ɺޫઢͱ͍ͬͨҙຯ΍ɺ‫ݐ‬ங෺1
Bφ = μ0 ρ0 vr,
2(3.11)

Ͱͷྊͱ͍͏ҙຯ͕͋Γɺ࿈ଓͯ͠࿈ͳͬͨ΋ͷͱ͍͏ΠϝʔδͱͳΔΑ͏Ͱ͋ΔɻՃ଎‫͏͍Ͱث‬ϏʔϜͱͱͳΔɻ͜ͷΑ͏ͳ Er , Bφ ಺Ͱͷՙిཻࢠ͕ड͚Δ͸ɺ୯Ұͷछྨͷཻࢠʢిࢠ΍ཅࢠͳͲʣɺ͋Δ͍͸ྗ͸ɺLorentz ྗΑΓ࣍ͷΑ͏ʹ༩͑ΒΕΔɻ

ෳ਺ͷछྨͷཻࢠ͕ɺ΄΅ಉ͡଎౓Λ࣋ͬͯಉҰํ޲

Fr

=
=
=
=

e(Er − vBφ )


1 1
1
2
e
ρ0 r −
ρ0 v r2ε0
2 ε0 μ0


v2
ρ0 r
1− 2
e
2ε0
c
1 e ρ0r.
2 ε0 γ 2ʹਐߦ͢ΔूஂͰ͋Δͱߟ͑Δ͜ͱ͕Ͱ͖ΔɻՃ଎‫ث‬ͰϏʔϜͱ͍͏ͱ͖ʹ͸ɺਐߦํ޲ʹूஂͰӡಈ͢Δͱ͍͏͜ͱΛࢦ͍ͯ͠Δͱ͍͑Δɻ͜ͷͨΊɺཻࢠͷ෼෍͕ਐߦํ޲ʹҰ༷ʹ࿈ଓͰ͋Δ͔ɺ͋Δ͍͸մʹͳ͍ͬͯΔ͔ʹ͸ؔ܎ͳ͘ɺϏʔϜͱ͍͏‫ݴ‬༿Λ࢖ͬ

(3.12)

͍ͯΔɻERL ͰͷిࢠϏʔϜ͸ɺਐߦํ޲ʹରͯ͠΋͋Δմͱͳ͓ͬͯΓɺ͜ΕΛόϯνͱ‫Ϳݺ‬ɻຊςΩετͰ͸ɺόϯνɺ͋Δ͍͸όϯνԽ͞ΕͨϏʔϜͱ͜͜ͰɺLorentz Ҽࢠ͍͏ද‫ݱ‬Λɺཻࢠ෼෍͕͋Δେ͖͞Λ͍࣋ͬͯΔ͜ͱ

1

γ=
,
1 − β2(3.13)

Λ‫ڧ‬ௐ͢Δͱ͖ʹ࢖͏ɻηΫγϣϯ 5.3 Ͱɺۭؒిՙ
ޮՌΛ‫ؚ‬Μͩ૬ର࿦తͳ৔߹ͷϏʔϜΤϯϕϩʔϓํ

β = v/cɺc2 = 1/(ε0 μ0 ) Λ༻͍ͨɻࣜ (3.12) ΑΓɺ2ఔࣜΛ‫ٻ‬ΊΔࡍʹ͸ɺ؆୯ͷͨΊʹɺόϯνԽ͞ΕͯϏʔϜ಺ͷՙిཻࢠ͕ड͚Δྗ͸ɺ1/γ ʹґଘ͢Δ͍ͳ͍ਐߦํ޲ʹҰ༷ͳϏʔϜΛߟ͑Δɻ͜Εʹର͜͠ͱ͕Θ͔ΔɻߴΤωϧΪʔϏʔϜɺ͢ͳΘͪɺγ ͕ͯɺࣹӨΤϛολϯεͷ૿‫ݮ‬Λߟ͑Δࡍʹ͸ɺϏʔϜे෼େ͖͍৔߹ʹ͸ɺϏʔϜ಺Ͱͷ Coulomb ૬‫࡞ޓ‬ΤϯϕϩʔϓํఔࣜΛ֦ுͯ͠ɺόϯνԽ͞ΕͨӨ‫ڹ‬༻Λແࢹ͢Δ͜ͱ͕Ͱ͖Δɻγ ͸ϏʔϜͷӡಈΤωϧΛిྲྀ෼෍ͷਐߦํ޲ʹର͢Δґଘੑͱͯ͠औΓೖΕΪʔΛ T ͱ͢Δͱɺ

Δɻਐߦํ޲ʹҰ༷ͳϏʔϜͱόϯνԽ͞ΕͨϏʔϜ

T = (γ − 1)mc2 ,(3.14)

Λਤ 36 ʹࣔ͢ɻ㸲㸫 όϯν௕ σz ͸ɺelectron beamz

electron bunch=(x − x)2 ,(3.16)σy2=

(y − y) ,(3.17)σz2=

(z − z)2 ,(3.18)σx22

ͱද͞ΕΔɻԣํ޲ͷϏʔϜαΠζ͸ɺ‫࣮ݱ‬ͷՃ଎‫ث‬zͰ͸ɺεΫϦʔϯϞχλʔ΍ϫΠϠʔεΩϟφʔͳͲʹΑͬͯଌఆ͞ΕΔɻόϯν௕͸ɺภ޲ RF ۭಎΛ༻͍ͯɺόϯν௕ʹԣํ޲ͷ଎౓มௐΛ༩͑ɺࣗ༝ۭؒ

ਤ 36: Ұ༷࿈ଓϏʔϜ (্ਤ) ͱɺόϯνԽ͞ΕͨΛ૸Βͤͨ‫ʹޙ‬ɺεΫϦʔϯϞχλʔͰԣํ޲ͷҐஔϏʔϜ (Լਤ)ɻ

Λଌఆ͢Δ͜ͱʹΑΓɺଌఆ͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷͱ͖ɺ࣌ؒํ޲ͷ෼෍͕ RF ి࣓৔ʹΑΔΩοΫʹ

ERL Ͱ͸୹͍ύϧε෯Λ࣋ͭޫΛੜ੒͢ΔͨΊʹɺΑͬͯɺԣํ޲ͷ෼෍ʹม‫͞׵‬ΕΔɻ

όϯν௕Λ୹͘͢Δ͜ͱ͕ཁ‫͞ٻ‬ΕΔɻ·ͨɺՃ଎ۭ
ಎͰϏʔϜΛՃ଎͢Δࡍʹ͸ɺ͋Δఔ౓୹͍όϯν

3.5.2

௕͕ඞཁͱ͞ΕΔɻ͜Ε͸ɺRF ి࣓৔ʹΑΔՃ଎Ͱ

Τϛολϯε͸ɺి࣓৔͕࣌ؒͱͱ΋ʹมԽ͢ΔͨΊɺόϯν௕͕ϏʔϜͷ඼࣭Λද͢ॏཁͳύϥϝλͱͯ͠ɺΤϛο༗‫͋Ͱݶ‬Δ৔߹ʹ͸ɺόϯν಺ͰͷҐஔʹΑͬͯΤωλϯε͕͋ΔɻΤϛολϯε͸ɺҐ૬ۭؒதͰͷϏʔϧΪʔ͕ࠩੜ͡ΔͨΊͰ͋Δɻೖࣹ‫Ͱث‬ཁ‫͞ٻ‬ΕΔΤ

Ϝͷ෼෍͕઎ΊΔ໘ੵʹΑΓఆٛ͞ΕΔɻΤϛολϯ

ωϧΪʔ֦͕Γ͸ɺ1 % ఔ౓Ͱ͋ΔͷͰɺ͜ΕΛ࣮‫ݱ‬ε͕খ͍͞ϏʔϜͰ͸ɺϏʔϜαΠζ͕খ͘͞ɺ·ͨ͢ΔͨΊʹ͸ɺՃ଎ۭಎʹೖΔલʹ͋Δఔ౓όϯν௕ͦͷ֦͕Γ΋খ͍͜͞ͱͳΔɻ͜͜Ͱ͸ɺҐ૬ۭؒΛ୹͓ͯ͘͘͠ඞཁ͕͋Δɻ͜ͷͨΊʹɺERL ೖࣹͰͷପԁ෼෍ͷ໘ੵ͔ΒΤϛολϯεΛߟ͑ɺͦͷ‫ޙ‬‫Ͱث‬͸ɺిࢠॐͱ௒఻ಋՃ଎ۭಎͷؒʹɺόϯνϟʔʹɺ‫࣮ݱ‬ͷՃ଎‫ʹࡍ࣮Ͱث‬ଌఆ͞ΕΔࣹӨΤϛολϯ

ۭಎΛઃஔͯ͠ɺόϯν௕Λѹॖ͍ͯ͠Δɻεʹ͍ͭͯߟ͑Δɻ·ͣ͸ɺҐ૬ۭؒ (x, x ) ্Ͱͷ෼෍͕

3.5

ax2 + 2bxx + cx2 = 1,ϏʔϜͷੑ࣭Λද͢ύϥϝλ(3.19)

ϏʔϜ͸ଟ਺ͷཻࢠ͔Βߏ੒͞Ε͓ͯΓɺͦΕΒͷͰ‫ه‬ड़͞ΕΔପԁ಺ͰҰ༷Ͱ͋Δ৔߹Λߟ͑Δɻ͜͜

ੑ࣭͸ɺ6 ࣍‫ݩ‬Ґ૬ۭؒͰͷ෼෍ʹΑͬͯද‫͞ݱ‬ΕΔɻ

Ͱɺa, b, c ͸ପԁΛ‫ه‬ड़͢ΔύϥϝλͰ͋Δɻ͜ͷͱ͔͠͠ͳ͕Βɺཻࢠ਺Λ N ͱ͢Δͱɺͦͷ෼෍͸ 6N

͖ɺҐ૬ۭؒͷ໘ੵ͸ɺ‫ݸ‬ͷ஋Λ࣋ͭ͜ͱʹͳΔɻ౰વͷ͜ͱͰ͋Δ͕ɺ͜ΕAx =ΒͷҐ૬ۭؒͷ৘ใ͸ඇৗʹॏཁͰ͋ΓɺϏʔϜͷੑ࣭Λ‫׬‬શʹද‫͍ͯ͠ݱ‬Δ͕ɺ࣮ࡍʹϏʔϜಉ࢜ͷ඼࣭Λൺֱ͢Δࡍʹ͸ɺ‫͔ͭز‬ͷύϥϝλͰද‫΄͖ͨͰݱ‬Πζɺόϯν௕ɺΤϛολϯεɺΤωϧΪʔ֦͕ΓɺϏʔϜޫֶؔ਺ͳͲΛ༻͍Δɻ͜͜Ͱ͸ɺϏʔϜͷ෼෍͔Βɺ͜ΕΒͷύϥϝλͷ‫ٻ‬ΊํΛ؆୯ʹ঺հ
͢Δɻ

3.5.1

(3.20)

Ͱ༩͑ΒΕΔɻ͜ͷ໘ੵΑΓΤϛολϯε͸

͏͕ศརͰ͋Δɻͦ͜Ͱɺଟ͘ͷ৔߹ 6 ࣍‫ݩ‬Ґ૬ۭؒͰͷ෼෍͔Β‫͞ࢉܭ‬ΕΔɺԣํ޲ͷϏʔϜͷ rms απ,(ac − b2 )1/2

εx =

Ax
,
π

(3.21)

ͱఆٛ͞ΕΔɻ

Liouville ͷఆཧΑΓɺҐஔ࠲ඪ x ͱྗֶతӡಈྔPx ͱ͔ΒͳΔҐ૬ۭؒͰͷ໘ੵ͸ɺ 
dxdPx = const.

(3.22)

ͱ͍͏Α͏ʹෆมྔͱͳΔɻ͜ͷͱ͖ɺҐ૬ۭؒϏʔϜαΠζɺόϯν௕ϏʔϜͷ෼෍ʹର͢Δ෺ཧྔͷฏ‫ۉ‬Λ  Ͱද͢ͱ͢Δͱɺԣํ޲ͷ rms ϏʔϜαΠζ (σx , σy )ɺrms(x, x ) Ͱͷ໘ੵ͸ɺ
 
 
1
1
dxdPx =
dxdPx , (3.23)
Ax =
P
γβmc

㸲㸫 



dxdPx ͸ҰఆͰϫʔ͕૿େ͢Δͱ͍͏͜ͱʹΑΔɻԾʹҐ૬ۭؒͷ໘͋ΔͷͰɺAx ͸ 1/γβ ʹґଘ͢Δ͜ͱ͕Θ͔Δɻి

ੵ͕ಉ͡Ͱ΋ɺ෼෍͕ඇઢ‫ޮܗ‬ՌͳͲʹΑͬͯ࿪ΈɺࢠஷଂϦϯάͷΑ͏ʹɺϏʔϜͷΤωϧΪʔ͕มԽ͠

rms αΠζ (σx , σx ) ͕૿େ͍ͯ͠Δ৔߹ʹ͸ɺ֤ిͳ͍৔߹ʹ͸ (βγ = const.)ɺAx ͸Ұఆɺͭ·ΓΤࢠ͔Βͷ์ࣹޫ΋֦͕ͬͯ͠·͍ɺ์ࣹύϫʔͷ‫ݮ‬গͱද͞ΔɻLiouville ͷఆཧΑΓϛολϯε εx ͸ҰఆʹͳΔɻ͔͠͠ɺERL ೖࣹ‫ث‬ΛҾ͖‫͢͜ى‬ɻ͢ͳΘͪɺҐ૬ۭؒͰͷ෼෍͕࿪ΜͰͷΑ͏ʹՃ଎ʹ൐͍ΤωϧΪʔ͕มԽ͍ͯ͘͠৔߹ʹ

͍Δ৔߹ɺҐ૬ۭؒͰͷ໘ੵ͕େ͖͍৔߹ͱಉ༷ʹ์͸ɺβγ = const. Ͱ͋ΔͷͰɺΤϛολϯε εx ͸ෆࣹޫͷ࣭ͷ௿ԼΛট͘ɻ͜ͷΑ͏ʹɺ์ࣹޫ‫ݯ‬ͷً౓มྔͱ͸ͳΒͳ͍ɻͦ͜ͰɺERL ೖࣹ‫Ͳͳث‬ͷΤω͸ rms ΤϛολϯεͰܾ·Γɺrms Τϛολϯε͕ϧΪʔͷมԽ͢ΔՃ଎‫Ͱث‬͸ɺΤωϧΪʔʹґଘ͠ͳখ͍͞΄Ͳً౓͕ߴ͘ͳΔɻ͜ͷͨΊɺϏʔϜͷ඼࣭͍ΤϛολϯεΛ࢖༻͢Δͷ͕ศརͰ͋Δɻ͜Ε͕ɺΛද͢ई౓ͱͯ͠ɺҐ૬ۭؒͰͷ໘ੵͰఆٛ͞ΕͨΤ‫֨ن‬ԽΤϛολϯεͰ͋ΓɺϛολϯεΛ༻͍ΔΑΓɺrms ΤϛολϯεΛ༻͍ͨ

εnx = γβεx ,(3.24)

ํ͕ྑ͍ɻ͜͜·Ͱ͸ɺϏʔϜ͸ਐߦํ޲ʹରͯ͠Ұ༷Ͱ͋Δ
ͱߟ͖͑ͯͨɻERL ೖࣹ‫Ͱث‬͸ిࢠϏʔϜ͸όϯνԽͱఆٛ͞ΕΔɻҐ૬ۭؒͰͷ໘ੵͷ୅ΘΓʹɺҰൠʹΑ͘࢖༻͞Ε͞Ε͍ͯΔͨΊɺ࣍ʹόϯνԽ͞ΕͨϏʔϜͷΤϛο

ΔΤϛολϯεͷఆٛʹɺrms Τϛολϯε͕͋Δɻλϯεʹ͍ͭͯߟ͑ΔɻόϯνԽ͞ΕͨϏʔϜͷ৔‫֨ن‬Խ rms Τϛολϯε͸ɺx, Px ͷฏ‫ۉ‬ೋ৐ภࠩ߹ɺόϯνΛਐߦํ޲ʹεϥΠεͯ͠ɺͦͷεϥΠε

ͱ݁߹߲͔ΒɺຖͷҐ૬ۭؒΛߟ͑Δ͜ͱ͕Ͱ͖Δɻ͢ͳΘͪɺεϥ

εn,x

=
=

ΠεຖʹΤϛολϯεΛఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ͜Ε

1  2
x Px2  − xPx 2mc
βγ x2 x2  − xx  = βγεx ,ΛεϥΠεΤϛολϯεͱ‫Ϳݺ‬ɻ·ͨɺόϯνશମΛਐߦํ޲ʹਨ௚ͳ໘ʹରࣹͯ͠Өͨ͠Ґ૬ۭؒ΋ߟ͑

(3.25)

Δ͜ͱ͕Ͱ͖Δɻ͜ͷҐ૬ۭ͔ؒΒ‫ٻ‬ΊΒΕΔΤϛο
λϯεΛࣹӨΤϛολϯεͱ͍͏ɻࣹӨΤϛολϯε

ͱఆٛ͞ΕΔɻ͜͜ͰɺΤϛολϯεͷࣜͷதͰͷ͸ɺ֤εϥΠεΤϛολϯεͷॏͶ߹ΘͤʹͳΔɻ͜

x2  ͸ฏ‫ۉ‬ೋ৐ภࠩΛද͓ͯ͠Γɺ‫͚ॻʹີݫ‬͹ɺͷࣹӨΤϛολϯε͕ॏཁͱͳΔͷ͸ɺRF ి࣓৔Λ(x − x)  ͷ͜ͱͰ͋Δɻ͜͜Ͱ͸ɺΤϛολϯε
ͷఆٛࣜ಺Ͱ͸ɺ‫ه‬ड़ͷ؆୯ԽͷͨΊʹ্‫ه‬ͷΑ͏ʹ‫ه‬ड़͢ΔͷͰɺ࣮ࡍʹ‫͢ࢉܭ‬Δࡍʹ͸ɺదٓ‫ࣜͳີݫ‬௨աͨ͠৔߹΍ɺۭؒిՙޮՌͷӨ‫͕͋ڹ‬Δ৔߹Ͱ͋͢Δ৔߹ɺRF ి࣓৔͕࣌ؒʹґଘ͢ΔͨΊɺόϯνʹಡΈସ͑ͯཉ͍͠ɻ

εϥΠεͷҐஔʹΑͬͯड͚Δྗ͕ҟͳͬͯ͘Δɻ·

Hamiltonian ʹΑͬͯ‫ه‬ड़͞ΕΔ‫Ͱܥ‬͸ɺLiouvilleͷఆཧ͕੒ΓཱͪɺҐ૬ۭؒͰͷཻࢠूஂ͕઎ΊΔ໘ͨɺۭؒిՙޮՌ͕͋Δ৔߹ʹ͸ɺόϯνதԝͱ୺ͷੵ͸อଘ͢Δɻ͔͠͠ɺҐ૬ۭؒͰͷ໘ੵ͕อଘ͢Δ

ྗ͕ҟͳͬͯ͘Δɻ͜ͷΑ͏ͳ৔߹ɺόϯνεϥΠε

৔߹Ͱ΋ɺrms Τϛολϯε͸อଘ͠ͳ͍৔߹͕͋

ͷҐஔʹΑͬͯɺҐ૬ۭؒͰͷ෼෍͕มԽ͢Δɻ͢ͳΔͷͰ஫ҙ͕ඞཁͰ͋ΔɻHamiltonian ʹඇઢ‫ܗ‬੒෼ΘͪɺεϥΠεʹΑͬͯΤϛολϯε͕ҟͳΔ͜ͱʹ͕‫·ؚ‬ΕΔ৔߹ʹ͸ɺҐ૬ۭؒͰͷ෼෍͕ᎇΕͨΓ͢ͳΔɻ͜ͷͱ͖ɺࣹӨ͞ΕͨҐ૬ۭؒͰ͸ɺ֤εϥΠΔɻ͜ͷͱ͖໘ੵ͸อଘ͞ΕΔ͕ɺrms Τϛολϯε

εͷҐ૬ۭؒ෼෍͕Ұக͠ͳ͍ͨΊ໘ੵ͕૿େ͢Δɻ

͸૿େ͢Δɻrms Τϛολϯε͕อଘ͢Δͷ͸ɺ֎ྗͭ·ΓɺࣹӨΤϛολϯεͷ૿େ͕ੜ͡Δɻਤ 37 ʹ͕ઢ‫͔ͭͰܗ‬ଞͷํ޲ͱͷӡಈͷ݁߹͕ͳ͍৔߹Ͱ

όϯνԽ͞ΕͨϏʔϜʹ͍ͭͯͷεϥΠεΤϛολϯ͋ΔɻεͱࣹӨΤϛολϯεͷؔ܎Λࣔ͢ɻεϥΠεΤϛο

2

ΔɻRF ి࣓৔Λ༗‫ݶ‬ͷόϯν௕Λ࣋ͭϏʔϜ͕௨ա෦෼Ͱ͸ిՙີ౓ͷҧ͍ʹΑΓɺۭؒిՙޮՌʹΑΔ

ଟ͘ͷ৔߹ɺΤϛολϯεͷఆٛͱͯ͠ rms Τϛολϯε͸ɺ‫ݪ‬ཧతʹ‫ݮ‬গͤ͞Δ͜ͱ͸Ͱ͖ͳ͍͕ɺࣹ

λϯε͕༻͍ΒΕ͍ͯΔ͕ɺ࣮ࡍʹ rms Τϛολϯε

ӨΤϛολϯε͸֤εϥΠεͷҐ૬ۭؒ෼෍ΛҰக͕͞์ࣹޫͷ඼࣭΁༩͑ΔӨ‫ͳ͏Ͳ͕ڹ‬Δ͔Λߟ͑Δɻ

ͤΔ͜ͱʹΑΓɺ‫ݮ‬গͤ͞Δ͜ͱ͕ՄೳͰ͋Δɻ͜ΕిࢠϏʔϜ͔Β์ࣹޫΛൃੜͤ͞Δ৔߹ɺΤϛολϯ

͕ࣹӨΤϛολϯεͷิঈͰ͋ΔɻۭؒిՙޮՌʹΑε͕খ͍͞ํ͕์ࣹޫͷً౓্͕ঢ͢Δ͜ͱ͕஌ΒΕΓ૿େࣹͨ͠ӨΤϛολϯεͷิঈํ๏ʹ͍ͭͯ͸ɺ

͍ͯΔɻ͜Ε͸ɺϏʔϜ಺ͷ֤ʑͷిࢠ͕์ࣹ͢Δ์

ηΫγϣϯ 6.2 ͰΑΓৄࡉʹ‫ݕ‬౼͢Δɻ

ࣹޫ͕খ͘͞·ͱ·͍ͬͯΔ΄ͲΑ͘‫ׯ‬ব͠ɺ์ࣹύ㸲㸫 Λ༻͍ΔͱΤϛολϯε߲͸

slice emittance
x’

x’

x’
x

εxc =

x

x


 2
x2c x2c  − xc xc  ,(3.29)

ͱ‫͞ࢉܭ‬ΕΔɻ͜ΕΛ༻͍ΔͱϏʔϜޫֶؔ਺͸ɺ
projected emittance
x’
x

α̂x

=

β̂x

=

γ̂x

=

electron bunchਤ 37: όϯνԽ͞ΕͨϏʔϜͰͷࣹӨΤϛολϯεɻxc xc ,εxc
βz x2c 
,
εxc
x2
c 
,
εxc βz −(3.30)(3.31)(3.32)

ۭؒిՙޮՌ΍ RF ి࣓৔ͳͲʹΑΓɺεϥΠεΤͱද͞ΕΔɻy ʹ͍ͭͯ΋ಉ༷ʹ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖

ϛολϯεʹෆҰக͕ੜ͍ͯ͡Δͱ͖ɺࣹӨΤϛολΔɻ͜͜Ͱɺ‫֨ن‬Խ͞Εཻͨࢠ଎౓ β ɺLorentz Ҽࢠϯε͸૿େ͢Δɻγ ͱɺϏʔϜޫֶؔ਺ͷ ”β”ɺ”γ” Λ۠ผ͢ΔͨΊʹɺϏʔϜޫֶؔ਺ʹ͸ ˆ Λ͚ͭͨɻ͜ΕΒΛ༻͍Δ

3.5.3

ΤωϧΪʔ֦͕Γ͜ͱͰɺཻࢠτϥοΩϯάίʔυΛ༻͍ͯ ERL ೖࣹϏʔϜ಺ͷཻࢠ͸ɺ΄΅ಉҰͷΤωϧΪʔΛ࣋ͬͯ
͍Δ͕ɺશͯͷཻࢠ͕‫׬‬શʹಉ͡ΤωϧΪʔΛ࣋ͬͯ
͍ΔΘ͚Ͱ͸ͳ͍ɻ͢ͳΘͪɺ‫࣮ݱ‬ͷϏʔϜ͸͋Δ༗
‫ݶ‬ͷΤωϧΪʔ֦͕ΓΛ࣋ͬͯΔɻϏʔϜ಺ͷཻࢠͰ‫ث‬ͷγϛϡϨʔγϣϯΛߦͬͨ‫ʹޙ‬ɺ࠷‫ޙ‬ͷཻࢠ෼෍͔ΒϏʔϜޫֶؔ਺Λ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δɻ͜ΕΒΛ༻͍ͯɺERL पճ෦ͰͷϏʔϜޫֶؔ਺‫ࢉܭ‬ͷॳ‫ظ‬஋ͱ͢Δ͜ͱ͕Ͱ͖Δɻ

ΤωϧΪʔ͕ࠩ͋Δ৔߹ʹ͸ɺ࣓৔தͰҟͳΔ‫ي‬ಓΛඳ͘͜ͱʹͳΓɺ͜Ε͕͞·͟·ͳӨ‫ڹ‬ΛҾ͖‫͜ى‬͢ɻྫ͑͹ɺϏʔϜ಺ͷཻࢠ͕༗‫ݶ‬ͷΤωϧΪʔࠩΛ࣋ͭ৔߹ɺภ޲ి࣓ੴΛ௨ա͢Δͱ‫ي‬ಓ͕ࠩੜ͡ɺԣ
ํ޲ͷҐஔͷ͕ࠩੜ͡Δɻ͜ͷΑ͏ͳΤωϧΪʔࠩʹΑͬͯੜ͡Δԣํ޲ͷͣΕͷେ͖͞Λදؔ͢਺͕ɺ෼
ࢄؔ਺Ͱ͋Δɻ෼ࢄؔ਺͕θϩͰͳ͍ՕॴͰ͸ɺ෼ࢄؔ਺ʹΤωϧΪʔ֦͕ΓΛֻ͚ͨ෼͚ͩɺԣํ޲ͷ෼෍ͷ֦͕Γ͕ੜ͡ΔɻERL ೖࣹ‫Ͱث‬͸ɺΤωϧΪʔ֦͕Γ͸ɺిࢠϏʔϜੜ੒࣌ʹੜ͡Δ΋ͷͷଞʹɺRFۭಎͰͷಈి࣓৔ʹΑͬͯੜ͡Δ΋ͷɺਐߦํ޲ͷۭؒిՙޮՌʹΑͬͯੜ͡Δ΋ͷ͕͋Δɻ
ΤωϧΪʔ֦͕Γ͸ɺϩʔϨϯπҼࢠ γ ͔Βδ=(γ − γ)2 1/2
σE
=
,
E0
γ

(3.26)

ͱ‫͞ࢉܭ‬ΕΔɻ͜͜ͰɺσE ͸ rms ΤωϧΪʔ֦͕ΓɺE0 ͸ΤωϧΪʔͷฏ‫ Ͱۉ‬E0 = mc2 γ Ͱ͋Δɻ

3.5.4

ϏʔϜޫֶؔ਺

ཻࢠ෼෍͔ΒϏʔϜޫֶؔ਺ (Courant-Synder Pa-

rameters) α̂, β̂, γ̂ ͸ɺ࣍ͷΑ͏ʹ‫͞ࢉܭ‬ΕΔɻཻࢠͷฏ‫ۉ‬Ґஔ͔ΒͷͣΕɺxc=x − x,(3.27)xc=

βx − βx ,(3.28)㸲㸫 ૬ର࿦తͳϏʔϜͷ୯ཻࢠతऔΓ

4

ͱͳΔɻಉ༷ʹͯ͠ɺ࣓৔ͷϙςϯγϟϧ φm (r, z) ͕1
φm (r, z) = φm (0, z) − φm (0, z)r2
4
1 (4)+ φm (0, z)r4 − · · · ,64ѻ͍͜͜Ͱ͸ɺ૬ର࿦తͳϏʔϜʹରͯ͠ɺιϨϊΠυ
΍ RF ి࣓৔ʹΑΔϏʔϜʹର͢Δऩଋ࡞༻Λࣔͨ͢

(4.5)

ΊʹɺۭؒిՙޮՌ͕ͳ͍৔߹ͷӡಈํఔࣜΛಋग़͢ͱల։͞ΕΔͱ͢Δͱɺ1 ࣍ͷ߲Λ࢒͠ φm = −B ͱΔɻ؆୯ͷͨΊʹɺϏʔϜ͸όϯνԽ͞Ε͓ͯΒͣɺ

͢Δͱɺ࣓৔͸࣍ͷΑ͏ʹͳΔɻ

ਐߦํ޲ʹରͯ͠Ұ༷Ͱ͋Δͱ͍ͯ͠Δɻ·ͨɺ͜͜Bz (r, z) =Ͱ‫ٻ‬ΊΔϏʔϜΤϯϕϩʔϓํఔࣜ͸ηΫγϣϯ 6.1
Ͱ঺հ͢ΔɺۭؒిՙޮՌʹΑΔࣹӨΤϛολϯεͷ

Br (r, z) =มԽΛ֓‫͢؍‬ΔͨΊͷ‫ͳͱૅج‬Δɻ

∂φm
= B,
∂z
∂φm
1
−
= − B  r.
∂r
2
−

(4.6)
(4.7)

ԁपํ޲ͷి৔ɺ࣓৔͸ɺԁ౵ରশੑͷԾఆΑΓɺEθ = 0ɺBθ = 0 Ͱ͋Δɻӡಈํఔࣜͷಋग़

4.1

࣍ʹి࣓৔தͰͷՙిཻࢠͷӡಈΛߟ͑Δɻӡಈํ͜͜Ͱ͸ɺՙిཻࢠͷूஂͷӡಈΛ࿦͡ΔલʹɺϏʔఔࣜ͸ɺࣜ (3.8) ͔ΒɺϜΛߏ੒͢Δ‫ݸ‬ผͷՙిཻࢠͷӡಈΛߟ͑Δɻՙిཻ

dP
dv
dγ
= γm
+ mv
= q(E + v × B),
dt
dt
dt

ࢠ͸ɺ૬ର࿦తͳ଎౓Λ࣋ͭͱͯ͠ɺۙ࣠ޫઢۙࣅΛ(4.8)

༻͍ͯӡಈํఔࣜΛಋग़͢Δɻର৅ͱ͢Δ‫ܥ‬͸ɺz ࣠Ͱ༩͑ΒΕΔɻ͜ͷӡಈํఔࣜ͸ɺԁ౵࠲ඪ‫( ܥ‬r, θ, z)ʹԊͬͯԁ౵ରশੑΛ΋ͭͱ͢Δɻ͜ͷͱ͖ɺཻࢠͷͷ৔߹ʹ͸ɺ

ӡಈ͸ԁ౵࠲ඪ‫( Ͱܥ‬r, ṙ, θ, θ̇, z, ż) ʹΑͬͯද͢͜ͱ

d(γmṙ) − γmrθ̇2 = q(Er + rθ̇Bz − żBθ ), (4.9)
dt
1 d(γmr2 θ̇) = q(Eθ + żBr − ṙBz ), (4.10)
r dt
d(γmż) = q(Ez + ṙBθ − rθ̇Br ), (4.11)
dt

͕Ͱ͖Δɻ͜͜Ͱɺ˙ ͸࣌ؒʹΑΔඍ෼ d/dt Λද͠ɺ͸ z ʹΑΔඍ෼ ∂/∂z Λද͢΋ͷͱ͢Δɻཻࢠ͸૬ର࿦తͳ଎౓Λ΋ͬͯɺz ্࣠Λਐߦ͢Δ΋ͷͱ͢Δɻ
ӡಈํఔࣜΛ‫ٻ‬ΊΔʹ͋ͨͬͯɺۙ࣠ޫઢۙࣅΛ༻
͍Δ͜ͱʹ͢Δɻ·ͣɺ࣍ͷΑ͏ͳԾఆΛಋೖ͢Δɺ

ͱͳΔɻ͜Εʹԁ౵ରশੑ͕͋Δ৔߹ͷ E ͱ B Λ୅

1. r ͸֎෦࣓৔Λ࡞ΔίΠϧ΍మ৺ͷ‫ʹܘ‬ൺ΂ͯे
෼খ͍͞

ೖ͢Δͱɺ

2. r ΋े෼ʹখ͍͞ (r  1)m

3. vθ ΋े෼ʹখ͍͞ (rθ̇  ż)͢ͳΘͪɺՙిཻࢠ͸ରশ࣠Ͱ͋Δ z ࣠ͷۙ͘Λมಈ͢ΔͱԾఆ͢Δɻ͜ͷԾఆͷԼͰ͸ɺ֎෦ి࣓৔ͷߴ࣍ͷ߲Λແࢹ͢Δ͜ͱ͕ՄೳͱͳΔɻz ࣠ۙ๣Ͱɺ֎෦ి৔ʹΑΔ੩ిϙςϯγϟϧ͸࣍ͷΑ͏ʹల։Ͱ͖Δɻ
1
1
φ(r, z) = V − V  r2 + V (4) r4 − · · · .
464(4.12)(4.13)(4.14)

ͱͳΔɻࣜ (4.12) ͷதͷ θ̇ ͸ɺࣜ (4.13) ͔Βܾ·Δ͕ɺ͜͜Ͱ͸ࣜ (4.13) ͷ୅ΘΓʹɺ‫ʹܥ‬ԁ౵ରশੑ
͕͋ΔͱԾఆͯ͠ɺਖ਼४ӡಈྔͷอଘଇ͔Β‫ٻ‬ΊΔ͜

(4.1)

ͱʹ͢Δɻి࣓৔தͰͷՙిཻࢠͷ Hamiltonian ͸ɺ͜͜ͰɺV (z) ͸ z ্࣠ͷϙςϯγϟϧͰV (z) = φ(0, z),d1(γ ṙ) − mγrθ̇2 = qV  r + qrθ̇B,
dt
2
d
1
2
m (γr θ̇) = − q żB  r2 − rṙB,dt2qdm (γ ż) = −qV  + r2 θ̇B  ,
dt
2

H=c
(4.2)



m2 c2 + (p − qA) + qφ − mc2 ,(4.15)ͱද͞ΕΔɻ͜͜ͰɺA ͸ϕΫτϧϙςϯγϟϧɺP

Ͱද͞ΕΔɻ͜ͷͱ͖ɺਐߦํ޲͓Αͼ൒‫޲ํܘ‬ͷి͸ྗֶతӡಈྔɺp ͸ਖ਼४ӡಈྔͰɺp = P + qA Ͱ৔͸ɺ͋Δɻφ ͱ A ͕ԁ౵ରশੑΛ࣋ͭ৔߹ɺHamiltonianEz (r, z) =Er (r, z) =

∂φ
∂V
−
(4.3)
=−
= −V  ,
∂z
∂z
∂φ
r ∂Ez
1
−
= V  r, = −, (4.4)∂r22 ∂z΋ಉ༷ͷରশੑΛ࣋ͭɻैͬͯɺHamiltonian ํఔࣜ
ΑΓ

㸲㸫 

dpθ
∂H
=−
= 0,
dt
∂θ

(4.16)

͋Δ͍͸ɺ

ͱͳΓɺpθ ͸อଘྔͱͳΔɻ2

pθ = γmr θ̇ + qrAθ = const.

(4.17)࣓ଋ Ψ = B · dS Λ༻͍Δͱɺ
Ψ = (∇ × A) · dS = A · dl = 2πrAθ , (4.18)
ΑΓɺ͜ͷ Ψ Λ༻͍ΔͱɺAθ = Ψ/(2πr) ͱͳΔɻ·d(mc2 γ + qV ) = 0,dtͱͳΔɻࣜ (4.27) Λੵ෼͢Δͱɺ

(4.27)

mc2 γ + qV = const.

(4.28)

ͱͳΓɺੵ෼͸ఆ਺ͱͳΔɻ͜ͷఆ਺ʹ mc2 Λ෇͚Ճ͑Δͱͨɺz ࣠ͷۙ๣Ͱ Bz (r, z)  Bz (0, z) = B ͱ͢Δͱɺ

(γ − 1)mc2 + qV = T + U = const.

(4.29)z ࣠ۙ๣ͷ൒‫ ܘ‬r ͷԁ౵அ໘Λ؏࣓͘ଋ͸ɺͱͳΓɺ͜Ε͸ΤωϧΪʔอଘଇΛ͍ࣔͯ͠Δ͜ͱ͕Ψ = πr2 B,(4.19)

ͱॻ͚ΔɻAθ ͷ୅ΘΓʹ B Λ༻͍Δͱɺอଘଇ͸ɺqpθ = γmr θ̇ + Br2 = const.
22(γ − 1)mc2 + qV = 0,

qB
pθ
,
+
2γm γmr2

dθ
dt
qB
pθ
,
= θ̇
−
+
dz
dz
2γmβc mcβγr2(4.21)γ =1−

qV (z)|qV (z)|
=1+
,
mc2
mc2

ԁपํ޲ͷਖ਼४ӡಈྔͷอଘ͔Β‫ٻ‬Ίͨࣜ (4.21)(4.22)ͷ θ̇ Λ r ํ޲ͷӡಈํఔࣜ (4.12) ʹ୅ೖ͢Δͱɺd(γ ṙ)dt߹ɺԁपํ޲ͷਖ਼४ӡಈྔ͸อଘྔͱͳΓɺͦͷ஋͸ࣜ (4.20) ΑΓɺཻࢠҐஔ rɺθ̇ɺγ ͦͯ͠ B ͷॳ‫ظ‬஋==ʹΑܾͬͯ·Δɻཻࢠ͕ࣗ༝ۭؒ (B = 0) ͔Βग़ൃ
͢Δ৔߹ʹ͸ɺpθ = γmr2 θ̇ ͱͳΔɻ͞Βʹɺԁपํ޲ͷॳ‫ظ‬଎౓͕θϩ (θ̇ = 0) ͷ৔߹ʹ͸ɺpθ = 0 ͱ

rθ̇
q 
V r + (mγ θ̇ + qB)
2m
m
2
1 qB
p2θ
q 
+
,
V r−
2m
4γ m
γm2 r3(4.32)

ͱͳΔɻࠨลͷඍ෼Λ‫͢ࢉܭ‬ΔͱɺͳΔɻ࣍ʹɺz ํ޲ͷӡಈํఔࣜ (4.14) ͔Β γ ͱϙςϯd(γ ṙ) = γ̇ ṙ + γ r̈,dtγϟϧ V ͷؔ܎Λಋ͘ɻż  v = βc (vθ = rθ  v)

ͱͳΔɻࠨลͷ γ ͱ r ͷඍ෼͸ɺͱ͢Δͱɺࣜ (4.14) ͷӈลୈ 2 ߲Λແࢹ͢Δ͜ͱ͕Ͱ͖ɺd(γ ż) = −qV  ,dtͱͳΔɻͭ͗ʹࠨลΛ‫͢ࢉܭ‬Δͱɺ
m=(4.23)dz d(γ ż)dt dz
v(γ v + γv  )c2 (γ  β 2 + γβ  β),

=

ṙ

=

r̈

=

dz dγ
= γ  βc,
dt dz
dz dr
(4.34)= r βc,
dt dz

d 
γ
(r βc) = r β 2 c2 + r 3 c2 ,dtγΔͱɺࣜ (4.33) ͸ɺ(4.24)

d(γ ṙ) = c2 (γβ 2 r + γ  r ),dt(4.35)

ͱͳΔɻ͜ΕΑΓɺr ํ޲ͷӡಈํఔࣜ (4.32) ͸ɺ(4.25)c2 (γβ 2 r + γ  r ) =ͱ‫·ٻ‬Δɻ͜ΕΑΓɺࣜ (4.23) ͸ɺ

mc2 γ  = −qV  ,γ̇

(4.33)

ͱͳΔɻ͜͜Ͱɺγ  /γ 3 = ββ  Λ༻͍ͨɻ͜ΕΛ༻͍ͱͳΔɻ͜͜Ͱɺβ   0ɺ͓Αͼ β  1 ͱ͢Δͱɺd(γ ż) = c2 γ  ,dt(4.31)ͱද͞ΕΔɻ

ͱͳΔɻ͜͜Ͱߟ͍͑ͯΔ‫ʹܥ‬ԁ౵ରশੑ͕͋Δ৔d(γ ż) =dt=(4.30)

ͱͳΔɻ͜ͷͱ͖ γ ͸͋Δ͍͸ɺż = dz/dt  βc ͱͯ͠ɺθ =

͖ V = 0 ͱ͢Δͱɺఆ਺͸θϩͱͳΓɺ

(4.20)

ͱͳΔɻ͜ͷͱ͖ɺθ̇ ͸θ̇ = −

Θ͔Δɻ͜͜ͰɺT ͸ӡಈΤωϧΪʔɺU ͸ϙςϯγϟϧΤωϧΪʔͰ͋ΔɻT = 0ɺͭ·Γ γ = 1 ͷͱ

(4.26)

㸲㸫 

q 
1
V r−
2m
γ
p2 1
+ θ2 3 ,
γm r

qB
2m

2

r
(4.36)

ͱͳΔɻ͜ͷࣜ͸ V  ͷ߲Λ‫ؚ‬Ή͕ɺΤωϧΪʔอଘ͜ͷͱ͖Ґ૬ۭؒͷ໘ੵ͸ɺଇ͔Β‫ٻ‬Ίͨࣜ (4.31) Λ༻͍Δ͜ͱΑΓɺ

qV



2 

= −mc γ ,A = επ =(4.37)π,(ac − b2 )1/2

ͱද͞ΕΔɻ͜͜Ͱɺε = 1/(ac − b2 )1/2 ͸Τϛολ

ͱɺγ Λ༻͍ͯද͢͜ͱ͕Ͱ͖Δɻࣜ (4.36) ʹ͜ͷϯεͰ͋Δɻ࣍ͷΑ͏ͳପԁͷ܎਺͔ΒͳΔߦྻV Λ୅ೖͯ͠੔ཧͨ͠΋ͷ͕ɺ͜͜Ͱ‫ٻ‬Ί͔ͨͬͨ૬ର࿦తͳཻࢠʹର͢Δۙ࣠ޫઢۙࣅͷ΋ͱͰͷӡಈ

σ̄ =

ํఔࣜ

r +

γ 
γ 
qB
r
r
+
r+
γβ 2
2γβ 2
2mcβγ
p2
1
− 2 2θ 2 2 3 = 0,m c γ β r(4.40)

1
ε2

c

−b

−b

a

,

(4.41)

Λ༻͍Δͱɺପԁͷࣜ (4.40) ͸rσ̄ −1 r = 1,t(4.38)ͱද͞ΕΔɻ͜͜Ͱɺ

Ͱ͋Δɻࠨลୈ 2 ߲͸அ೤߲ɺୈ 3 ߲͸Ճ଎ి৔ʹͦͯ͠ୈ 5 ߲͸ԁपํ޲ӡಈྔʹΑΔ߲Ͱ͋Δɻ

r

,

r

=

r

t

=

(r, r ),ΑΔऩଋͷ߲ɺୈ 4 ߲͸ιϨϊΠυʹΑΔऩଋͷ߲ɺrҎ্ͷΑ͏ʹɺ͜͜Ͱ͸ۙ࣠ޫઢۙࣅΛ༻͍ͯɺ૬

(4.42)

ର࿦తͳ଎౓Λ࣋ͭ 1 ཻࢠͷӡಈํఔࣜΛಋ͍ͨɻ

Ͱ͋Δɻߦྻ σ̄ ͸ɺr ͱ r ʹ͍ͭͯͷೋ৐ภ͔ࠩΒ࣮ࡍͷϏʔϜ͸ཻࢠ෼෍Λ࣋ͭͨΊɺͦͷӡಈΛߟߏ੒͞ΕΔ σ ߦྻͱ͑Δࡍʹ͸ཻࢠ෼෍ʹΑΔӨ‫ڹ‬Λߟྀ͢Δඞཁ͕͋ΔɻϏʔϜͷूஂͱͯ͠ͷӡಈΛߟ͑Δ৔߹ɺ‫ݸ‬ผͷཻࢠͷ࠲ඪΛશͯ‫ه‬ड़͢ΔͷͰ͸ͳ͘ɺཻࢠ෼෍ʹର
͢Δ rms ϏʔϜαΠζͷมԽΛߟ͑Δํ͕ศརͰ͋
Δɻԣํ޲ͷϏʔϜͷӡಈ͸ɺϏʔϜΤϯϕϩʔϓͱ‫ݺ‬͹ΕΔϏʔϜͷେ͖͞Λ‫ه‬ड़͢Δแབྷؔ਺ʹΑͬͯ‫ه‬ड़͢Δ͜ͱ͕Ͱ͖ΔɻϏʔϜΤϯϕϩʔϓͷ࣌ؒൃలΛ‫ه‬ड़͢ΔํఔࣜΛϏʔϜΤϯϕϩʔϓํఔࣜͱ‫Ϳݺ‬ɻϏʔϜ͕൒‫ ܘ‬rm ͷԁ‫ܗ‬அ໘Λ΋ͪɺిՙີ౓
͸ϏʔϜ಺ͰҰ༷ͰɺϏʔϜ಺ͷཻ֤ࢠͷ‫ي‬ಓ͸ަࠩ͠ͳ͍ͱԾఆ͢ΔͱɺϏʔϜαΠζͷมԽ rm (z) ͸ɺࣜ (4.38) ͷதͷ r Λ rm Ͱஔ͖‫ͨ͑׵‬ϏʔϜΤϯϕ

Ͱ͋Δɻ͜͜Ͱɺr =ۭؒதͰͷཻࢠ෼෍ʹΑΔӨ‫ྀ͞ߟ͕ڹ‬Ε͍ͯͳ͍ɻ

r  =

ϏʔϜ͕༗‫ͳݶ‬ΤϛολϯεΛ࣋ͭ৔߹ɺϏʔϜΤϯ
௥Ճ͢Δඞཁ͕͋Δɻ࣍ʹɺϏʔϜཻࢠͷ෼෍͕Ґ૬্ۭؒͰପԁ‫ܗ‬ঢ়Λ࣋ͭͱͯ͠ɺϏʔϜΤϯϕϩʔϓؔ਺ͱΤϛολϯεͷؔ܎ࣜΛಋग़͢Δɻ

σrr
σr2

= ε4 σ̄,(4.43)

ͱ͍͏ؔ܎͕͋Δɻ‫֨ن‬Խ͞Εཻͨࢠ෼෍ີ౓ ρ(r, r )( ρdrdr = 1) Λ༻͍Δͱɺ
σr2 =ρ(r − r)2 drdr ,(4.44)

σr2 =
ρ(r − r )2 drdr ,(4.45)
σrr =
ρ(r − r)ρ(r − r )drdr , (4.46)

ϩʔϓํఔࣜͰද͞ΕΔɻ͔͠͠ɺ͜ͷࣜͰ͸ɺҐ૬ϕϩʔϓํఔࣜʹΤϛολϯεʹΑΔ૿େΛද߲͢Λ

σr2
σr  rσ=ρrdrdr ,(4.47)

ρr drdr ,(4.48)

ͱͨ͠ɻ؆୯ͷͨΊʹɺి࣓৔ʹΑΔऩଋ࡞༻ͷͳ͍ࣗ༝ۭ
ؒͰͷɺΤϛολϯεʹΑΔϏʔϜΤϯϕϩʔϓͷมԽΛߟ͑Δɻ(r1 , r1 ) Λཻ࣋ͭࢠ͕ɺ௕͞ z ͷࣗ༝ۭؒΛਐΜͩͱ͖ɺग़‫Ͱޱ‬ͷ࠲ඪ͸ɺࣗ༝ۭؒͰͷసૹ

4.2Τϛολϯε߲ߦྻɺM=ԣํ޲ͷҐ૬্ۭؒ (r, r = dr/dz) Ͱཻࢠ෼෍͕
ପԁ‫ܗ‬ঢ়Λ࣋ͭ৔߹ɺͦͷପԁ͸܎਺ aɺbɺc Λ༻͍ͯ࣍ͷΑ͏ʹද͢͜ͱ͕Ͱ͖Δɻ

ar2 + 2brr + cr2 = 1,

1 z
0 1

,

(4.49)

Λ༻͍ͯɺ

(4.39)

㸲㸫 

r2
r2

=M

r1
r1

,

(4.50)

ͱද͞ΕΔɻ͜ͷసૹߦྻΛ༻͍ͯɺࣗ༝ۭؒΛਐΜͱͳΔɻ͜ΕΑΓɺࣗ༝ۭؒͰͷΤϛολϯεʹΑΔ

ͩͱ͖ͷପԁ෼෍ͷύϥϝλ (a, b, c) ͷมԽΛ‫ٻ‬ΊΤϯϕϩʔϓͷมԽ͸ɺ

Δɻసૹߦྻ M ʹରͯ͠ɺߦྻ σ̄ ͸ɺR −σ¯2 = M σ¯1 tM,(4.51)

ε2= 0,
R3(4.61)ͱද͞ΕΔɻ͜Ε͕ΤϛολϯεʹΑΔ߲Ͱ͋ΓɺΤͷΑ͏ʹసૹ͞ΕΔɻ͜͜Ͱɺσ¯1 ͸ M ͷೖ‫Ͱޱ‬ͷϛολϯε͕θϩͰͳ͍৔߹ʹ͸ɺϏʔϜΤϯϕϩʔ஋ɺσ¯2 ͸ग़‫Ͱޱ‬ͷ஋Ͱ͋Δɻࣗ༝ۭؒೖ‫ޱ‬ͷύϥϝλ

ϓํఔࣜ (4.38) ʹ͜ͷ߲Λ෇͚Ճ͑Δඞཁ͕͋Δɻ

−b1
a1

c1
−b1

1
ε2

σ¯1 =

,

(4.52)

4.3

͔Βɺࣗ༝ۭؒग़‫ޱ‬ͷύϥϝλ͸ɺసૹߦྻΛ༻͍ͯɺ

σ¯2

=

1
ε2

c2

−b2

−b2

a2

͜͜Ͱ͸ɺιϨϊΠυి࣓ੴʹΑΔऩଋ࡞༻Λߟ͑Δɻγ ͕Ұఆͱ͠ɺ·ͨॳ‫ظ‬ͷԁपํ޲ͷӡಈྔΛθϩͱ͢Δ (pθ = 0)ɻιϨϊΠυͷ࣓৔Λ B ͱ͢

=

1
ε2

1 z

c1

−b1

1

0

0

−b1

a1

z

1

=

1
ε2

c1 − b1 z + z(a1 z − b1 )a 1 z − b11ιϨϊΠυʹΑΔऩଋ࡞༻

a1 z − b1a1Δͱɺۙ࣠ޫઢۙࣅͷ΋ͱͰͷཻࢠͷӡಈํఔࣜ͸ɺࣜ (4.38) ΑΓɺr + k 2 (z)r = 0,(4.62),
ͱͳΔɻ͜͜Ͱɺ

(4.53)k 2 (z) =ͱද͢͞ΕΔɻ͜ΕΑΓɺ௕͞ z ͷࣗ༝ۭؒͰ͸ɺପԁ෼෍ͷύϥϝλ͸

qB(z)
2mcβγ

2

,

(4.63)

ͱͨ͠ɻ͜ͷӡಈํఔࣜΛιϨϊΠυͷೖ‫( ޱ‬z1 ) ͔

a2

=

a1 ,

(4.54)b2=

b1 − a1 z,(4.55)

c2=

a1 z 2 − 2b1 z + c1 ,(4.56)

Βग़‫( ޱ‬z2 ) ·Ͱੵ෼͢Δͱɺೖ‫ͱޱ‬ग़‫Ͱޱ‬ͷϏʔϜͷ܏͖ͷࠩ͸ɺ

r (z2 ) − r (z1 ) = − z2

k 2 (z)dz,(4.64)z1ͱసૹ͞ΕΔɻ͜ΕΒͷύϥϝλΛ༻͍ͯɺϏʔϜΤϯϕϩʔϓͷมԽΛ‫͢ࢉܭ‬Δɻࣗ༝ۭؒग़‫Ͱޱ‬ͷϏʔϜ൒‫ܘ‬ͷ࠷େ஋͸ɺ

(4.57)

dR2
ε 1
R2 =
= √
dz

2

c2

c2 ,

΅͍ͯ͠Δͱ͍͏͜ͱΛ͍ࣔͯ͠Δɻࣜ (4.64) ͕ࣔ͢Α͏ʹɺ͜ͷੵ෼͸ k 2 ͷੵ෼ɺͭ·Γ B(z)2 ʹͭ

(4.58)

ͱͳΔɻࣜ (4.56) Λ 1 ճඍ෼ͨ͠ c2 = 2(b1 − a1 z) =−2b1 Λ༻͍Δͱɺ=͍ͯͷੵ෼Ͱ͋ΔͨΊɺιϨϊΠυ͸࣓৔ͷ޲͖ʹؔ܎ͳ͘ऩଋྗΛ‫ͳʹͱ͜͢΅ٴ‬ΔɻιϨϊΠυʹΑΔऩଋͷ΋͏Ұͭͷಛ௃ͱͯ͠ɺxyۭؒͰͷ෼෍Λऩଋͱͱ΋ʹճసͤ͞Δͱ͍͏ޮՌ͕

εb2
ε 1 
√ c = −√ ,2 c2 2c2(4.59)

1 −3/2ε
1/2 
− c2 c22 + c2 c2
2
2
ε2
,
R23

͋Δɻԁपํ޲ͷ࣌ؒൃల͸ɺࣜ (4.22) ʹΑΓද͞ΕΔɻιϨϊΠυͷೖ‫Ͱޱ‬ɺθ = 0ɺpθ = 0 ͱͯ͠ɺθ Λੵ෼͢Δͱɺཻࢠ͸ϏʔϜͷग़‫Ͱޱ‬ɺ
 z2
 z2
qB(z)
k(z)dz,θr = −=−z1
z1 2mcβγ

ͱ‫·ٻ‬Δɻ͜ΕΛ͞Βʹඍ෼͢Δͱɺ2 ֊ඍ෼͸=ͱͳΓɺ͜ͷͱ͖ιϨϊΠυग़‫Ͱޱ‬ͷ܏͖͸ɺೖ‫Ͱޱ‬ͷ܏͖ʹൺ΂ͯখ͘͞ͳΓɺιϨϊΠυ͸ऩଋྗΛ‫ٴ‬ͱද͞ΕΔɻ͜ΕΛ 1 ճඍ෼͢Δͱɺ

R2

ಓ͕ιϨϊΠυ಺Ͱ z ࣠Λԣ੾Βͳ͍ (r ≥ 0) ͱ͢Δ
ͱɺr (z2 )−r (z1 ) < 0 ͱͳΔɻͭ·Γɺr (z2 ) < r (z1 )√R2 = σr,2 = ε c2 ,R2 =ͱ‫͞ࢉܭ‬ΕΔɻk 2 ͸ৗʹਖ਼Ͱ͋Δ͜ͱ͔Βɺཻࢠͷ‫ي‬(4.60)(4.65)

͚ͩճస͢Δ͜ͱʹͳΔɻ͜ͷੵ෼͸ɺB 2 Ͱ͸ͳ͘B ʹ͍ͭͯͷੵ෼ͳͷͰɺճసํ޲͸ B ͷූ߸ɺͭ㸲㸫 ·Γ࣓৔ͷ޲͖ʹґଘ͢Δɻ͜ͷΑ͏ʹɺιϨϊΠυ

electron bunch͸ϏʔϜʹରͯ͠ऩଋ࡞༻ͱͱ΋ʹ xy ۭؒͰͷճసΛҾ͖‫͢͜ى‬ɻιϨϊΠυʹΑΔऩଋ࡞༻ͷয఺‫ ཭ڑ‬f ͸ɺ

1
r
=− =
f
r

2  z2q2mcβγ

2B (z)dz,zlaser beam

(4.66)z1ͱఆٛ͞ΕΔɻιϨϊΠυͷத৺࣓৔ͷ‫͞ڧ‬Λ B0 ͱͯ͠ɺ༗ޮ࣓৔௕ɺ

Lsol =

1
B02

 z2

photo cathodeB 2 (z)dz,(4.67)

Λ༻͍Δͱɺয఺‫཭ڑ‬͸ɺ

1=fਤ 38: RF ޫӄ‫ࢠిۃ‬ॐͷ໛ࣜਤɻ

z1

q
2mcβγ

͜͜Ͱɺω ͸ RF ి࣓৔ͷ֯ৼಈ਺ɺk ͸೾਺Ͱ k =2

Lsol B02 ,(4.68)

2π/λ = ω/cɺλ ͸ RF ి࣓৔ͷ೾௕Ͱ͋Δɻ·ͨɺόϯνͷத৺࠲ඪΛ z0 ͱ͢ΔͱɺҐஔ z ʹ͋Δόϯν಺ͷཻࢠͷɺόϯνத৺͔Βͷ૬ରҐஔ͸ɺs = z −z0ͱද͞ΕɺB 2 ͷґଘੑΛ࣋ͭ͜ͱ͕Θ͔Δɻ

Ͱද͞ΕΔɻి৔ͷ࠷େ஋ E0 ͸ɺz ͷؔ਺Ͱۭಎͷೖ‫ޱ‬ɺग़‫ޱ‬΍‫͍ڱ͕ܘޱ‬෦෼Ҏ֎Ͱ͸ఆ਺ͱͳΓɺ·4.4ۭͨಎ֎Ͱ͸θϩͱͳΔɻ͜͜Ͱ͸ɺ؆୯ͷͨΊʹɺಈి࣓৔ʹΑΔऩଋൃࢄ࡞༻

ERL ೖࣹ‫Ͱث‬ͷॏཁͳ෺ཧͱͯ͠ɺRF ి࣓৔Λར༻ͨ͠Ճ଎ɺόϯνԽ͕͋Δɻ͜ΕΒ͸ɺϚΠΫϩ೾ΛՃ଎ۭಎ಺ʹಋೖͯ͠‫ͤ͞ৼڞ‬ɺۭಎ಺ʹಈి࣓৔Λ‫ܗ‬੒͠ɺͦΕΒͱిࢠϏʔϜΛ૬‫͞༻࡞ޓ‬Δ͜ͱ

E0 ͸ఆ਺Ͱ͋Δͱ͢Δɻz ࣠ۙ๣Ͱ͸ɺࣜ (2.6) ͓Αͼࣜ (2.7) ΑΓɺEr (z, t) ͓Αͼ Bθ (z, t) ͸ɺr ∂E0Er (z, t) = −cos(kz) cos(ωt)
2 ∂zʹΑͬͯߦΘΕΔɻ͜ΕΒͷಈి࣓৔͸ RF (Radio−E0 sin(kz) cos(ωt) ,(4.70)

Frequency) ి࣓৔ͱ‫ݺ‬͹ΕΔɻిࢠϏʔϜΛՃ଎͢rω
E0 cos(ks) sin(ωt),
2c2(4.71)Bθ (z, t) =Δࡍʹ͸ɺిࢠϏʔϜ͕ۭಎΛ௨ա͢Δͱ͖ʹɺి࣓৔ͷ௖఺ (࠷େ஋Λ༩͑Δͱ͖) ʹλΠϛϯάΛ߹Θͱ‫͞ࢉܭ‬ΕΔɻ൒‫޲ํܘ‬ͷӡಈํఔࣜ͸ɺ

ͤΔඞཁ͕͋Δɻ·ͨɺిࢠϏʔϜΛόϯνԽͤ͞Δ

dPr= q (Er − vBθ ) ,dt৔߹ʹ͸ɺηΫγϣϯ 4.5 Ͱઆ໌͢ΔΑ͏ʹɺి࣓৔ͷθϩ఺ͱަࠩ͢ΔΑ͏ͳλΠϛϯάͰόϯνΛ௨աͤ͞ɺόϯνͷલ‫Ͱޙ‬଎౓ࠩΛੜͤ͡͞Δɻ͜͜Ͱ͸ɺERL ೖࣹ‫Ͱث‬ॏཁͱͳΔԣํ޲ͷӡಈʹண໨͍ͯ͠ΔͷͰɺRF ి࣓৔ʹΑΔԣํ޲ͷऩଋൃࢄ࡞༻ʹ͍ͭͯͷΈ঺հ͢Δɻਐߦํ޲ʹ͍ͭͯͷϏʔϜμΠφϛΫεʹ͍ͭͯ͸ɺઢ‫ܗ‬Ճ଎‫ث‬ͷ‫ڭ‬Պॻ΍ଞͷࢀߟจ‫ݙ‬Λࢀরͯ͠ཉ͍͠ɻ
·ͣɺRF ి࣓৔ʹΑΔԣํ޲ͷऩଋൃࢄ࡞༻Λௐ

(4.72)

Ͱ͋Δ͔Βɺ͜Εʹ্‫ه‬ͷ Er ͱ Bθ Λ୅ೖ͢ΔͱɺdPr
qr ∂E0= −cos(kz) cos(ωt)
dt2∂z−E0 sin(kz) cos(ωt)ω− βE0 cos(ks) sin(ωt) , (4.73)cͱͳΔɻϏʔϜཻࢠͷ RF ి࣓৔ʹ͍ͭͯͷҐ૬ɺ΂ΔͨΊʹɺҰͭͷۭಎ͕͋ΔͱԾఆ͠ɺͦͷۭಎ಺ϕ = ωt − kz,(4.74)

ʹԁ౵ରশੑΛ࣋ͬͨఆࡏ೾ (TM01 Ϟʔυ) ͕ग़དྷ
͍ͯΔͱ͢ΔɻͦͷதΛిࢠϏʔϜ͕௨աͨ͠ࡍʹɺΛಋೖ͢Δͱɺ

ԣํ޲ʹͲͷΑ͏ͳӨ‫͕ڹ‬ੜ͡Δ͔Λߟ͑Δɻ۩ମతdPrdt

ʹ͸ɺRF ޫӄ‫ࢠిۃ‬ॐ͕ͦΕʹ͋ͨΔɻRF ޫӄ‫ۃ‬ిࢠॐͷ໛ࣜਤΛਤ 38 ʹࣔ͢ɻۭಎ಺Ͱͷ z ্࣠ͷి৔ͷ z ੒෼͸ɺEz (z, t) = E0 (z) cos(kz) cos(ωt),(4.69)

㸲㸫 

=


1 ∂E0
r
kqE0 −
cos(ks) cos(ks + ϕ)2
kE0 ∂z1+ (1 + β) sin(2ks + ϕ)2
1
(4.75)− (1 − β) sin(ϕ) ,2ͱͳΔɻ͜ΕΑΓɺ൒‫޲ํܘ‬ͷྗ͸Ґ૬ ϕ ʹґଘɺ͢ҟͳΔ͜ͱΛද͓ͯ͠ΓɺࣹӨΤϛολϯεͷ૿େΛͳΘͪɺόϯν಺Ͱͷཻࢠͷ૬ରతͳҐஔʹґଘ͠Ҿ͖‫ͳʹͱ͜͢͜ى‬Δɻόϯν͕Ґ૬ۭؒ಺Ͱ͋Δ෼

ͯมԽ͢Δɻ͜ͷࣜ͸ɺRF ిࢠॐʹ‫ݶ‬ΒͣɺҰൠͷ෍Λ࣋ͭͱ͠ɺΔPx ͱ xɺͦͯ͠ s ʹ͍ͭͯͷೋ৐RF Ճ଎ۭಎʹద༻͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷԣํ޲ͷऩଋൃࢄͷޮՌ͸ɺ௿ΤωϧΪʔͷΠΦϯઢ‫ܗ‬Ճ଎ฏ‫͔ۉ‬Β RF ి࣓৔ͷൃࢄ࡞༻ʹΑΔΤϛολϯεͷ

‫ث‬΍ɺߴ͍Ճ଎ޯ഑Λ࣋ͭిࢠϏʔϜ༻ೖࣹ‫ث‬΍όϯͯ͠ͷฏ‫͢ͱۉ‬Δͱɺx ํ޲ͷ‫֨ن‬ԽΤϛολϯεͷ

νϟʔɺRF ిࢠॐͳͲͰॏཁͱͳΔɻ͜͜ͰɺϏʔ૿େྔ͸ɺ૿େྔΛ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δɻ Λཻࢠ෼෍ʹର଎΍͔ʹ β  1 ʹՃ଎͞ΕΔͱߟ͑Δɻ͜ͷͱ͖ɺ
∂E0 /∂z ͷ߲͸ɺ௨ৗͦΕ΄ͲॏཁͰ͸ͳ͍ͨΊɺٞΔεnx,rfdPrr= keE0 sin ϕ,
dt
2

=
=

࿦Λ؆୯ʹ͢ΔͨΊʹɺ͜͜Ͱ͸͜ͷ߲Λলུ͢Δɻ͜ΕΒͷԾఆͷԼͰ͸ɺ൒‫޲ํܘ‬ͷӡಈํఔࣜΛ2
x2 ΔPx2  − xΔPx 1/2mc
eE0 k 2σ σz | sin ϕ01 |,(4.81)
2mc2 xϜ͸ిࢠϏʔϜ (q = e) Ͱ͋Δͱͯ͠ɺిࢠϏʔϜ͸ͱ‫͞ࢉܭ‬ΕΔɻ͜͜Ͱɺσx = x2 1/2 ɺσz = s2 1/2

(4.76)

ͱͨ͠ɻ͜ͷΑ͏ʹɺόϯνத৺ͰͷҐ૬ ϕ01 ʹΑͬͯɺΤϛολϯεͷ૿େ͕ੜ͡Δ͜ͱ͕ࣔ͞Εͨɻͱۙࣅ͢Δ͜ͱ͕Ͱ͖Δɻ͜ΕΑΓɺҐ૬͕ −π/2 <࣍ʹɺԣํ޲ͷӡಈྔͷόϯνத৺͔Βͷ‫ ཭ڑ‬s ʹ

ϕ < 0 ͷ৔߹ʹ͸ɺRF ి࣓৔ʹΑΔ൒‫޲ํܘ‬ͷྗ͸ɺൃࢄྗͱͳΔɻ࣍ʹɺ͜ͷํఔࣜΛੵ෼ͯ͠ɺ൒

ର͢Δґଘੑ͔Βɺ্‫ه‬ͷ৔߹ͱಉ༷ʹΤϛολϯε
ͷมԽΛ‫ݕ‬౼͢Δɻԣํ޲ͷӡಈྔ Px ͸ɺ‫޲ํܘ‬ͷӡಈྔΛ‫ٻ‬ΊΔɻॳ‫ظ‬৚݅ͱͯ͠ɺz = 0 Ͱɺ

∂Px
∂Px
x+
z
∂x
∂z
∂ 2 Px+xz + · · · ,∂x∂zPx (x, s) = Px (0, 0) +Pr = 0 ͱ͠ɺ·ͨ൒‫޲ํܘ‬ͷཻࢠͷҐஔ͸ҰఆʹอͨΕΔ (r  const.) ͱ͠ɺRF ۭಎͷೖ‫͔ޱ‬Βग़‫·ޱ‬Ͱੵ෼͢Δͱɺ൒‫޲ํܘ‬ͷӡಈྔ Pr ͸ɺ

Pr =

erE0
cos ϕ1 ,2cͱల։͢Δ͜ͱ͕Ͱ͖Δͱ͢Δɻ͜ͷͱ͖ɺӈลୈ 3(4.77)

ͱͳΔɻ͜͜Ͱɺϕ1 ͸ۭಎೖ‫Ͱޱ‬ͷཻࢠͷҐ૬Ͱ͋Δɻ࣍ʹɺ͜ͷҐ૬ ϕ1 Λɺόϯνͷத৺ͰͷҐ૬ ϕ01ͱͦͷपΓͰͷඍখͳҐ૬ Δϕ Ͱɺ

ϕ1 = ϕ01 + Δϕ,(4.82)

߲͸ RF ి࣓৔ʹΑΔΩοΫΛද͠ɺୈ 4 ߲͸ऩଋൃࢄ࡞༻Λද͢ɻਤ 39 ʹɺҐ૬ۭؒதͰͷ RF ి࣓৔ʹΑΔΩοΫͱऩଋൃࢄ࡞༻ͷӨ‫ڹ‬Λࣔ͢ɻΩοΫʹΑΔӨ‫ڹ‬͸ɺεϥΠεʹΑͬͯҐ૬ۭؒͰͷ෼෍͕Px ํ޲ʹฏߦʹҠಈ͢Δ͜ͱͰ͋Δ͕ɺͦΕʹର͠
(4.78)

ͱల։Ͱ͖Δͱ͢Δɻ͜ͷͱ͖ɺඍখͳҐ૬͸ Δϕ −k(z − z0 ) = −ks ͱද͢͜ͱ͕Ͱ͖Δɻ͜ΕΛ༻͍ͯऩଋൃࢄ࡞༻ʹΑΔӨ‫ڹ‬͸ɺཻࢠͷҐஔ x ʹ΋ґଘ͢ΔɻΩοΫͷӨ‫ڹ‬͸ɺϏʔϜϥΠϯͷԼྲྀଆͰΩϟϯηϧՄೳͰ͋Δͷʹରͯ͠ɺऩଋൃࢄ࡞༻ʹΑΔӨ‫ڹ‬͸Ұൠతʹ͸ΩϟϯηϧͰ͖ͳ͍ͱ͍͏େ͖ͳҧ͍

Δͱɺόϯνத৺ͷཻࢠ (x, z0 ) ͱɺத৺͔Β s ͚͕ͩ͋ΔɻΩοΫͱऩଋൃࢄ࡞༻ʹΑΔΤϛολϯεͷ

཭Εཻͨࢠ (x, z) ͷӡಈྔͷࠩ͸ɺ૿େ͸ɺ

ΔPx = Px (x, z) − Pz (x, s0 ),(4.79)

ͱͳΔɻ͜͜Ͱɺ‫Ͱޙ‬ఆٛ͢Δԣํ޲ͷ rms ΤϛολϯεΛ x ํ޲ʹରͯ͠‫͢ࢉܭ‬ΔͨΊʹɺr Λ x ͰΔεkick=Δεf ocus=
1  ∂Px 
σx σ z ,mc  ∂z 
 2 
1  ∂ Px  2σ σz ,
mc  ∂s∂x  x(4.83)(4.84)

ஔ͖‫ͨ͑׵‬ɻࣜ (4.77) ΑΓɺ0 < ϕ0 < π/2 ͷͱ͖ӡͱද͢͜ͱ͕Ͱ͖Δɻ͜ΕΒΑΓɺRF ۭಎͰͷ‫֨ن‬ಈྔͷࠩ͸ԽΤϛολϯεͷ૿େ͸ɺ

eE0 k
ΔPx =
| sin ϕ01 |xs,2c(4.80)

ε2nx = ε20 + (Δεkick )2 + (Δεf ocus )2 ,(4.85)

ͱ‫͞ࢉܭ‬ΕΔɻ͜ΕΑΓɺx ํ޲ͷӡಈྔͷࠩ͸ɺԣͱ͢Δ͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺε0 ͸ RF ۭಎೖ‫Ͱޱ‬ํ޲ͷҐஔ x ͱɺόϯνத৺͔Βͷਐߦํ޲ͷҐஔͷΤϛολϯεͰ͋Δɻs ͷੵͰද͞ΕΔ͜ͱ͕Θ͔Δɻs ʹґଘ͢Δͱ͍͏͜ͷΑ͏ʹɺ͜͜Ͱ͸؆୯ͳϞσϧΛ༻͍ͯɺRF͜ͱ͸ɺ͢ͳΘͪόϯνεϥΠεʹΑͬͯɺൃࢄྗ͕

ి࣓৔ʹΑࣹͬͯӨΤϛολϯεͷ૿େ͕Ҿ͖‫͜͞ى‬

㸲㸫 

tail

bunched beamheadͨΊɺϨʔβʔͷύϧε෯͸ɺཁ‫͞ٻ‬ΕΔిՙྔɺόβϯν௕ɺΤϛολϯεʹ߹Θͤͯ࠷దͳ஋ʹ͠ͳ͚Ε͹ͳΒͳ͍ɻ

(a) kickPx(b) focusingPxిࢠॐͰੜ੒͞Εͨόϯν͸ɺۭؒిՙޮՌʹΑͬͯ‫૿ʹܹٸ‬େ͢ΔͨΊɺೖࣹ‫ث‬ͷग़‫͍୹Ͱޱ‬όϯν௕ͷϏʔϜΛಘΔʹ͸ɺిࢠॐͷ‫ޙ‬ΖʹόϯνϟʔۭಎΛઃஔͯ͠ɺ଎౓มௐʹΑͬͯόϯν௕Λ୹͘͢Δඞxx

ཁ͕͋Δɻ͜ͷΑ͏ͳόϯνϯάͷํ๏ΛυϦϑτό
ϯνϯά (drift bunching)ɺ͋Δ͍͸଎౓όϯνϯά
ͱ͍͏ɻ·ͨɺ΋͏Ұͭͷόϯνϯάͷํ๏ͱͯ͠ɺ
࣓৔όϯνϯά͕͋Δɻ͜Ε͸ɺ࣓৔தͰӡಈྔͷҟਤ 39: RF ి࣓৔ʹΑΔΩοΫͱൃࢄऩଋ࡞༻͕Ґ૬

ͳΔཻࢠ͸ҟͳΔ‫ي‬ಓΛऔΔͱ͍͏ੑ࣭Λ༻͍ͯΔɻ

ۭؒ෼෍ʹ‫͢΅ٴ‬Ө‫ڹ‬ɻ(a) ΩοΫʹΑΔεϥΠεͷ

όϯν಺ͷཻࢠʹద౰ͳӡಈྔͷࠩΛੜ͓ͤͯ͡͞

มԽɺ(b) ൃࢄऩଋ࡞༻ʹΑΔεϥΠεͷมԽɻ͚͹ɺద੾ͳ࣓৔தͷ‫ي‬ಓΛ௨ͯ͠΍Δ͜ͱʹΑΓɺ͋Δ஍఺Ͱͷόϯν௕Λ୹͘͢Δ͜ͱ͕ՄೳͱͳΔɻΕΔ͜ͱΛ‫͖ͨͯݟ‬ɻ࣮ࡍͷ ERL ೖࣹ‫Ͱث‬͸ɺෳ਺͜ͷͨΊʹ͸ɺӡಈྔࠩΛੜͤ͡͞ΔͨΊͷ RF ి࣓ͷ RF Ճ଎ۭಎ͕ಋೖ͞ΕΔͨΊɺ௿͍Τϛολϯ

৔ɺͦͯ͠ద੾ͳ‫ي‬ಓΛ‫ܗ‬੒͢ΔͨΊͷి࣓ੴ‫͕ܥ‬ඞεΛ࣋ͭϏʔϜΛੜ੒͢ΔͨΊʹ͸ɺ͜ΕΒͷҐ૬΍ཁͱͳΔɻՃ଎ޯ഑Λௐ੔͢Δඞཁ͕͋Δɻ͞Βʹɺ࣮ࡍͷՃ଎͜ͷ 2 ͭͷόϯνϯάͷํ๏Ͱ͸ɺͦΕͧΕ࣍ͷి৔ͷ෼෍͸ҰൠʹՃ଎ۭಎͷ‫ܗ‬ঢ়ʹґଘ͠ɺ·ۭͨΑ͏ͳಛ௃͕͋ΔɻυϦϑτόϯνϯάͷ৔߹͸ɺόؒిՙޮՌͷӨ‫ڹ‬΋ߟྀ͠ͳ͚Ε͹ͳΒͳ͍ͨΊɺ͜ϯν಺ͷ଎౓͕ࠩॏཁͰ͋ΔͨΊɺϏʔϜͷΤωϧ

ΕΒͷଟ਺ͷύϥϝλΛղੳతʹ࠷దԽ͢Δͷ͸೉͠Ϊʔ͕े෼ߴ͍৔߹ (β ∼ c) ͷ৔߹ʹ͸ɺे෼ͳ଎

͍ɻ͜ͷͨΊɺ௨ৗ͸ཻࢠτϥοΩϯάʹΑΔγϛϡ

౓͕ࠩಘΒΕͣɺ͏·͘‫ػ‬ೳͤ͞Δ͜ͱ͕ग़དྷͳ͍ɻ

Ϩʔγϣϯͷ‫܁‬Γฦ͠ʹΑΔύϥϝλͷ࠷దԽ͕ߦΘ͜ͷͨΊɺυϦϑτόϯνϯά͸ΤωϧΪʔͷ௿͍ྖΕΔɻҬͰ༻͍ΒΕΔɻ·ͨɺυϦϑτόϯνϯάͷ৔߹ʹ·ͨɺ௿ΤϛολϯεϏʔϜΛಘΔ্ͰɺRF ۭಎ͸ɺ଎౓มௐΛ༩͑Δόϯνϟʔۭಎ͕͋Ε͹Α͍ͷͷઃஔਫ਼౓΋ॏཁͰ͋Δɻઃஔ‫ʹࠩޡ‬ΑΓɺRF ి࣓Ͱɺͦͷ૷ஔ͸ίϯύΫτʹͳΔɻैͬͯɺϏʔϜͷ৔ͷத৺࣠ʹରͯ͠ɺϏʔϜ‫ي‬ಓͷΦϑηοτ΍܏͖ΤωϧΪʔ͕·ͩ௿͍ͱ͍͏͜ͱɺͦͯ͠খ‫ن‬໛ͳγ͕ੜ͡Δ͕ɺ͜ΕΒ͸Τϛολϯεͷ૿େΛҾ͖‫͜ى‬εςϜͰ͋Δͱ͍͏͜ͱͰɺೖࣹ‫Ͱث‬͸ओʹυϦϑτ

͢ɻ࣮ࡍͷ ERL ೖࣹ‫Ͱث‬͸ɺ͜ΕΒͷઃஔ‫ͭʹࠩޡ‬όϯνϯά͕༻͍ΒΕ͍ͯΔɻ࣓৔όϯνϯάͷ৔߹͍ͯ΋ߟྀ͠ͳ͚Ε͹ͳΒͳ͍ɻʹ͸ɺӡಈྔͷࠩʹΑΔ࣓৔தͰͷ‫ي‬ಓ௕ͷมԽΛར༻͢ΔͨΊɺυϦϑτόϯνϯάʹൺ΂Δͱେ‫ن‬໛ͳγεςϜͱͳΔɻ͔͠͠ɺ଎౓มௐΛར༻͍ͯ͠ͳ͍

4.5

όϯνϟʔʹΑΔόϯν௕ͷѹॖͨΊɺߴ͍ΤωϧΪʔͷϏʔϜͰ΋͜ͷํ๏͸༗ޮͰ

ERL ೖࣹ‫༻ث‬ͷޫӄ‫ࢠిۃ‬ॐ͔Β͸ɺύϧεঢ়ͷϨʔβʔΛ༻͍ͯɺόϯνԽ͞ΕͨϏʔϜ͕औΓग़͞ΕΔɻ୹͍όϯν௕ΛಘΔͨΊʹ͸ɺϨʔβʔͷύϧε෯Λ୹͘͢Ε͹ྑ͍͕ɺ࣮ࡍʹ͸ۭؒిՙޮՌ
͕͋ΔͨΊɺੜ੒௚‫ޙ‬ͷόϯν௕͸อଘ͞Εͳ͍ɻಛʹɺΧιʔυۙ๣ͷిࢠϏʔϜ͕ੜ੒͞Εͨ௚‫Ͱޙ‬͸ɺϏʔϜͷΤωϧΪʔ͕௿͍ͨΊʹɺۭؒిՙޮՌ
ʹΑΔόϯν௕͕Ҿ͖৳͹͞ΕΔޮՌ͕‫ݦ‬ஶʹ‫ݱ‬ΕΔɻಉ͡ిՙྔͷόϯνΛੜ੒͢Δ৔߹ɺϨʔβʔͷύϧεΛ୹͘͢Δͱɺిՙີ౓͕૿େ͢Δ͜ͱʹͳ͋ΔɻERL पճ෦Ͱ͸ɺόϯν௕Λѹॖ͢ΔͨΊʹ࣓৔όϯνϯά͕༻͍ΒΕΔɻӡಈྔࠩ͸ɺओՃ଎ۭ
ಎʹΑͬͯ༩͑ΒΕɺपճ෦ͷϏʔϜޫֶؔ਺Λ࠷ద
Խ͢Δ͜ͱʹΑΓɺૠೖޫ‫ݯ‬ͷҐஔͰόϯν௕Λ୹͘͢Δ͜ͱ͕ՄೳͰ͋Δɻ࣍ʹυϦϑτόϯνϯάͷ֓೦ʹ͍ͭͯɺ؆୯ͳϞσϧΛ༻͍ͯΛ঺հ͢ΔɻERL ೖࣹ‫Ͱث‬͸͋·Γ࢖ΘΕͳ͍࣓৔όϯνϯάʹ͍ͭͯ͸ɺଞͷจ‫ݙ‬Λࢀরͯ͠ཉ͍͠ɻυϦϑτόϯνϯάͰ͸ɺόϯνʹ଎౓มௐΛ༩͑ΔͨΊʹɺόϯνϟʔۭಎΛ௨աͤ͞Δɻ

ΓɺۭؒిՙޮՌ͕ΑΓ‫ݱ͘ڧ‬ΕΔ͜ͱʹͳΔɻ͜ͷ㸲㸫 

(2) buncher(1) entrance of buncher

RF fieldβ

deceratePz
drift

accelerate

accelerate(3) exit of buncher(4) after drift spaceslower ββfasterz
deceratedriftLਤ 40: RF ి࣓৔ʹΑΔόϯνϯάɻόϯνத৺ͰRF ి࣓৔͕θϩʹͳΔΑ͏ʹͯ͠ɺόϯνʹ଎౓มௐΛֻ͚Δɻόϯνલํ͸‫ݮ‬଎͞Εɺόϯν‫ํޙ‬͸Ճ଎͞ΕΔͨΊɺ͋Δ‫ ཭ڑ‬L ͚ͩࣗ༝ۭؒΛਐߦͨ͠‫ޙ‬ɺόϯν௕͸࠷খʹͳΔɻ

ΑΔόϯνϯάɻόϯνೖ‫Ͱޱ‬଎౓ࠩͷͳ͔ͬͨϏʔϜ͕ɺόϯνϟʔͰ଎౓มௐΛड͚ɺͦͷ෼෍͸܏͘ɻࣗ༝ۭؒͰ͸ Pz ͸มԽ͠ͳ͍͕ɺz ͸ϏʔϜ͕ਐߦ
͢Δʹै͍มԽ͢Δɻࣗ༝ۭؒΛ͋Δ‫཭ڑ‬ਐΉͱҐ૬ۭؒதͰͷ z ͷ෼෍͸࠷খʹͳΔɻ

όϯνϟʔ಺Ͱͷ RF ి࣓৔ΛEz = E0 sin(ωt),ਤ 41: ਐߦํ޲ͷҐ૬ۭؒ (z, Pz ) Ͱͷ RF ి࣓৔ʹ

(4.86)

ͷύϥϝλΛ࠷దԽ͢Δඞཁ͕͋Δɻ

ͱ͢Δͱɺਤ 40 ʹࣔ͢Α͏ʹɺωt = 0 ͱ͖ʹόϯνͷத৺͕௨ա͢ΔΑ͏ʹ͢Δɻ͜ͷͱ͖ɺόϯνத৺ΑΓલʹ͍Δཻࢠ͸‫ݮ‬଎͞Εɺ‫ޙ‬Ζʹ͍Δཻࢠ͸Ճ଎͞ΕΔɻҐ૬ ωt ͕θϩۙ๣ʹ͋Δͱ͢ΔͱɺEz ͸࣍ͷΑ͏ʹEz ∼ E0 ωt,(4.87)

ͱɺωt ͷ 1 ࣍ͷؔ਺ͱͯ͠ల։͢Δ͜ͱ͕Ͱ͖Δɻόϯνϟʔۭಎ௨ա‫ޙ‬͸ɺਤ 41 ʹࣔ͢Α͏ʹɺ(z, Pz )Ґ૬ۭؒதͰɺ෼෍͕܏͘͜ͱʹͳΔɻࣗ༝ۭؒΛ௨
ա͢Δͱ͖͸ɺPz ͸มԽͤͣɺz ͷΈ͕ Pz ʹґଘͯ͠มԽ͢Δɻద౰ͳ‫཭ڑ‬ਐߦ͢Δͱɺਤ 41 ʹࣔ͢Α͏ʹɺ෼෍ͷۭؒత֦͕Γɺͭ·Γόϯν௕͸࠷খͱͳΔɻ͜ͷΑ͏ʹɺυϦϑτόϯνϯάʹΑΓɺόϯν௕Λѹॖ͢Δ͜ͱ͕ՄೳͰ͋ΔɻҎ্ͷϞσϧͰ͸ɺۭಎ಺Ͱͷి৔͕ E0 ͰҰ༷Ͱ
͋Δͱͯ͠؆୯Խ͕ͨ͠ɺ‫ʹ࣮ݱ‬͸ɺRF ి࣓৔ͷඇઢ‫ܗ‬ੑ΍ɺ૬ର࿦తͳޮՌʹΑΓɺ଎౓มௐ͸ඇઢ‫ܗ‬
ͱͳΔɻ͜ͷͨΊɺࣗ༝ۭؒΛ࠷దͳ‫཭ڑ‬ਐΜͩ‫Ͱޙ‬΋ɺόϯν௕͸༗‫·ཹʹݶ‬Γɺθϩͱ͸ͳΒͳ͍ɻ·ͨɺ্Ͱઆ໌ͨ͠Α͏ʹɺۭؒిՙޮՌʹΑΔόϯν
௕ΛҾ͖৳͹͢ޮՌ͕͋ΔͨΊɺ‫࣮ݱ‬ͷೖࣹ‫Ͱث‬ͷόϯν௕ͷѹॖ͸ෳࡶʹͳΓɺղੳతʹόϯν௕ͷ࠷খ஋Λ༧ଌ͢Δͷ͸೉͍͠ɻ͜ͷͨΊɺೖࣹ‫ث‬ͷઃ‫Ͱܭ‬͸ɺۭؒిՙޮՌΛ‫ؚ‬Μͩ਺஋‫ࢉܭ‬ίʔυΛ༻͍ͯɺՃ଎ి৔ͷ‫͞ڧ‬΍Ґ૬ɺՃ଎ۭಎؒͷ‫Ͳͳ཭ڑ‬ͷଟ਺㸲㸫 ғ·Ͱ͜ͷྖҬʹೖΔͱ͢Δ͔ʹ͍ͭͯ͸ɺ‫Ͱޙ‬ಋೖۭؒిՙޮՌ

5

͢Δ Debey ௕͕ͦͷࢦ਑ͱͳΔɻۭؒతʹ‫׈‬Β͔ͳ͜͜Ͱ͸ɺۭؒిՙޮՌͷӨ‫঺͍ͯͭʹڹ‬հ͢Δɻ

·ͣ͸ɺର৅ͱ͢ΔϏʔϜ͕ۭؒిՙޮՌ͕ॏཁͳ
ྖҬʹ͋Δ͔ɺ͋Δ͍͸ΤϛολϯεʹΑΔޮՌ͕
ॏཁͳྖҬ͋Δ͔Λ൑ఆ͢ΔͨΊͷύϥϝλͱͯ͠ɺ

Debye ௕Λಋೖ͢Δɻͦͷ‫ޙ‬ɺۭؒిՙޮՌʹΑΔྗ͕ۭؒతʹ‫׈‬Β͔Ͱ͋Δ৔߹ʹ͍ͭͯɺηΫγϣϯ
4.1 Ͱ‫ٻ‬ΊͨϏʔϜΤϯϕϩʔϓํఔࣜΛ֦ு͢Δɻ

ϙςϯγϟϧ͸ɺిՙີ౓෼෍ ρ(r, t) ͔Βɺρ(r, t)φs (r, t) =dV
4π0 |r − r  |(5.4)ͱ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δɻଟ͘ͷۭؒిՙ‫ࢉܭ‬ίʔυͰ͸ɺۭؒతʹ‫׈‬Β͔ͳྗͷΈΛऔΓೖΕۭͯؒిՙޮՌΛ‫͍ͯ͠ࢉܭ‬Δ͕ɺ৔߹ʹΑͬͯ͸͜ͷԾఆ͕੒ΓཱͨͣɺিಥྗʹΑΔޮՌ͕ࢧ഑తʹͳΔ৔߹΋ࡏΓಘΔɻਖ਼ۭؒ͘͠ి

5.1

ՙ‫ࢉܭ‬ίʔυΛ࢖༻͢ΔͨΊʹ͸ɺର৅ͱ͢ΔϏʔۭؒిՙޮՌͷ෼ྨϏʔϜ͸ՙిཻࢠ͕ूஂͰ΄΅ಉ͡଎౓Ͱӡಈ͢
Δঢ়ଶͰ͋Δɻՙిཻࢠؒʹ͸౰વͳ͕Β Coulomnb
૬‫͕༻࡞ޓ‬ಇ͖ɺಉ͡ిՙූ߸ͷཻࢠؒͰ͸ੱྗ͕ಇ͖ɺҧ͏ిՙූ߸ͷཻࢠؒͰ͸Ҿྗ͕ಇ͘ɻϏʔϜͷΑ͏ʹඇৗʹଟ͘ͷՙిཻࢠΛ‫ؚ‬Ή৔߹ʹ͸ɺ཭Εͨ‫͋ʹ཭ڑ‬Δཻࢠूஂ͔Βͷ૬‫༻࡞ޓ‬ͷશମ͸ۭؒతʹ‫׈‬Β͔ͳྗͱͯ͠ߟ͑Δ͜ͱ͕Ͱ͖Δ͕ɺ͜ͷଞʹɺۙ઀ཻࢠཻͨ͠ࢠ͕͍Δ৔߹ʹ͸িಥతͳྗ΋ߟ͑Δ
ඞཁ͕͋Δɻ͜ͷঢ়‫گ‬Λ΋͏গ͠ৄ͘͠આ໌͢Δɻ·ͣɺϏʔϜͷ੩ࢭ‫Ͱܥ‬ϏʔϜ಺ͷཻࢠͷ଎౓͕ࠩඇ૬
ର࿦తͰ͋Δͱ͢Δɻ·ͨɺϏʔϜ͸ N ‫ݸ‬ͷཻࢠ͔ΒͳΓɺཻ֤ࢠ͸ిՙ q Λ࣋ͭͱ͢Δɻ͜ͷͱ͖ɺཻࢠ j ͱཻࢠ i ͷ‫཭ڑ‬Λ r ij = r i − r j ͱ͢Δͱɺཻࢠj ཻ͕ࢠ i ʹ‫ ͢΅ٴ‬Coulomb ྗ͸ɺ
q 2 r ij
F ij =
34π0 rij͋Δɻ͜͜Ͱ͸ɺϓϥζϚ෺ཧͰͷ‫ج‬ຊύϥϝλͰ͋Δ

Debye ௕ͷ֓೦ΛిࢠϏʔϜʹద༻͠ɺۭؒిՙޮ
Ռ͕‫׈‬Β͔ͳྗͱͯ͠औΓѻ͑Δ͔ɺ͋Δ͍͸িಥతͳྗ΋ߟྀ͢Δඞཁ͕͋Δ͔Λ෼ྨ͢Δɻ·ͨɺۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜͷγϛϡϨʔγϣϯͰ͸ɺଟ͘ͷ৔߹ϚΫϩཻࢠΛ࢖ཻͬͨࢠτϥοΩϯά‫͕ࢉܭ‬ߦΘΕΔ͕ɺϚΫϩཻࢠͷ਺͕Ͳͷఔ౓ඞཁ͔Λ೺Ѳ͢ΔͨΊʹ΋ɺDebye ௕ͷߟ͑͸༗ޮͰ͋Δɻ͜͜Ͱ͸ɺ·ͣதੑϓϥζϚதͰͷ Debye ःณͷޮՌΛ‫ݟ‬ͨ‫ʹޙ‬ɺ૬ର࿦తϏʔϜͰͷ Debye ௕ɺϓϥζϚৼಈ਺Λಋೖ͢Δɻ࠷‫ʹޙ‬ɺDebye ௕Λར༻ͯ͠ɺిࢠ
ϏʔϜͷۭؒిՙޮՌΛ෼ྨ͢Δɻ
·ͣɺதੑͳϓϥζϚʹରͯ͠ͷ Debey ௕Λಋೖ

(5.1)

͠ɺͦͷޮՌΛ঺հ͢ΔɻҰՁͷਖ਼ͷΠΦϯͱిࢠ͔ΒͳΔதੑͳϓϥζϚΛߟ͑ΔɻϓϥζϚ͸Թ౓ TͰฏߧঢ়ଶʹ͋Δͱ͢Δɻ؆୯ͷͨΊʹɺ‫࠲ٿ‬ඪͰߟͱͳΓɺཻࢠ i ʹର͢Δશ Coulomb ྗ͸ɺN
q 2 r ijFi =3
4π0 rijϜʹ͓͍ͯɺͲͪΒͷྗ͕ࢧ഑తͳͷ͔Λ஌Δඞཁ͕

͑ɺϓϥζϚͷத৺ (r = 0) ʹςετిՙ q ͕ஔ͔Ε

(5.2)

ͨͱ͢Δɻ͜ͷͱ͖ɺిՙ෼෍͸தੑ͔Β֎Εɺϙςϯγϟϧ φ(r) ͕ੜ͡ΔɻΠΦϯͱిࢠͷ෼෍͸ɺ೤j=iͱද͞ΕΔɻCoulomb ྗͷੑ࣭ͱͯ͠ɺ͋Δఔ౓ཻࢠi ͔Β‫཭͕཭ڑ‬Εཻͨࢠ͔Βͷྗ΋ɺF i ʹ‫د‬༩͢Δɻ͜ΕΒཻࢠ͔Βͷଟ਺ͷখ͍͞ྗ͕ू·Δ͜ͱͰɺۭ

ฏߧঢ়ଶͱԾఆͨ͠ͷͰɺMaxwell ෼෍Ͱද͞Εɺ


qφ(r)
(5.5)
n(r) = n0 exp −kB Tؒతʹ‫׈‬Β͔ͳϙςϯγϟϧ φs (r, t) ͕‫ܗ‬੒͞ΕΔͱ

ͱͳΔɻ͜͜Ͱɺn0 = n(0) Ͱ͋Δɻ͜ͷϙςϯγϟ

Έͳ͢͜ͱ͕Ͱ͖Δɻͨͩ͠ɺཻࢠ i ʹ‫ۃ‬Ί͍ͯۙ‫ڑ‬ϧ͸ɺΠΦϯͱిࢠ͔Β࡞ΒΕΔిՙ෼෍ͷͣΕʹ཭ʹ΋‫͕͋ࢠཻ͔ͭز‬Δ͜ͱ͕ߟ͑ΒΕɺ͜ΕΒͷྗΑͬͯੜ͡ɺPoisson ํఔࣜq∇2 φ(r) = − ni (r) − ne (r)
0

q
qφ(r)
= − n0 exp −
0
kB T

qφ(r)
− exp
kB T

͸ཻࢠ i ͱͷҐஔؔ܎ʹ‫͘ڧ‬ґଘͨ͠িಥతͳྗͱͳ
Δɻ͜͜Ͱɺ‫ࣜܗ‬తʹશ Coulomb ྗΛۭؒతʹ‫׈‬Β͔ͳ߲ͱিಥతͳ߲ʹ෼͚ΔͱɺFi =q 2  r ij∂
φs (r, t) +
3
∂r
4π0
rij

(5.3)

(5.6)

ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺӈลୈ 2 ߲͸ཻࢠ i ʹʹΑͬͯ‫͞ࢉܭ‬ΕΔɻ͜͜Ͱɺӈลୈ 1 ߲͸ΠΦϯʹ

‫ۃ‬Ί͍ͯۙҐஔʹ͋Δཻࢠͷ࿨ͱ͢ΔɻͲͷఔ౓ͷൣΑΔ߲ɺୈ 2 ߲͸ిࢠʹΑΔ߲Ͱ͋Δɻqφ(r)  kB T㸲㸫 ͱͯ͠ɺࢦ਺ؔ਺Λల։͠ɺ1 ࣍ͷ߲ͷΈΛ࢒͢ͱɺ
φ(r)q 2qφ(r)
2
=2 2
(5.7)
∇ φ(r) = 0 kB T
λD
ͱͳΔɻ͜͜ͰɺλD ͸ඇ૬ର࿦తͳ৔߹ͷ Debye ௕
Ͱ͋Γɺ

λD =

0 k B T
q 2 n02(5.8)

ͱఆٛ͞ΕΔɻ·ͨɺ೤ฏߧঢ়ଶͰͷ rms ଎౓ v¯x =(kB T /m)1/2 ͱɺϓϥζϚप೾਺ ωp = (q 2 n/0 m)1/2
Λ༻͍ͯɺ
v¯x
(5.9)
λD =
ωp
ͱఆٛ͢Δ͜ͱ΋Ͱ͖Δɻࣜ (5.7) ͷղ͸ɺ√2r
q 1
φ(r) =
exp −4π0 rλD෼ྨΛྨਪ͢Δ͜ͱ͕Ͱ͖ΔɻిࢠϏʔϜͷ৔߹ʹ͸ɺҟछిՙʹΑΔःณͷޮՌ͸ͳ͍͕ɺཻࢠؒͷ૬‫ۭ͕ؒ༻࡞ޓ‬తʹ‫׈‬Β͔Ͱ͋Δ͔ɺ͋Δ͍͸িಥతͰ͋Δ͔Λࣔ͢ͷʹ Debye ௕Λ࢖͏͜ͱ͕Ͱ͖Δɻ࣍ʹ૬ର࿦తͳ଎౓Λ࣋ͭిࢠϏʔϜʹରͯ͠ͷϓϥζϚप೾਺ͱ Debye ௕Λఆٛ͢ΔɻηΫγϣϯ

3.2 Ͱࣔͨ͠Α͏ʹɺ૬ର࿦తͳ଎౓Λ࣋ͭిࢠϏʔϜͷԣํ޲ͷӡಈํఔࣜ͸ɺẍ = ωp2 /2 = Fs /γm ͱॻ͚Δɻ͜͜ͰɺϏʔϜ͸ԁ౵அ໘Λ࣋ͭ࿈ଓϏʔϜͱ͢ΔͱɺۭؒిՙޮՌʹΑΔྗ͸ࣜ (3.12) ΑΓɺ

Fs = q 2 nr/(20 γ 2 ) ͱද͞ΕΔɻ͜ΕΑΓ૬ର࿦తͳ
৔߹ͷϓϥζϚप೾਺Λ

(5.10)

ͱͳΔɻr  λd ͷྖҬͰ͸ɺࢦ਺ؔ਺͸΄΅ 1 ͱͳ

q2 n
0 γ 3 m

ωp =(5.12)

ͱఆٛ͢Δ͜ͱ͕Ͱ͖Δɻ͜͜Ͱఆٛͨ͠ϓϥζϚप೾਺Λ༻͍ͯɺ૬ର࿦తͳ৔߹ͷ Debye ௕ΛΓɺϙςϯγϟϧ͸

q 1
φ(r) =
4π0 r

1/2

λD =(5.11)v¯x=ωp

0 mγ 3 v¯x2
q2 n

1/2

(5.13)

ͱͳΓɺਅۭதͰͷిՙ q ͕࡞Δϙςϯγϟϧͱ౳͠

ͱఆٛ͢Δɻฏ‫ۉ‬଎౓ v¯x ͸ɺϏʔϜ͕࣮‫ܥࣨݧ‬ͷԹ͘ͳΔɻ͜ͷ͜ͱ͸ɺDebye ௕ΑΓཻࢠؒͷ‫͕཭ڑ‬౓Ͱ೤ฏߧঢ়ଶʹ͋Δͱ͢Δͱɺγmv¯x 2 = kB T ͷؔ୹͍৔߹ (r  λ) ʹ͸ిՙ q ͕࡞Δϙςϯγϟϧʹ

܎͔Βɺv¯x = (kB T /γm)1/2 ͱ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ

౳͘͠ͳΔ͕ɺDebye ௕ΑΓཻࢠؒͷ‫͕཭ڑ‬௕͍৔͜ΕΛ্ͷࣜʹ୅ೖ͢Δͱɺ૬ର࿦తͳ৔߹ͷ Debye

λD ) ʹ͸ϙςϯγϟϧ͸ࢦ਺ؔ਺తʹ‫ݮ‬গ͢Δͱ͍͏͜ͱΛ͍ࣔͯ͠Δɻਤ 42 ʹࣔ͢Α͏ʹɺ

௕͸ɺ

߹ (r

Debye ௕ΑΓ֎ଆͰ͸ɺདͷిՙͷͱ͖ͱൺ΂ͯి৔͕ःณ͞Ε͍ͯΔɻ͜Ε͕ϓϥζϚதͰͷ Debye ः
ณͰ͋Δɻ

λD =

0 γ 2 kB T
q2 n

1/2

(5.14)

ͱͳΔɻ࠷‫ʹޙ‬ɺ͜͜Ͱఆٛͨ͠૬ର࿦తͳϏʔϜʹର͢Δ

Debye ௕Λ༻͍ͯɺϏʔϜ಺ͰͷۭؒిՙޮՌͷӨ‫ڹ‬Λ෼ྨ͢Δɻ·ͣ͸ɺϏʔϜ಺ͷཻࢠؒฏ‫཭ڑۉ‬Λ lp
ͱͯ͠ɺλDlp ͷ৔߹Λߟ͑Δɻ͜ͷ৔߹ɺDebye௕ͷ಺ଆʹ͸े෼ʹଟ͘ͷཻࢠ͕‫·ؚ‬ΕΔ͜ͱʹͳφ (r/λD)Γɺిՙ෼෍͋Δ͍͸ͦΕ͕࡞Δి࣓৔͸े෼ʹۭؒ
Shielded potentialతʹ‫׈‬Β͔Ͱ͋Δͱߟ͑ΒΕΔɻϏʔϜͷ൒‫ܘ‬Λ a ͱ͢ΔͱɺDebye ௕ʹର͢Δେ͖͞ͷൺ͔Βɺ࣍ͷΑ͏

Coulomb potentialͳ 3 ͭͷྖҬ͕ߟ͑ΒΕΔɻ

0

1

2

1. λD
a: িಥతͳྗ͕‫͍ڧ‬ɺΤϛολϯε͕ࢧ
഑తͳྖҬ

3

r/λD

2. λD  a: ूஂతͳޮՌʹΑΔۭؒతʹ‫׈‬Β͔ͳྗ͕ࢧ഑తͳྖҬਤ 42: ϓϥζϚதͰͷ Debye ःณͷޮՌɻDebye ௕ΑΓ֎ଆͰ͸ɺϙςϯγϟϧ͕དͷిՙͷͱ͖ʹൺ΂

3. λD ∼ a: িಥతͳྗɺۭؒతʹ‫׈‬Β͔ͳྗͷ྆ํ͕ॏཁͯখ͘͞ͳΔɻిࢠϏʔϜ͸ඇதੑϓϥζϚͱߟ͑Δ͜ͱ͕Ͱ͖ɺ
͜ΕΒͷٞ࿦͔Β Debye ௕ʹΑΔཻࢠؒ૬‫༻࡞ޓ‬ͷ

1. ͷ λDa ͷ৔߹͸ɺϏʔϜΤωϧΪʔ͕ߴ͍ɺ͋
Δ͍͸ϏʔϜͷԹ౓͕े෼ߴ͍৔߹ʹ‫͖ى‬ΔɻϏʔϜ㸲㸫 ಺ͷཻࢠ͸ Debye ௕Ҏ಺ʹऩ·ΔͨΊɺϏʔϜͷूஂతͳޮՌʹΑΔۭؒతʹ‫׈‬Β͔ͳྗΑΓ΋ɺ਺ଟ͘ͷཻࢠ͔Βड͚Δিಥతͳྗ͕ࢧ഑తͱͳΔɻ·ͨɺিಥతͳྗʹΑΓϏʔϜ಺ͷཻࢠؒͰΤωϧΪʔͷ΍Γ

observerऔΓ͕े෼ʹߦΘΕΔͨΊɺϏʔϜ͸೤ฏߧঢ়ଶʹୡ

Debye length͠ɺͦͷ෼෍͸ Maxwell-Bolzmann ෼෍Ͱද͞ΕΔɻ
͜ͷΑ͏ͳঢ়‫Ͱگ‬͸ɺ౷‫ֶ͔ྗܭ‬ΒͷྨਪʹΑΓɺ‫ݸ‬ผͷཻࢠؒͷ૬‫༻࡞ޓ‬͸ແࢹ͢Δ͜ͱ͕Ͱ͖ɺϏʔϜ
Λ 6 ࣍‫ݩ‬ͷҐ૬ۭؒ (r, P ) Λ࣋ͭ୯ཻࢠͷूஂͱͯ͠ѻ͏͜ͱ͕Ͱ͖Δɻ͜ͷ৔߹ʹ͸ɺϏʔϜͷ࣋ͭΤϛολϯε͕ॏཁͳ໾ׂΛՌͨ͢ɻ࣍ʹɺ2. ͷ λD  a ͷ৔߹͸ɺϏʔϜΛߏ੒͢Δ
ཻࢠͷଟ͘͸ Debye ௕ͷ֎ʹ͋ΔͨΊɺۭؒతʹ‫׈‬Β͔ͳྗ͕ࢧ഑తͱͳΔɻ͜ͷ৔߹ʹ͸ɺ‫ݸ‬ผͷཻࢠਤ 43: λD ∼ lp ͷ৔߹ͷ૬‫༻࡞ޓ‬ɻண໨͢Δཻࢠͷ‫ۃ‬ۙ๣ʹ͍Δগ਺ͷཻࢠ͔Βͷྗ͕ࢧ഑తͱͳΔɻ

5.2

ۭؒిՙޮՌͷӨ‫ڹ‬ؒͰͷ૬‫༻࡞ޓ‬ΑΓɺϏʔϜͷूஂతͳޮՌʹΑΔྗ

ԣํ޲Ͱ͸ɺηΫγϣϯ 3.3 Ͱ‫ͨݟ‬Α͏ʹɺిࢠϏʔ͕ॏཁͱͳΔɻۭؒిՙޮՌʹΑΔྗ͸ۭؒతʹ‫׈‬ΒϜͷαΠζΛ޿͛Δํ޲ʹɺͭ·Γੱྗͱۭͯؒ͠ి

͔Ͱ͋ΔͱΈͳ͢͜ͱ͕Ͱ͖ɺ֎͔ΒՃ͑ͨྗͱಉ༷

ՙޮՌ͕࡞༻͢Δɻ·ͨɺਐߦํ޲Ͱ΋ಉ༷ͷੱྗޮʹѻ͏͜ͱ͕Ͱ͖Δɻ͜ͷͨΊɺཻࢠͷӡಈ͸ɺ֎৔ՌΛ༩͑Δ͕ɺ΋͏ҰͭॏཁͳޮՌͱͯ͠ɺϏʔϜʹதͰͷ୯ཻࢠͷ Hamiltonian ʹΑͬͯ‫ه‬ड़͞ΕΔ͜ͱ

ΤωϧΪʔ֦͕ΓΛ༩͑Δͱ͍͏͜ͱ͕͋ΔɻϏʔϜ͕ՄೳͰ͋Γɺ·ͨɺͦͷ 6 ࣍‫ݩ‬Ґ૬ۭؒ (r, P ) தͷΤωϧΪʔ֦͕Γ͕มԽ͢Δ৔߹ɺภ޲ి࣓ੴ͔ΒͰͷ Liouville ͷఆཧ΋੒Γཱͭͱߟ͑Δ͜ͱ͕Ͱ͖ߏ੒͞ΕΔ߹ྲྀ෦ͳͲͷ෼ࢄؔ਺͋Δ෦෼Ͱ͸Ϗʔ

ΔɻηΫγϣϯ 5.3 Ͱߟ͑ΔɺϏʔϜ಺Ͱ‫ي‬ಓ͕ަࠩϜαΠζͷ૿େΛҾ͖‫͢͜ى‬ɻ·ͨɺόϯνԽ͞Εͨ͠ͳ͍ϏʔϜ (laminar beam) ͷ৔߹ʹ͸ɺۭؒిՙ

ϏʔϜͰ͸ɺόϯνεϥΠεͷҐஔʹґଘͨ͠ΤωϧޮՌ͸Ґஔͷઢ‫͢ͱ਺ؔͳܗ‬Δ͜ͱ͕Ͱ͖Δɻ͜ͷΑΪʔ֦͕Γ͕ੜ͡ΔͨΊɺࣹӨΤϛολϯεͷ૿େΛ͏ͳ৔߹ʹ͸ɺrms Τϛολϯε΋อଘ͢ΔɻҾ͖‫͢͜ى‬ɻ͜ͷӨ‫͍ͯͭʹڹ‬͸ɺηΫγϣϯ 6.2 Ͱ

3. ͷ৔߹͸ɺۭؒతʹ‫׈‬Β͔ͳྗ͚ͩͰͳ͘ɺি
ಥతͳྗ΋ߟྀ͢Δඞཁ͕͋ΓɺऔΓѻ͍͸ෳࡶʹͳ
Δɻিಥతͳྗ͸ཻࢠ෼෍ΛมԽͤ͞ΔͨΊɺۭؒి঺հ͢Δɻ

ՙʹΑΔྗ΋ඇઢ‫ͳͱܗ‬Δɻ͔͠͠ɺ্‫ه‬ͷ 2 ͭͷ৔

5.3

߹ͱಉ༷ʹɺ6 ࣍‫ݩ‬Ґ૬ۭؒ (r, P ) Ͱͷཻࢠ෼෍ͱ͠
ͯɺͦͷӡಈΛද‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δɻ

ۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜΤϯϕϩʔϓํఔࣜ

ۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜΤϯϕϩʔϓํఔࣜΛ࠷‫ʹޙ‬ɺλD ∼ lp ͷ৔߹Λߟ͑Δɻ͜ͷ৔߹ɺDebye௕Ҏ಺ʹೖΔཻࢠͷ਺͸‫ݶ‬ΒΕͯ͘Δɻ͜ͷΑ͏ͳঢ়‫ٻ‬ΊΔͨΊʹɺ͜͜Ͱ͸࿈ଓϏʔϜΛߟ͑ɺ࣍ͷΑ͏ͳԾఆΛಋೖ͢Δɻ‫گ‬͸ɺϏʔϜͷԹ౓͕‫ʹ୺ۃ‬௿͍৔߹͔ɺཻࢠີ౓͕
ߴ͍৔߹ʹੜ͡Δɻ͋ΔཻࢠΛߟ͑ΔͱɺͦͷपΓʹ͍Δগ਺ͷඇৗʹۙ઀ཻͨ͠ࢠ͔Βͷྗ͸ɺଞͷ཭Εͨଟ਺ͷཻࢠ͔Βͷྗʹൺ΂ͯɺ‫ͳʹͱ͍͜ڧ‬Δɻ
͢ͳΘͪɺۙ઀ͨ͠গ਺ͷཻࢠͱͷॠؒతͳ૬‫༻࡞ޓ‬ʢিಥʣ͕ॏཁͱͳΔɻ͜ͷΑ͏ͳঢ়‫گ‬Λਤ 43 ʹࣔͨ͠ɻΑ͘࢖༻͞ΕΔۭؒిՙ‫ࢉܭ‬ίʔυͰ͸ɺিಥతͳ૬‫༻࡞ޓ‬͸ߟྀ͞Ε͍ͯͳ͍ͨΊɺϏʔϜ͕͜ͷྖҬʹ͋Δ৔߹ʹ͸ɺ஫ҙ͕ඞཁͰ͋Δɻ

1. ϏʔϜ͸൒‫ ܘ‬a ͷԁ‫ܗ‬அ໘Λ࣋ͪɺ൒‫ ܘ‬b ͷಉ৺ԁυϦϑτ؅ͷதΛਐߦ͢Δɻ

2. ϏʔϜ্࣠ͱυϦϑτ؅ͷน໘ͷؒͰͷϙςϯγϟϧͷࠩ͸ɺӡಈΤωϧΪʔʹൺ΂ͯे෼খ͍͞ɻ

3. ಉ৺ԁυϦϑτ؅น໘Ͱͷ‫ి૾ڸ‬ՙ͕ϏʔϜʹ‫ٴ‬΅͢ޮՌ͸ແࢹ͢Δɻ
4. ϏʔϜͷཻࢠີ౓͸ɺϏʔϜ಺Ͱ͸Ұ༷ɺϏʔϜ֎Ͱ͸θϩͱ͢Δɻ2. ΑΓϏʔϜ಺ͷཻࢠͷਐߦํ޲ͷ଎౓͸શͯಉ͡ͱ͢Δ͜ͱ͕Ͱ͖ɺ͜ͷͱ͖ϏʔϜ಺ͷిྲྀີ౓͸Ұ༷Ͱ͋ΔͱͰ͖Δɻ㸲㸫 

5. શͯͷϏʔϜཻࢠ͸‫ي‬ಓ্ʹԊͬͯӡಈ͠ɺͦͷ‫ي‬ಓ͸ަࠩ͠ͳ͍ɻ

ͱ‫͞ࢉܭ‬ΕΔɻ࣍ʹɺ࣓৔͸ɺAmpere ͷ๏ଇΑΓԁपํ޲੒෼͚ͩʹͳΓɺ

6. ϏʔϜ͸҆ఆঢ়ଶʹ͋Δͱ͢Δ (∂/∂t = 0)ɻϏʔϜਐߦํ޲ʹԊͬͨ͋ΒΏΔҐஔͰɺϏʔϜஅ໘͸࣌ؒʹґଘͯ͠มԽ͠ͳ͍ɻ

Bθ (r)=Bθ (r)=Ir, (r ≤ a),
2πa2
I
μ0
, (r > a)2πr
μ0(5.21)(5.22)

7. ཻࢠͷ‫ي‬ಓ͸ɺۙ࣠ޫઢۙࣅʹै͏ʢr ͸খ͍͞ʣͱ‫͞ࢉܭ‬ΕΔɻϏʔϜͷిՙີ౓ ρ ͱిྲྀີ౓ J ͸ɺͦͷ఺ͰͷΔϙςϯγϟϧΛ‫͢ࢉܭ‬ΔɻਅۭνΣϯόʔน໘্(r = b) Ͱϙςϯγϟϧ͕θϩ (φ = 0) ͱ͢ΔͱɺϏʔϜͷ଎౓Λ v ͱ͢Δͱɺ࿈ଓͷࣜΑΓɺ

J = ρv

ࣜ (5.19)ɺ(5.20) Λੵ෼͢Δ͜ͱʹΑΓɺి৔͕࡞

(5.15)

φ(r) =

Vs 1 + 2 logͱɺిྲྀ͸ɺ

φ(r)(5.16)=

2Vs logbr

Vs =ՙʹΑΓɺϏʔϜ্࣠ (r = 0) ͱϏʔϜͷ୺ (r = a)ɺ
ͷ͕ࠩੜ͡ΔɻϏʔϜͷӡಈΤωϧΪʔ͕ҰఆͰ͋Δ
ͱԾఆ͢Δͱɺ্࣠ͷཻࢠͷӡಈΤωϧΪʔ͸ɺϏʔϜͷ୺ʹ͍ΔཻࢠΑΓ΋খ͘͞ͳΔɻैͬͯɺཻࢠͷ଎౓͸ r ͷؔ਺ v(r) ͱͳΔɻҰൠʹɺρɺv ɺJ ͷ͏ͪҰ͕ͭ༩͑ΒΕΕ͹ɺଞͷύϥϝλ͸ Maxwell ํఔࣜͱӡಈํఔࣜʹΑΓηϧϑίϯγεςϯτʹܾΓɺr ʹґଘ͠ͳ͍ͱ͢Δɻۙ࣠ޫઢۙࣅͷԼͰ͸ɺ

vr  v ɺvθ  v ɺvz  v Ͱ͋ΔͷͰɺr ํ޲ͷϙς

ρ0 a2
I
=
40
4π0 βc(5.25)

࣍ʹɺϏʔϜ͕࡞Δి࣓৔தͰͷӡಈํఔࣜΛߟ͑Δɻr ํ޲ͷӡಈํఔࣜ͸ɺࣜ (4.9) ͷதͰɺ֎෦ి
࣓৔ʹΑΔՃ‫ݮ‬଎͕ͳ͘ (γ = const.)ɺ·ͨɺԁपํ޲ͷ଎౓ rθ̇ ͕े෼খ͍͞ͱͯ͠ɺqrθ̇Bz ͷ߲Λແࢹ͢Δͱɺd(γmṙ) = mγ r̈ = qEr − q żBθdt

(5.26)

ͱͳΔɻ͜ΕʹɺϏʔϜ͕࡞Δి࣓৔ɺࣜ (5.19) ͷEr ͱࣜ (5.21) ͷ Bθ Λ୅ೖ͢Δͱɺ

ϯγϟϧͷมԽ͸ཻࢠͷӡಈΤωϧΪʔʹൺ΂ͯখ͍͞ͱߟ͑Δ͜ͱ͕Ͱ͖ΔͷͰɺͦΕ΄Ͳେ͖ͳࠩΛ༩

(5.24)

Ͱ͋Δɻ

ఆ͞ΕΔ͕ɺ͜͜ͰͷϞσϧͰ͸ٞ࿦Λ؆୯ʹ͢ΔͨΊʹɺϏʔϜஅ໘্Ͱ͸ ρɺv ɺJ ͸શͯҰ༷Ͱ͋(a ≤ r)ͱͳΔɻ͜͜Ͱɺ

ͱ‫ه‬ड़͞ΕΔɻ͜͜Ͱɺvz  v ͱԾఆ͍ͯ͠ΔɻۭؒిਅۭνΣϯόʔͷน໘্ (r = b) ্Ͱ͸ϙςϯγϟϧ

(0 ≤ r ≤ a),(5.23)

ͱؔ܎෇͚ΒΕΔɻ·ͨɺిՙͷઢີ౓Λ ρL ͱ͢Δ

I = ρL v

b
r2
− 2
a a

mγ r̈ =

qIr
(1 − β 2 )2π0 a2 βc(5.27)

͑ͳ͍ɻैͬͯɺ͜͜Ͱͷٞ࿦Ͱ͜ͷۙࣅΛ༻͍Δ͜

ͱͳΔɻ͜͜Ͱɺ0 μ0 = c−2 ͱ ż = v = βc Λ࢖༻ͱ͸ͦΕ΄Ͳѱ͘ͳ͍ɻ͜ͷԾఆͷ΋ͱͰ͸ɺϏʔϜͨ͠ɻࠨลͷ࣌ؒʹΑΔඍ෼Λ z ʹΑΔඍ෼Ͱද‫͢ݱ‬ిྲྀ͔Βిྲྀີ౓͸ ρ0 = I/(a2 πv) ͱද͞ΕΔɻ͜ͷΔͱɺ

ͱ͖ిྲྀີ౓ɺిՙີ౓͸ɺ0 ≤ r ≤ a ͷ৔߹ɺI,
Jz = J =πa2I,ρ = ρ0 =
πa2 v

r̈ =

(5.17)(5.18)

r > a ͷ৔߹͸ɺJ = 0ɺρ = 0 ͱͳΔɻԁ౵‫ܗ‬ϏʔϜͰిՙີ౓͕Ұఆͷ৔߹ɺGauss ͷ๏ଇΑΓి৔͸൒‫޲ํܘ‬੒෼͚ͩʹͳΓɺEr (r) =Er (r) =ρ0 r
Ir=, (r ≤ a), (5.19)20
2π0 a2 v
ρ0 a2I=, (r > a), (5.20)20 r
2π0 vr

dz d
dt dz

dz dr
dt dz

= β 2 c2 r

(5.28)

ͱͳΔɻ͜ΕΑΓɺۭؒిՙޮՌΛ‫ؚ‬Μͩӡಈํఔ
ࣜ͸ɺ

r =

qIr
2π0 mc3 β 3 γ 3 a2

(5.29)

ͱͳΔɻ͜Ε͕ɺۭؒిՙޮՌʹΑΔϏʔϜͷൃࢄΛද‫͢ݱ‬Δɻ͜͜ͰɺۭؒిՙޮՌͷӨ‫ڹ‬Λද‫͢ݱ‬Δͨ
Ίͷ‫͔ͭز‬ͷύϥϝλΛಋೖ͢ΔɻLasown ʹΑΓಋೖ͞Εͨແ࣍‫ݩ‬ύϥϝλͰ͋ΔҰൠԽ͞ΕͨύʔϏΞϯε (generalized perveance) ͸ɺ

㸲㸫 

K=

ωp2 a2I 2
2νB==I0 β 3 γ 3β2γ32β 2 c2(5.30)

ͷΑ͏ʹఆٛ͞ΕΔɻ͜͜ͰɺI0 ͸ۭؒిՙͷಛ௃

2σx = 2σy ͱͳΔɻσx = σy = σ ͱ͠ɺ·ͨ r = σ ͱిྲྀ (characteristic current)ɺ

͢Δͱɺࣜ (5.34) ΑΓɺrms ϏʔϜαΠζʹ͍ͭͯͷI0 =4π0 mc3,qϏʔϜΤϯϕϩʔϓํఔࣜ͸ɺ(5.31)σ  =

νB ͸ Budker ύϥϝλʔɺ
νB =

I
,
I0 β

K1
4 σ

(5.37)

ͱͳΔɻ

(5.32)

͜ͷ݁ՌΛ࢖͏ͱɺ֎෦ి࣓৔͕͋Δ৔߹ͷϏʔϜΤϯϕϩʔϓํఔࣜ͸ɺࣜ (4.38) ͷதͰ r = rm ͱ͠ωp ͸ϓϥζϚप೾਺ɺͨ΋ͷʹɺΤϛολϯεʹΑΔ߲Λදࣜ͢ (4.61) ͱɺ

qI
ωp2 =
π0 mcβγ 3 a2

(5.33)

ۭؒిՙʹΑΔൃࢄΛද߲͢ͷࣜ (5.35) ΛՃ͑ͨ΋ͷͱͳΔɻҰൠʹ֎෦ి࣓৔͕༩͑ΒΕͨ৔߹ʹ͸ɺͰ͋ΔɻҰൠԽ͞ΕͨύʔϏΞϯεΛ༻͍ΔͱɺཻۭؒࢠͷΤωϧΪʔͷมԽ͕ੜ͡ΔͨΊɺγ = const. ͱిՙޮՌΛ‫ؚ‬Μͩ r ํ޲ͷӡಈํఔࣜ͸ɺ͸ͳΒͳ͍ɻ͜ͷ৔߹ɺΤϛολϯε ε ͸ΤωϧΪʔ

r =

K
r
a2

ʹґଘͯ͠มԽͯ͠͠·͏ͨΊɺ͋·ΓศརͰ͸ͳ

(5.34)

͍ɻͦ͜ͷͰɺε ͷ୅ΘΓʹཻࢠͷΤωϧΪʔ͕มԽͯ͠΋ෆมͱͳΔ‫֨ن‬ԽΤϛολϯε εn = βγε Λ༻ͱද͞ΕΔɻ
͜͜·Ͱ͸ɺۭؒిՙޮՌͷ͋Δ৔߹ͷϏʔϜ಺͍Δɻ࠷ऴతʹ‫ٻ‬Ί͍ͨϏʔϜΤϯϕϩʔϓํఔࣜ͸

γ 
γ 
qB
rm
rm +
rm +
2
γβ
2γβ 2
2mcβγ
1
p2
ε2 1
1
− 2 2θ 2 2 3 − 3n 2 3 − K= 0 (5.38)
m c γ β rm
β γ rmrmͰͷՙిཻࢠͷӡಈํఔࣜΛ‫ٻ‬Ί͖͕ͯͨɺ͜ΕΛ


rm
+

ϏʔϜΤϯϕϩʔϓΛ‫ه‬ड़͢Δํఔࣜʹ֦ு͢Δɻࣜ

(5.34) ͸ɺϏʔϜ൒‫ ܘ‬a ͕ z ͷؔ਺ͱͯ͠༩͑ΒΕΕ͹ɺϏʔϜ಺ (r ≤ a) ͷཻࢠʹରͯ͠ղ͘͜ͱ͕Ͱ͖
ΔɻϏʔϜΛߏ੒͢Δՙిཻࢠͷ‫ي‬ಓ͕ɺ‫ࠩަʹ͍ޓ‬͠ͳ͍ (laminar ﬂow) ͱԾఆ͢Δͱɺࣜ (5.34) ΑΓɺͱͳΔɻ·ͨɺϏʔϜͷ൒‫ ܘ‬rm ͷ୅ΘΓʹɺrms ϏʔϜαΠζ σ Λ༻͍Δͱɺશͯͷཻࢠͷ‫ي‬ಓ͸ɺr/a Ͱεέʔϧ͞ΕΔಉ͡‫ܗ‬ͷ

γ 
γ 
qB
σ
σ
+
σ+
2
2
γβ
2γβ
2mcβγ
p2
1
ε2 1
K1
− 2 2θ 2 2 3 − 3n 2 3 −=0m c γ β σ
β γ σ
4 σσ  +‫ي‬ಓΛ࣋ͭɻ͜ͷ͜ͱΑΓɺϏʔϜͷ୺ʹ͋Δ r = aΛ΋ཻͭࢠ͸ɺ࣌ؒൃల‫ޙ‬΋ͣͬͱϏʔϜͷ୺ʹ͍Δ
͜ͱʹͳΔɻ͢ͳΘͪɺࣜ (5.34) Ͱ r = a = rm ͱ͠(5.39)

ͨ΋ͷ͕ɺࣗ༝ۭؒͰͷϏʔϜαΠζ rm ͷ࣌ؒൃలͱͳΔɻΛ‫ه‬ड़͢ΔɺϏʔϜΤϯϕϩʔϓํఔࣜ

rm
=K

1
rm

࣍ͷηΫγϣϯͰɺ͜ͷۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔ

(5.35)

ͱͳΔɻ·ͨɺ࣮ࡍͷϏʔϜΛѻ͏৔߹ʹ͸ɺϏʔϜϜΤϯϕϩʔϓํఔ͔ࣜΒɺΤϛολϯεͷ࣌ؒൃలΛ঺հ͢Δɻ

ͷ൒‫ ܘ‬a ͷ୅ΘΓʹ rms ϏʔϜαΠζ σx ͋Δ͍͸σy Λ࢖͏ํ͕ศརͰ͋Δɻ͜͜Ͱ͸ɺԁ‫ܗ‬அ໘Λ΋ͭϏʔϜΛߟ͑ɺͦͷதͰͷཻࢠີ౓͸Ұఆͱ͍ͯ͠
ΔͷͰɺ໘ີ౓ ρS0 ͕ҰఆͰ͋Δͱ͢Δͱɺrms ϏʔϜαΠζ͸ɺ

σx2

=
=
=

 2π
1
rdθ
drx2 ρS0
ρS0
0
 2π
 a
1
2
r3 dr
cos θdθπa2 00
 a 2
2(5.36)

ͱ‫͞ࢉܭ‬ΕΔɻ͜͜Ͱɺx = r cos θ Λ༻͍ͨɻσy ΋ಉ༷ʹ‫͞ࢉܭ‬Εɺԁ‫ܗ‬அ໘Λ΋ͭϏʔϜͰ͸ɺa =㸲㸫 ۭؒిՙޮՌʹΑΔࣹӨΤϛολ

6ͱల։Ͱ͖Δͱ͢Δɻ͜ΕΛࣜ (6.2) ʹ୅ೖ͢Δͱɺδσ ʹ͍ͭͯͷํఔࣜ͸ϯεͷ૿େδσ  + 2Kr δσ = 0͜͜Ͱ͸ɺηΫγϣϯ 5.3 Ͱಋೖͨ͠ɺۭؒిՙޮ
ՌΛ‫ؚ‬ΜͩϏʔϜΤϯϕϩʔϓํఔࣜΛ࢖ͬͯɺฏߧ

ͱͳΔɻ͜Ε͸ɺ֯ৼಈ਺

√

(6.5)

2Kr Λ࣋ͭɺ୯ৼಈΛঢ়ଶ͔ΒͷඍখͳৼಈΛߟ͑ɺΤϛολϯε͕૿‫͢ݮ‬

ද͢ɻKr ͸εϥΠεͷҐஔ ζ ʹґଘ͠ͳ͍ɻॳ‫ظ‬৚Δ͜ͱΛࣔ͢ɻͦͷ‫ޙ‬ɺιϨϊΠυʹΑΔࣹӨΤϛο݅ σ(0, ζ) = σeq (ζ) + δσ(ζ) = σ0 ɺσ  (0, ζ) = 0 Λ࣋λϯεͷิঈͷ‫ݪ‬ཧʹ͍ͭͯઆ໌͢Δɻ

ͭ৔߹ɺࣜ (6.5) ͷղ͸ɺ

6.1

Τϛολϯεͷ૿‫ݮ‬͜͜Ͱ͸ɺSeraﬁni ͱ Rosenzweig ͕ࣔͨ͠ํ๏ [26,27] Λ༻͍ͯɺࣹӨΤϛολϯε͕૿‫͢ݮ‬Δ͜ͱΛࣔ͢ɻࣹӨΤϛολϯεͷ૿‫ݮ‬Λࣔ͢Ϟσϧͱͯ͠ɺԁ౵ରশੑΛ࣋ͪɺ૬ର࿦తͳ଎౓Ͱӡಈ͢Δ࿈ଓϏʔϜΛߟ͑Δɻ͜ͷΑ͏ͳϏʔϜͷӡಈ͸ɺۭؒిՙޮՌ
Λ‫ؚ‬ΜͩϏʔϜΤϯϕϩʔϓํఔࣜ (5.38) Λ༻͍ͯɺ

1
γ 
ε2n 1
I
σ +
σ
+
K
σ
−
−
=0
r
γβ 2
β 3 γ 2 σ3
2I0 β 3 γ 3 σ(6.1)σ(z, ζ)=σ  (z, ζ)=σeq (ζ) + δσ(ζ) cos( 2Kr z), (6.6)
− 2Kr δσ(ζ) sin( 2Kr z) (6.7)ͱද͞ΕΔɻҐ૬ۭؒͰ͸ɺ͜ͷӡಈ͸ σeq Λத৺ͱͨ͠ପԁͰද͞ΕΔɻ࣍ʹɺ(σ, σ  ) Λ༻͍ͯɺࣹӨΤϛολϯε͕૿‫͢ݮ‬Δ͜ͱΛ؆୯ͳϞσϧΛྫͱͯ͠঺հ͢ΔɻϏʔϜεϥΠεͰͷฏ‫ۉ‬Λ  ͱ͢Δͱɺ(σ, σ  ) Λ༻͍ͯ rmsΤϛολϯε͸ɺ

ε(z) =

σ 2 σ 2  − σσ  2

(6.8)

ͱ‫͢ࢉܭ‬Δ͜ͱ͕Ͱ͖Δɻ͜͜Ͱ͸ɺٞ࿦Λ؆୯ʹ͢ͱද͞ΕΔɻ͜͜Ͱɺσ ͸ r ํ޲ͷ rms ϏʔϜαΠζΔͨΊʹɺϏʔϜ͸ 3 ͭͷεϥΠε͔Βߏ੒͞ΕΔ

Ͱ͋ΓɺKr ͸ιϨϊΠυɺ͋Δ͍͸ RF ి࣓৔ʹΑ

ͱߟ͑Δɻ֤εϥΠε͸ϏʔϜిྲྀ I1 ɺI2 ɺI3 Λ࣋

Δऩଋ࡞༻Λද͢܎਺Ͱ͋Δɻ͜͜Ͱ͸ɺ؆୯ͷͨΊͭͱ͠ɺI1 < I2 < I3 ͱ͢Δɻ·ͨɺϏʔϜͷॳ‫ظ‬ʹҰ༷ͳऩଋྗ Kr = const. ͷԼͰͷӡಈΛߟ͑Δɻ஋͸ͲͷεϥΠεͰ΋ (σ, σ  ) = (σ0 , 0) Λ࣋ͭͱԾఆ

·ͨɺॳ‫ظ‬ͷԁपํ޲ӡಈྔ͸θϩ (pθ = 0) ͱ͢Δɻ͢Δɻ͜ͷͱ͖ɺ3 ͭͷεϥΠε͔Β‫͞ࢉܭ‬ΕΔࣹӨϏʔϜ͸֎෦ి࣓৔ʹΑΓՃ଎͞Εͳ͍ (γ  = 0) ͱΤϛολϯε͸θϩͰ͋Δɻ֤εϥΠεͷத৺ϏʔϜԾఆ͠ɺ·ͨɺϏʔϜ಺Ͱ͸ۭؒిՙޮՌ͕ࢧ഑తͰ

αΠζ σeq ͸ɺεϥΠεͷϏʔϜిྲྀ I(ζ) ʹґଘ͢

͋Δͱͯ͠ɺӈลୈ 2 ߲ͱɺӈลୈ 4 ߲ͷΤϛολϯ

ΔͨΊɺਤ 44 ʹࣔ͢Α͏ʹҐ૬ۭؒதͰͷପԁͷதεʹΑΔ߲Λແࢹ͢Δɻ͜ͷͱ͖ɺϏʔϜΤϯϕϩʔ৺Ґஔ͸εϥΠεʹΑͬͯҟͳΔɻҐ૬ۭؒͰͷӡಈ√͸ɺσeq Λத৺ʹପԁΛඳ͕͘ɺδσ ͷ֯ৼಈ਺ 2Kr

ϓํఔࣜ͸ɺσ  + Kr σ −I(ζ) 1=0
2I0 β 3 γ 3 σ͸ɺϏʔϜిྲྀʹ͸ґଘ͠ͳ͍ͨΊɺ֤εϥΠε͸ಉ(6.2)

ͱͳΔɻ͜͜ͰɺϏʔϜ͸όϯνԽ͞Ε͍ͯΔͱ͠
ͯɺόϯνͷத৺͔Βͷਐߦํ޲ͷ‫཭ڑ‬Λ ζ ͱͯ͠ɺόϯνͷεϥΠεຖʹϏʔϜిྲྀ I(ζ) Λ࣋ͭͱߟ͑
͍ͯΔɻσ  = 0(Brillouin ﬂow) ͱ͢Δͱɺ͋ΔεϥΠεͰͷϏʔϜαΠζ͸ɺ


σeq =

I(ζ)
2Kr I0 β 3 γ 3֤͡ৼಈ਺Ͱৼಈ͢Δ͜ͱʹͳΔɻਤ 45 ʹɺ֤εϥ√Πε͕Ґ૬ 2Kr z = 0, π/2, 3π/2, 2π Λ࣋ͭͱ͖ͷҐ૬ۭؒ಺ͰͷมԽΛࣔ͢ɻҐ૬͕ 0 ͷͱ͖͸ɺ̏ͭͷεϥΠε͕࡞Δ໘ੵ͸θϩͰ͋ΔɻҐ૬͕ π/2 ʹͳΔͱεϥΠε͕࡞Δ໘ੵ͸࠷େʹͳΔɻͦͷ‫ޙ‬ɺҐ૬ π Ͱθϩʹͳͬͨ‫ޙ‬ɺ࠶౓ 3π/2 Ͱ࠷େ஋ΛऔΓɺ·ͨθϩʹ໭ΔΑ͏ʹͳΔɻ
͜͜ͰऔΓѻͬͨ؆୯ͳϞσϧͷ৔߹ʹ͸ɺۭؒి(6.3)

ՙޮՌͷӨ‫ʹڹ‬ΑΓɺϏʔϜ͕ਐߦ͢Δʹै͍ࣹӨΤϛολϯεͷ૿‫͖ى͕ݮ‬Δ͜ͱΛࣔͨ͠ɻಉ༷ͷࣹӨͱ‫͞ࢉܭ‬ΕΔɻϏʔϜ͸͜ͷ σeq ͷपΓͰඍখͳৼ෯

Τϛολϯεͷ૿‫ݮ‬͸ɺDC ిࢠॐ΍ೖࣹ‫ʹث‬ରͯ͠δσ Ͱӡಈ͢Δͱߟ͑ɺ1 ࣍ͷ߲ͷΈΛ࢒ͯ͠ɺͷ‫ػࢉܭ‬γϛϡϨʔγϣϯͰ΋‫؍‬ଌ͞ΕΔ [28]ɻ

σ(z, ζ) = σsq (ζ) + δσ(z)

(6.4)

㸲㸫 

σ’

σ’
1/2

(2Kr) z=π/2

I1

I2

I3

I1

I2

I3

σ

σ
1/2

σeq1

σeq2

(2Kr) z=0, 2π

σeq3

1/2

(2Kr) z=3π/2

ਤ 44: ̏ͭͷεϥΠε͕͋Δ৔߹ͷҐ૬ۭؒ (σ, σ  )ɻ֤εϥΠεͷϏʔϜిྲྀ͸ɺI1 ɺI2 ɺI3 ɻਤ 45: ̏ͭͷεϥΠε͕͋Δ৔߹ͷࣹӨΤϛολϯε
√2Kr z = 0, π/2, 3π/2, 2π

ͷมԽɻ֤εϥΠε͕Ґ૬Λ࣋ͭͱ͖ͷҐ૬ۭؒ಺ͰͷมԽΛࣔ͢ɻ

6.2

ιϨϊΠυʹΑΔࣹӨΤϛολϯεͷ
ิঈઌ୺͓Αͼऴ୺ʹൺ΂ͯɺΑΓҐ૬ۭؒ෼෍ͷ܏͖͕େ͖͘ͳΔɻιϨϊΠυೖ‫ࣗͰ·ޱ‬༝ۭؒΛਐߦͨ͠ηΫγϣϯ 6.1 Ͱ‫ͨݟ‬Α͏ʹɺҰ༷ͳऩଋྗͷதͰ͸ɺࣹӨΤϛολϯε͕૿‫͢ݮ‬Δ͜ͱ͕Θ͔ͬͨɻͱ͖ɺ֤όϯνεϥΠεͷҐ૬ۭؒ෼෍ʹ͸ͣΕ͕ੜ͡ɺࣹӨΤϛολϯεͷ૿େ͕ൃੜ͢ΔɻERL ೖࣹ‫ث‬ͷ৔߹ʹ΋ɺిࢠॐͰੜ੒͞Εͨόϯν͸ɺόϯνͷલ‫ిۭؒͰޙ‬ՙޮՌʹΑΔԣํ޲ͷൃࢄؒ෼෍͕ɺਤ 46 ͷ (c) Ͱ͋ΔɻιϨϊΠυʹΑΔྗ

ྗ͕ҟͳΔͨΊɺ֤εϥΠεͰͷҐ૬ۭؒ෼෍ͷͣΕ͸ɺιϨϊΠυͷ࣓৔த৺͔Βͷ‫ʹ཭ڑ‬ൺྫͨ͠ऩଋ

͕ੜ͡ɺࣹӨΤϛολϯε͕૿େ͍ͯ͘͠ɻ͜ΕΒΛྗͱͳΓɺҐ૬ۭؒͰͷ෼෍ͷճసΛҾ͖‫ͱ͜͢͜ى‬཈੍͢ΔͨΊʹɺిࢠॐͷ‫ʹޙ‬ιϨϊΠυΛઃஔ͠ऩʹͳΔɻద੾ͳιϨϊΠυͷ‫͞ڧ‬ΛબͿ͜ͱʹΑΓɺଋྗΛ༩͑ɺࣹӨΤϛολϯεΛճ෮ͤ͞ΔΑ͏ʹ͠ਤ 46(c) ʹࣔ͢Α͏ͳํ޲ʹ෼෍Λճసͤ͞Δ͜ͱ͕

͍ͯΔɻ͜͜Ͱ͸ɺERL ೖࣹ‫༻ث‬ͷిࢠॐͰ࢖ΘΕ

Ͱ͖Δɻͨͩ͠ɺιϨϊΠυ௚‫Ͱޙ‬͸·֤ͩεϥΠε͜ͷόϯν͕ιϨϊΠυΛ௨աͨ͠ͱ͖ͷҐ૬ۭΔιϨϊΠυʹΑΔࣹӨΤϛολϯεͷิঈͷ‫ݪ‬ཧʹຖͷ෼෍ͷෆҰக͸࢒͍ͬͯΔͨΊʹɺࣹӨΤϛολ͍ͭͯઆ໌͢ΔɻηΫγϣϯ 6.1 Ͱͷٞ࿦ͱͷҧ͍͸ɺϯε͸࠷খʹ౸ୡ͍ͯ͠ͳ͍ɻ

ERL ೖࣹ‫ث‬ͷ৔߹ʹ͸ɺҰ༷ͳऩଋྗ͕༩͑ΒΕΔ
Θ͚Ͱ͸ͳ͘ɺιϨϊΠυͷ͋Δ෦෼ͰͷΈऩଋྗ͕༩͑ΒΕΔͱ͍͏͜ͱͰ͋Δɻ͜ͷͨΊɺ࣮ࡍͷೖࣹ

46(d) ʹࣔ͢Α͏ʹ֤εϥΠεͷ෼෍͕Ұக͠ɺࣹӨΤϛολϯε͕࠷খʹͳΔɻҎ্͕ιϨϊΠυʹΑΔ

‫ث‬ͷઃ‫Ͱܭ‬͸ɺιϨϊΠυͷҐஔ͕ॏཁͱͳΔɻࣹӨΤϛολϯεิঈͷ‫ݪ‬ཧͰ͋Δɻ

ιϨϊΠυͷग़‫͔ޱ‬Βɺద੾ͳ‫཭ڑ‬ΛਐΉͱɺਤ·ͣɺόϯνԽ͞ΕͨϏʔϜͰͷɺԣํ޲ۭؒిՙ

͜͜Ͱɺ஫ҙ͕ඞཁͳͷ͸ɺࣹӨΤϛολϯε͕࠷ྗʹΑΔࣹӨΤϛολϯεͷ૿େΛߟ͑Δɻ͜ͷ༷ࢠখʹͳΔҐஔ͸ɺۭؒిՙྗͷ‫͞ڧ‬ɺιϨϊΠυͷ‫ڧ‬Λɺਤ 46 ͷ (a)ɺ(b) ʹࣔ͢ɻ(a) ͸ిࢠॐ௚‫ޙ‬ͷԣ͞ͱҐஔɺͦͯ͠ΤϛολϯεΛଌఆ͢ΔҐஔʹґଘํ޲ͷҐ૬ۭؒ෼෍Ͱɺx ͷ֦͕Γ͸Χιʔυද໘Ͱ

͢Δͱ͍͏͜ͱͰ͋Δɻ͞Βʹɺ࠷খʹͳͬͨҐஔ͔ͷԹ౓౳ʹΑͬͯੜͨ͡΋ͷͰ͋Δɻిࢠॐ௚‫Ͱޙ‬͸Βόϯν͕ਐߦ͢Δͱɺ·ࣹͨӨΤϛολϯεͷ૿େΤωϧΪʔ͕௿͍ͨΊɺۭؒిՙޮՌʹΑΔൃࢄྗͷ͕ੜ࢝͡ΊΔͱ͍͏͜ͱ͕͋Δɻ͜ͷͨΊɺͲ͜Ͱࣹ

Ө‫͘ڧ͕ڹ‬ಇ͖ɺҐ૬ۭؒͰͷ෼෍ͷճసͱม‫͕ܗ‬ੜӨΤϛολϯεΛ࠷খʹ͢Δ͔ͱ͍͏͜ͱ͕ॏཁͱͳ

͡ΔɻόϯνԽ͞ΕͨϏʔϜͰ͸ɺόϯνதԝͱલ‫ޙ‬ΔɻERL ೖࣹ‫Ͱث‬͸ɺೖࣹ‫ث‬ͷ‫ʹޙ‬ଓ͘पճ෦ͷओͰ͸ిՙີ౓͕ҟͳΔͨΊʹɺҟͳΔൃࢄྗʹΑΓɺՃ଎ۭಎͷҐஔͰࣹӨΤϛολϯε͕࠷খʹͳΔΑ͏֤εϥΠεͰ͸Ґ૬ۭؒ෼෍ʹͣΕ͕ੜ͡Δ͜ͱʹʹ͢Δͷ͕๬·͍͠ɻ͜Ε͸ɺΤωϧΪʔ͕ߴ͘ͳΕ

ͳΔɻόϯνதԝͰ͸ిՙີ౓͕ߴ͍ͨΊɺόϯνͷ͹ۭؒిՙྗʹΑΔԣํ޲ͷྗ͸૬ରతʹऑ͘ͳΓɺ㸲㸫 ͦͷྗΛແࢹͰ͖ΔΑ͏ʹͳΔͨΊɺՃ଎ۭಎͷҐஔͰ࠷খʹͳΔΑ͏ʹ͓͚ͯ͠͹ɺ࠷খΤϛολϯεΛ
อ࣋ͨ͠··पճ෦ΛਐΉ͜ͱ͕Ͱ͖Δ͔ΒͰ͋Δɻ
‫࣮ݱ‬ͷ ERL ೖࣹ‫Ͱث‬͸ɺRF ి࣓৔ʹΑΔࣹӨΤϛολϯεͷ૿େͳͲ΋͋Γɺ࠷খΤϛολϯεΛಘΔͨΊͷઃ‫ʹܭ‬͸ɺ‫ػࢉܭ‬γϛϡϨʔγϣϯʹΑΔํ
๏͕༻͍ΒΕΔɻηΫγϣϯ 7 Ͱ͸ɺۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜͷ‫ػࢉܭ‬γϛϡϨʔγϣϯʹ͍ͭͯ঺հ
͢Δɻ
(a)

(b)

x’

x’

x

x

electron bunchz(c)slice 1slice 2x’

projected emittance
(d)
x

x’
x

ਤ 46: ιϨϊΠυి࣓ੴʹΑΔࣹӨΤϛολϯεิঈͷ‫ݪ‬ཧɻ(a) ిࢠॐ͔ΒిࢠϏʔϜ͕ੜ੒͞Εͨ௚‫ޙ‬ɻσx ͸ిࢠੜ੒࣌ʹੜͨ͡΋ͷͰ͋Δɻ(b) ιϨϊΠυೖ‫ޱ‬ɻۭؒిՙޮՌʹΑΓɺεϥΠεΤϛολ
ϯεͷෆҰகɺͭ·ΓࣹӨΤϛολϯεͷ૿େ͕‫͖ى‬Δɻ(c) ιϨϊΠυग़‫ޱ‬ɻൃࢄͷେ͖͞ʹൺྫͨ͠ऩ
ଋྗΛड͚ɺҐ૬ۭؒͰͷ෼෍ͷճస͕‫͖ى‬Δɻ(d)ιϨϊΠυԼྲྀɻ͋Δద੾ͳ‫ࣗ཭ڑ‬༝ؒΛਐΉͱɺεϥΠεΤϛολϯεͷෆҰக͕ղফ͞ΕɺࣹӨΤϛολϯε͕࠷খʹͳΔɻ㸲㸫 Ͱ͸ɺਐߦํ޲ɺԣํ޲ͱ΋ʹిՙີ౓͕ҰఆͱԾఆۭؒిՙޮՌͷ਺஋‫ࢉܭ‬7

͍ͯ͠Δɻ͜ͷͱ͖ɺۭؒిՙޮՌʹΑΔඇઢ‫ిͳܗ‬ۭؒిՙޮՌ͸ɺ࿈ଓϏʔϜۙࣅ΍ಛผͳ෼෍ؔ਺

࣓৔͸ແࢹ͢Δ͜ͱ͕Ͱ͖ΔɻεϥΠεͷӡಈ͸ɺ֤ʹରͯ͠͸ղੳతʹѻ͏͜ͱ͕Ͱ͖Δ͕ɺ࣮ࡍͷՃεϥΠεͷ൒‫޲ํܘ‬ͷେ͖͞ (ΤϯϕϩʔϓʣͷมԽ଎‫ث‬தͰͷӨ‫ڹ‬Λղੳతʹѻ͏͜ͱ͸೉͍͠ɻͦͷͨͱਐߦํ޲ͷ෯ͷมԽʹΑΓ‫ه‬ड़͞ΕΔɻ͜ͷํ๏ͷΊɺଟ͘ͷ৔߹ʹ͸਺஋‫ʹࢉܭ‬ΑۭͬͯؒిՙޮՌͷར఺͸ɺόϯνΛԿ‫ݸ‬ͷεϥΠεʹ෼ׂ͢Δ͔ʹґଘӨ‫͕ڹ‬ௐ΂ΒΕ͍ͯΔɻ͜͜Ͱ͸ɺۭؒిՙޮՌ‫ࢉܭ‬

͢Δ͕ɺ‫ج‬ຊతʹऔΓѻ͏εϥΠε਺͸গͳ͍ͨΊɺίʔυʹ͍ͭͯ঺հͨ͠‫ޙ‬ɺ‫͔ͭز‬ͷۭؒిՙ‫ࢉܭ‬๏‫ߴ͕ࢉܭ‬଎Ͱ͋Δͱ͍͏͜ͱͰ͋Δɻ͜ͷͨΊɺԁ౵ʹ͍ͭͯ঺հ͢Δɻ

ରশੑ͕͋ΔϏʔϜʹରͯ͠ɺύϥϝλ୳ࡧͳͲͷߴ଎ͳ‫͕ࢉܭ‬ཁ‫͞ٻ‬ΕΔ৔߹ʹద͍ͯ͠Δɻ
Ϩϕϧ 2 ͷ৔߹͸ɺϏʔϜΛϚΫϩཻࢠͷूஂͱۭؒ͠ిՙޮՌΛ‫ؚ‬ΜͩγϛϡϨʔγϣϯίʔυͯऔΓѻ͏ɻۭؒిՙޮՌͷ‫ʹࢉܭ‬͸ɺཻࢠ෼෍Λ࣮

ERL Ͱ͸ɺయ‫ܕ‬తʹ 1.3 GHz ͷप೾਺Ͱɺ100 mAͰͷిՙ෼෍͔Β੩ి৔Λ‫͢ࢉܭ‬Δํ๏Ͱ͋Δɻ੩ࢭͷϏʔϜΛߟ͍͑ͯΔɻ͜ͷͱ͖ɺҰͭͷిࢠόϯν‫Ͱܥ‬ͷ੩ి৔͕‫ޙͨͬ·ٻ‬͸ɺ‫ ٯ‬Lorentz ม‫ʹ׵‬ΑΓ಺ʹ͸ɺQ = −77 pC ͷిՙ͕‫·ؚ‬ΕΔ͜ͱʹͳ੩ࢭ‫Ͱܥ‬ͷ੩ి৔Λ࣮‫Ͱܥࣨݧ‬ͷి࣓৔ʹม‫͠׵‬ɺཻΔɻ͜ΕΑΓɺిࢠόϯν಺ͷిՙ਺Λ‫͢ࢉܭ‬Δͱɺࢠʹ૬‫༻࡞ޓ‬Λ༩͑ɺ࣌ؒൃలͤ͞Δɻ੩ి৔ͷ‫ࢉܭ‬N = Q/e = 4.8 × 10 ͱ๲େͳ਺ͱͳΔɻԾʹɺ਺஋‫͜Ͱࢉܭ‬ΕΒશͯͷిࢠؒͷ Coulomb ૬‫༻࡞ޓ‬Λ௚͸ɺిՙີ౓͔Β Poisson ํఔࣜΛղ͘ํ๏΍ɺ੩ࢭ઀‫͢ࢉܭ‬ΔͱɺͦΕ͸ 1 λΠϜεςοϓ͋ͨΓɺ∼ N

2ཻࢠʹ͍ͭͯ‫͢ࢉܭ‬Δͱ͍͏ํ๏͕͋Δɻ͜ΕΒͷ‫ܭ‬ͱͳͬͯ͠·͍ɺ‫࣮ݱ‬తͰ͸ͳ͍ɻ͜ͷͨΊɺిࢠόࢉͰɺલఏͱͳΔͷ͕ɺϏʔϜͷ੩ࢭ‫ܥ‬ΛఆٛͰ͖Δ

ϯνͷ਺஋γϛϡϨʔγϣϯΛߦ͏ʹ͸ɺిࢠόϯνͱ͍͏͜ͱͰ͋ΔɻΤωϧΪʔ֦͕Γͷͳ͍ϏʔϜͷΛද‫͢ݱ‬ΔͨΊͷଞͷํ๏͕ඞཁͰ͋Δɻ͜ͷํ๏ͱ৔߹ʹ͸ɺ‫ʹີݫ‬ϏʔϜͷ੩ࢭ‫ܥ‬ΛఆٛͰ͖Δ͕ɺΤͯ͠͸ɺଟ͘ͷ৔߹࣍ͷ̎ͭͷํ๏͕༻͍ΒΕΔɻҰωϧΪʔ֦͕Γ͕༗‫ͳݶ‬৔߹ʹ͸ɺ‫ʹີݫ‬͸੩ࢭ‫ܥ‬Λͭ͸ɺిࢠόϯνΛਐߦํ޲ʹεϥΠεͯ͠ɺখ͍͞

ఆٛͰ͖ͳ͍ɻ͜ͷͨΊɺϨϕϧ 2 ͷํ๏Λ༻͍Δ৔

෯Λ࣋ͭଟ਺ͷԁ౵εϥΠεͷू·Γͱͯ͠ද‫͢ݱ‬Δ߹ʹ͸ɺΤωϧΪʔ֦͕ΓΛߟྀ͢Δඞཁ͕͋Δɻ͜

7.1

8‫͔ܥࣨݧ‬ΒϏʔϜͷ੩ࢭ‫ ʹܥ‬Lorentz ม‫͠׵‬ɺ੩ࢭ‫ܥ‬‫Ͱܥ‬ͷϚΫϩཻࢠؒͰͷ Coulomb ૬‫༻࡞ޓ‬Λશͯͷํ๏Ͱ͋Δɻ΋͏Ұͭ͸ɺಉ͡ e/m Λ࣋ͬͨϚΫϩͷํ๏ͷར఺͸ɺϚΫϩཻࢠͷ਺Λ૿΍ͤ͹ɺෳࡶͳཻࢠͷू·Γͱͯ͠‫ه‬ड़͢Δํ๏Ͱ͋Δɻిՙ Q Λ෼෍Λද‫͖Ͱݱ‬Δͱ͍͏͜ͱͰ͋Δɻ·ͨɺ੩ి৔ͷ࣋ͭόϯνΛɺNm ‫ݸ‬ͷϚΫϩཻࢠͰද‫͢ݱ‬ΔͱɺҰ‫ํࢉܭ‬๏΋‫͔ͭز‬બ୒Ͱ͖ΔͨΊɺൺֱతߴ଎ͳ‫ࢉܭ‬ͭͷϚΫϩཻࢠ͋ͨΓͷిՙ͸ Q/Nm ͱͳΔɻϚΫ΋ՄೳͰ͋Δɻͨͩ͠ɺ͜ͷํ๏Ͱ͸஗Ԇϙςϯγϟ

ϩཻࢠͷ‫਺ݸ‬ͷऔΓํʹ͍ͭͯ͸ɺDebye ௕Λ‫͠ࢉܭ‬ϧͷӨ‫ྀ͞ߟ͕ڹ‬Ε͍ͯͳ͍ͷͰɺΑΓ‫͢ࢉܭʹີݫ‬ͯɺۭؒిՙޮՌʹΑΔి࣓৔͕ۭؒతʹ‫׈‬Β͔ʹ‫ه‬Δʹ͸Ϩϕϧ 3 ͷํ๏Λ࢖༻͢Δඞཁ͕͋Δɻ

ड़Ͱ͖Δ͔Λߟྀ͓ͯ͘͠ඞཁ͕͋Δɻ

Ϩϕϧ 3 ͷ৔߹͸ɺLienard-Wiechert ϙςϯγϟ

ۭؒిՙޮՌΛ‫ؚ‬ΜͩϏʔϜγϛϡϨʔγϣϯ͸ɺ࣍ͷΑ͏ͳ̏ͭͷϨϕϧʹ෼͚ΒΕΔ [29]ɻ

ϧղ͘͜ͱͰɺ஗ԆޮՌΛऔΓೖΕΔɻ͜ͷͨΊɺϨ
ϕϧ 3 ͷํ๏Ͱ͸ɺ༨‫ͳܭ‬Ծఆ͕‫·ؚ‬Ε͍ͯͳ͍ͨΊ

1. ԁ౵εϥΠεΛ༻͍ͨ४ղੳతίʔυʢεϥΠε಺Ͱ͸ిՙີ౓͕Ұఆͱͯ͠ɺۭؒిՙޮՌʹΑ
Δి࣓৔ͷඇઢ‫ܗ‬ੑΛແࢹʣ
2. ϚΫϩཻࢠΛ༻͍ɺϏʔϜͷ੩ࢭ‫Ͱܥ‬੩ి৔Λ‫ܭ‬ʹɺ‫͕ࢉܭͳີݫ‬ՄೳͰ͋Δɻ͔͠͠ɺҰൠʹ‫ʹࢉܭ‬ֻ͕͔࣌ؒΔͨΊʹɺߴ଎ʹύϥϝλ୳ࡧΛ͍ͨ͠৔߹ͳͲʹ͸޲͔ͳ͍ɻ
ۭؒిՙ‫ࢉܭ‬Λ‫ؚ‬ΜͩγϛϡϨʔγϣϯίʔυͷ୅දతͳ΋ͷ͸ɺࢉ͢Δίʔυ

1. HOMDYN (Ϩϕϧ 1) [30]3. ஗ԆϙςϯγϟϧͷޮՌΛऔΓೖΕͨίʔυ(Lienard-Wiechert ϙςϯγϟϧΛ༻͍Δ)
Ϩϕϧ 1 ͷ৔߹͸ɺϏʔϜΛՙిཻࢠͷूஂͱͯ͠Ͱ
͸ͳ͘ɺԁ౵εϥΠεͷू·Γͱߟ͑ΔɻεϥΠε಺㸲㸫 

2. ASTRA (Ϩϕϧ 2ɺԁ౵ରশϏʔϜ) [31]3. Parmela (Ϩϕϧ 2ɺԁ౵ରশϏʔϜɺ3 ࣍‫ۭؒݩ‬ిՙޮՌ) [32]‫཭ڑ‬͸ɺ

4. GPT (Ϩϕϧ 2ɺ3 ࣍‫ిۭؒݩ‬ՙޮՌ) [33]5. TREDI (Ϩϕϧ 3) [34]r ji = r ji +6. PETAR (Ϩϕϧ 3) [35]Ͱ͋Δɻ͜ΕҎ֎ʹ΋ଟ͘ͷۭؒిՙ‫ࢉܭ‬Λ‫ؚ‬Μͩίʔυ͕։ൃ͞Ε͍ͯΔ͕ɺશͯΛ঺հ͢Δ͜ͱ͸Ͱγj2(r ji · β j )β j
γj + 1(7.2)

ͱͳΔɻ͜͜Ͱɺβ j ͸ཻࢠ j ͷ଎౓ϕΫτϧɺγj =
1/ 1 − |β j |2 ͸ Lorentz ҼࢠͰ͋Δɻཻࢠ j ͷ੩ࢭ‫Ͱܥ‬͸ɺ੩ి৔ͷΈ͕ଘࡏ͠ɺCoulomb ͷ๏ଇΑΓɺ͖ͳ͍ͷͰɺΑ͘࢖ΘΕ͍ͯΔίʔυΛ঺հ͢ΔʹཹE ji =

Ίͨɻ࣍ʹɺϚΫϩཻࢠΛ࢖ཻͬͨࢠ௥੻‫Ͱࢉܭ‬ͷۭؒి

Qr ji4π0 |r ji |3(7.3)

ՙͷ‫ํࢉܭ‬๏ʹ͍ͭͯɺҰൠʹΑ͘༻͍ΒΕ͍ͯΔϨͱ‫͞ࢉܭ‬ΕΔɻ͜ΕΛ‫ ٯ‬Lorentz ม‫ͯ͠׵‬ɺ࣮‫ܥࣨݧ‬ϕϧ 2 ͷํ๏ʹ͍ͭͯ঺հ͢Δɻ͜ΕΒͷۭؒిՙͰͷి࣓৔Λ‫ٻ‬ΊΔͱE ji = γj E ji −ͷ‫Ͱࢉܭ‬͸ɺਖ਼֬ʹ‫͢ࢉܭ‬Δ͜ͱ΋ඞཁͰ͋Δ͕ɺ·
ͨग़དྷΔ͚ͩૣ͘‫͢ࢉܭ‬Δ͜ͱ΋ॏཁͱͳͬͯ͘ΔɻͦΕͧΕͷۭؒిՙͷ‫ํࢉܭ‬๏ʹ͸ɺ௕ॴͱ୹ॴ͕͋
దͨ͠‫ࢉܭ‬ίʔυΛ࢖༻͢Δ͜ͱ͕ॏཁͰ͋Δɻ

=

B ji

Γɺର৅ͱ͢Δ‫͓ܥ‬Αͼॏཁͳ෺ཧ‫ݱ‬৅Λ‫ۃݟ‬Ίͯɺ

γj(β · E ji )β j ,γj + 1 jγj β j × E ji
c


(7.4)
(7.5)

ͱͳΔɻ͜ΕͰɺ࣮‫ ࢠཻͰܥࣨݧ‬j ཻ͕ࢠ i ͷҐஔʹ࡞Δి࣓৔͕‫ͨͬ·ٻ‬ɻϏʔϜ͸ N ‫ݸ‬ͷཻࢠ͔Βߏ

7.2

఺ిՙؒͷྗΛ‫͢ࢉܭ‬Δํ๏

੒͞ΕΔͱͨ͠ͷͰɺϏʔϜશମ͕ i ͷҐஔʹ࡞Δి

ۭؒిՙΛ‫ؚ‬ΜͩϏʔϜΛγϛϡϨʔτ͢ΔίʔυͰ͸ɺଟ͘ͷ৔߹ɺϚΫϩཻࢠΛ༻ཻ͍ͨࢠτϥοΩϯά͕࢖ΘΕΔɻ͜͜Ͱ͸ɺ఺ిՙؒͷ૬‫༻࡞ޓ‬Λ௚઀‫͢ࢉܭ‬Δํ๏Λ঺հ͢Δɻ఺ిՙؒͷྗͷ௚઀‫ࢉܭ‬࣓৔͸ɺࣗ෼ࣗ਎Λআ͍ͨ N − 1 ʹର্ͯ͠‫ه‬ͷ‫ܭ‬ࢉΛ‫܁‬Γฦ͠ɺ଍͠߹ΘͤΔ͜ͱʹΑΓɺ

 
γj


Ei =
γj E ji −(β · E ji )β j ,γj + 1 j
j=i

(7.6)

Ͱ͸ɺϏʔϜΛߏ੒͢Δ̎ͭͷϚΫϩཻࢠΛͦͷ੩ࢭ‫ ʹܥ‬Lorentz ม‫ʹޙͨ͠׵‬ɺCoulomb ͷ๏ଇ͔Β੩

Biి৔Λ‫͠ࢉܭ‬ɺ࠶ͼͦΕΛ࣮‫ ٯʹܥࣨݧ‬Lorentz ม‫׵‬= γj β j × E ji
j=i

c

(7.7)

͢Δ͜ͱʹΑΓɺۭؒిՙޮՌΛ‫͢ࢉܭ‬Δɻ௕ॴ͸ɺཻࢠͷ੩ࢭ‫Ͱܥ‬ͷ੩ి৔Λ‫ٻ‬ΊΔʹ͋ͨͬͯɺిՙີ
౓΍ి࣓৔ͷฏ‫ۉ‬ԽͳͲͷۙࣅΛಋೖ͍ͯ͠ͳ͍ͱ͍
͏͜ͱ͕‫͛ڍ‬ΒΕΔ͕ɺҰํɺཻࢠ਺͕૿͑ͨ৔߹ʹ͸‫͕ࢉܭ‬๲େʹ૿͑ΔͨΊʹɺ‫͕ؒ࣌ࢉܭ‬௕͘ͳΔͱ
͍͏୹ॴ͕͋Δɻ·ͨɺϚΫϩཻࢠͷ਺͕গͳ͍৔߹
ʹ͸ɺཻࢠؒ‫͍͕ۙ཭ڑ‬৔߹ʹ૬‫͕༻࡞ޓ‬ඇৗʹ‫͘ڧ‬ͱ‫͞ࢉܭ‬ΕΔɻ͜ͷి࣓৔͔Βཻࢠ i ʹର͢ΔྗΛ‫ܭ‬ࢉ͠ɺ࣍ͷ࣌ؒεςοϓͷཻࢠҐஔΛ‫͢ࢉܭ‬Δɻ͜ΕΛϚΫϩཻࢠશͯʹରͯ͠‫܁‬Γฦ͢ͱɺ1 εςοϓͷ‫ܭ‬ࢉ͕ऴྃ͢Δɻ͢ͳΘͪɺ1 εςοϓͷؒʹ N (N − 1)ͷ‫͕ࢉܭ‬ඞཁͱͳΔɻཻࢠ਺͕૿େͨ͠৔߹ʹ͸ɺ‫ܭ‬ࢉճ਺͸΄΅ N 2 ʹͳΔͨΊɺ‫૿͕ؒ࣌ࢉܭ‬େ͢Δɻ

ͳΔ৔߹͕͋Γɺ਺஋తͳϊΠζ͕ੜ͡Δͱ͍͏͜ͱ
͕͋ΔͷͰɺ஫ҙ͕ඞཁͰ͋Δɻ
ۭؒిՙޮՌΛ఺ిՙؒͷ૬‫͢ࢉܭͰ༻࡞ޓ‬ΔྲྀΕ

7.3

͸࣍ͷΑ͏ʹͳΔɻ͜͜ͰɺϏʔϜ͸ిՙ Q Λ࣋ͭϏʔϜͷ੩ࢭ‫ʹܥ‬ઃஔͨ֨͠ࢠ্Ͱ੩
ి৔Λ‫͢ࢉܭ‬Δํ๏

N ‫ݸ‬ͷϚΫϩཻࢠ͔Βߏ੒͞ΕΔͱ͢ΔɻೋͭͷϚ఺ిՙؒͷ૬‫༻࡞ޓ‬Λ௚઀‫͢ࢉܭ‬ΔίʔυͰ͸ɺۙΫϩཻࢠ i ͱ j Λߟ͑ɺϚΫϩཻࢠ j ͕ i ͷҐஔʹ࡞ࣅ͕‫·ؚ‬Ε͍ͯͳ͍ͱ͍͏ར఺͕͋Δ͕ɺཻࢠ਺͕

Δి࣓৔Λ‫ٻ‬ΊΔɻ࣮‫Ͱܥࣨݧ‬ೋͭͷཻࢠͷҐஔ͸ɺ૿Ճͨ͠ͱ͖ͷ‫ؒ࣌ࢉܭ‬ͷ໰୊ɺ·ཻͨࢠؒͷ‫ʹ཭ڑ‬r i ɺr j Ͱද͞ΕΔͱ͢Δͱɺཻࢠؒͷ‫཭ڑ‬͸Αͬͯ͸਺஋తͳϊΠζ͕ੜ͡Δͱ͍͏໰୊͕͋Δɻ

r ji = r i − r j఺ిՙؒͷ૬‫༻࡞ޓ‬Λ‫͢ࢉܭ‬Δํ๏ͷଞʹɺۭؒి(7.1)

ͱͳΔɻ͜ΕΛཻࢠ j ͷ੩ࢭ‫͔ܥ‬Β‫ݟ‬ΔͱɺཻࢠؒͷՙΛ‫͢ࢉܭ‬ΔͨΊʹΑ͘࢖ΘΕΔํ๏ͱͯ͠ɺϏʔϜͷ੩ࢭ‫Ͱܥ‬੩ి৔Λ‫͢ࢉܭ‬Δํ๏͕͋Δɻ͜ͷํ๏Ͱ͸ɺϚΫϩཻࢠ͔Βߏ੒͞ΕΔϏʔϜΛͦͷ੩ࢭ‫ܥ‬㸲㸫 ʹ Lorentz ม‫ʹޙͨ͠׵‬ɺϏʔϜΛ‫ؚ‬Ή্ۭؒʹ֨ࢠΈͳͤΔΑ͏ͳ‫͕͋ܥ‬Δͱߟ͑ΔɻͦͷΑ͏ͳ੩ࢭ‫ܥ‬Λ࡞੒͠ɺͦͷ֨ࢠ্Ͱిՙີ౓Λ‫͠ࢉܭ‬ɺͦΕ͔Βͷ࣮‫ʹܥࣨݧ‬ର͢Δ଎౓ β 0 ͸ɺγi β iβ0 = 
ni γi

Poisson ํఔࣜΛղ͘͜ͱʹΑΓɺ੩ి৔Λ‫ٻ‬ΊΔɻ‫ٻ‬Ίͨ֨ࢠ্ͷ੩ి৔Λ‫ ٯ‬Lorentz ม‫ʹ׵‬ΑΓɺ࣮‫ݧ‬ࣨ‫Ͱܥ‬ͷి࣓৔ʹม‫͠׵‬ɺ֤ϚΫϩཻࢠʹՃΘΔྗΛ‫ٻ‬ΊΔɻ͜ͷํ๏Ͱ͸ɺ্ۭؒʹ֨ࢠΛ࡞ΔͨΊɺ֨ࢠִؒ಺Ͱ͸ཻࢠ෼෍͕ฏ‫ۉ‬Խ͞Εɺ఺ిՙؒͷྗͷ‫ࢉܭ‬ͷΑ͏ͳҙਤ͠ͳ͍িಥతͳྗͷӨ‫ڹ‬Λආ͚ΒΕΔɻ͔͠͠ͳ͕Βɺ֨ࢠִؒʹΑͬͯ͸ࡉཻ͔͍ࢠ
෼෍ͷӨ‫͏·ͯͬ͠ͳ͘ͳ͑ݟ͕ڹ‬৔߹΋͋ΔͷͰɺ֨ࢠαΠζͱϚΫϩཻࢠ਺ͷબͼํ͕ॏཁͱͳΔɻ·(7.8)

ͱද͞ΕΔɻ͜͜ͰɺϚΫϩཻࢠʹ൪߸ i Λ͚ͭɺϚΫϩཻࢠ಺Ͱͷిࢠ਺Λ ni ɺ࣮‫Ͱܥࣨݧ‬ͷཻࢠͷ଎౓ϕΫτϧΛ β i ɺLorentz ҼࢠΛ γi ͱͨ͠ɻ·ͨɺ࿨͸ɺn = 1 ͔ΒϚΫϩཻࢠͷ૯਺ N ·Ͱ‫͞ࢉܭ‬ΕΔɻ
੩ࢭ‫͍ͯͭʹܥ‬ͷ‫֨ن‬Խӡಈྔ γ0 β 0 ɺ͓Αͼ LorentzҼࢠ γ0 ͸ɺ੩ࢭ‫ܥ‬ͷ଎౓ β 0 ͔ΒͨɺLorentz ม‫׵‬Λ͢Δʹ͋ͨͬͯϏʔϜͷ੩ࢭ‫ͱܥ‬

γ0 β 0 = ͍͏΋ͷΛߟ͕͑ͨɺϏʔϜͷΤωϧΪʔ֦͕Γ͕େ͖͍৔߹ɺϏʔϜͷฏ‫ۉ‬଎౓ͷ‫ ʹܥ‬Lorentz ม‫ͯ͠׵‬γ0 =΋ΤωϧΪʔͷͣΕཻͨࢠ͸ͦͷ੩ࢭ‫ʹܥ‬ରͯ͠଎౓

β0

,

(7.9)


1 + γ02 β 20(7.10)

1 − β 20Λ΋ͭ͜ͱʹͳΓɺ੩ి৔ͷΈ͕‫ݱ‬ΕΔલఏ͕ਖ਼͘͠

ͱ‫͞ࢉܭ‬ΕΔɻγ0 Λ༻͍Δͱɺ࣮‫Ͱܥࣨݧ‬ͷϚΫϩ

ͳ͘ͳΔɻ͜ͷͨΊɺ͜ͷํ๏͸ϏʔϜͷΤωϧΪʔཻࢠͷҐஔ r i ͸ɺLorentz ม‫ʹ׵‬ΑΓɺϏʔϜͷ੩ࢭ֦͕Γ͕େ͖͍৔߹ʹ͸࢖͑ͳ͍ͷͰɺ஫ҙ͕ඞཁͰ͋Δɻ‫Ͱܥ‬ͷҐஔ r i ɺϏʔϜ੩ࢭ‫ܥ‬ͷ֨ࢠ্Ͱ੩ి৔Λ‫͢ࢉܭ‬ΔྲྀΕ͸࣍ͷΑ͏ʹͳΔɻ

1. ϏʔϜͷ੩ࢭ‫ܥ‬Λ‫ٻ‬ΊΔ

r i

=

γ0 r i ,(7.11)

r i⊥

=

r i⊥

(7.12)

ʹม‫͞׵‬ΕΔɻ͜͜Ͱɺr  ͸ϏʔϜਐߦํ޲ͷҐஔ

2. ϏʔϜΛߏ੒͢ΔϚΫϩཻࢠͷҐஔͱ଎౓ (r i ,ui ) ΛɺϏʔϜͷ੩ࢭ‫ ʹܥ‬Lorentz ม‫͢׵‬Δ(r i , ui )3. ϏʔϜͷ੩ࢭ‫Ͱܥ‬ɺϏʔϜΛ‫ؚ‬Ήۭؒதʹ֨ࢠΛ࡞੒͠ɺ֨ࢠ఺্Ͱͷిՙີ౓ ρ Λ‫͢ࢉܭ‬Δ

࠲ඪɺr ⊥ ͸ਐߦํ޲ʹਨ௚ͳํ޲ͷ࠲ඪͰ͋Δɻ͜ΕΒೋͭΛ͋ΘͤΔͱɺཻࢠͷ࠲ඪ͸r i = r i +ϙςϯγϟϧ V Λ‫͢ࢉܭ‬Δ

ి৔ E i Λ‫ٻ‬ΊΔ

(7.13)

ͱม‫͞׵‬ΕΔɻ·ͨɺཻࢠͷ଎౓͸

4. ిՙີ౓ ρ ʹରͯ͠ Poisson ํఔࣜΛղ͖ɺ੩ి

5. ੩ిϙςϯγϟϧ V ͔Βɺཻ֤ࢠͷҐஔͰͷ੩

r i · γ0 β 0γ0 β 0γ0 + 1

ui

=

ui⊥

=

ui
,
(1 − β u · β 0 )ui⊥
γ0 (1 − β u · β 0 )(7.14)(7.15)

ͱม‫͞׵‬Εɺ·ͱΊΔͱu

6. ‫ٻ‬Ίͨ੩ి৔ E i Λ‫ ٯ‬Lorentz ม‫ͯ͠׵‬ɺ࣮‫ࣨݧ‬‫Ͱܥ‬ͷి࣓৔Λ‫ٻ‬ΊΔ
7. ࣮‫Ͱܥࣨݧ‬ͷి࣓৔͔ΒϚΫϩཻࢠʹՃΘΔྗΛ‫͠ࢉܭ‬ɺ࣍ͷ࣌ؒεςοϓͷཻࢠͷҐஔͱ଎౓Λ=c
×γu γ0 − γu β u · γ0 β 0γu β u · γ0 β 0− γu γ0 β 0 + γu β uγ0 + 1(7.16)

ͱද͢͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺβ u = u/cɺγu = 1/(1−‫͢ࢉܭ‬Δ͜ͷྲྀΕʹैͬͯɺ‫ํࢉܭ‬๏Λ঺հ͢Δɻ·ͣ࢝ΊʹɺϏʔϜͷ੩ࢭ‫ܥ‬Λఆٛ͢ΔɻϏʔϜͷΤωϧΪʔβ 2u )1/2 Ͱ͋Δɻ·ͨɺ੩ࢭ‫Ͱܥ‬ͷϚΫϩཻࢠͷ LorentzҼࢠ͸γu = γ0 γu (1 − β0 βu )֦͕Γ͕༗‫͋Ͱݶ‬Δ৔߹ʹ͸ɺ‫ʹີݫ‬ϚΫϩཻࢠ͕੩(7.17)

ࢭͯ͠‫͑ݟ‬Δ‫ܥ‬͸ଘࡏ͠ͳ͍ɻ͔͠͠ɺ͜͜Ͱ͸Ϗʔͱఆٛ͢Δ͜ͱ͕Ͱ͖Δ͕ɺ͜Ε͕͋Δఔ౓Ҏ্େ͖ϜͷΤωϧΪʔ֦͕Γ͕े෼খ͍͞ͱͯ͠ɺ੩ࢭ‫ͱܥ‬͍৔߹ʹ͸ɺ੩ࢭ‫Ͱܥ‬ͷཻࢠͷ଎౓͕ࠩແࢹͰ͖ͳ͘
ͳΔͨΊɺ஫ҙ͕ඞཁͰ͋Δɻ㸲㸫 ࣍ʹɺ੩ࢭ‫Ͱܥ‬ϏʔϜΛऔΓғΉ্ۭؒʹ‫ڥ‬քϘοΫε Ω Λઃఆ͠ɺͦͷதʹ֨ࢠΛ࡞Δɻͦͯ͠ɺཻࢠ෼෍ (r i , ui ) ͔Β֨ࢠ఺্Ͱͷిՙີ౓ ρ(r  ) Λ‫ٻ‬ΊͷपΓͷ֨ࢠ఺্ͷ஋͔Β಺ૠͯ͠‫͚ͳ͠ࢉܭ‬Ε͹ͳΒͳ͍ɻ࠷‫ʹޙ‬ɺ࣮‫͚͓ʹܥࣨݧ‬Δཻࢠ i ͷҐஔͰͷి࣓৔

Δɻ֨ࢠͷऔΓํ͸ɺ౳ִؒ֨ࢠ΍ඇ౳ִؒ֨ࢠͳͲ

Λ‫ٻ‬ΊΔͨΊʹɺ੩ࢭ‫Ͱܥ‬ͷ੩ి৔ E i Λ‫ ٯ‬Lorentz

͕͋Δɻ֨ࢠΛઃஔ͢ΔࡍʹॏཁͱͳΔύϥϝλ͸ɺม‫͢׵‬Δɻi Ͱͷి࣓৔͸ɺ
γ0E i = γ0 E i −(β 0 · E i )β 0 (7.20)
γ0 + 1֨ࢠͷ૯਺Ͱ͋Δɻ֨ࢠͷ਺͕૿͑Ε͹ɺΑΓۭؒతʹࡉ͔͍৘ใ͕‫·ؚ‬ΕΔ͜ͱʹͳΔ͕ɺ͋·Γʹ֨ࢠ਺Λ૿΍͢ͱɺిՙີ౓ ρ(r  ) ͕ෆ࿈ଓʢҰͭͷηϧ

Bi಺ʹཻࢠ͕਺‫͔͠ݸ‬ೖΒͳ͘ͳΔʣʹͳΓɺ·ͨ֨ࢠ఺ͷ਺͕૿͑ΔͨΊ௕͍‫͕ؒ࣌ࢉܭ‬ඞཁͱͳΔɻैͬ
ͯɺϚΫϩཻࢠͷ૯਺ͱཁ‫͢ٻ‬Δਫ਼౓͔Βɺ֨ࢠͷ਺ͱִؒʢ͋Δ͍͸‫ڥ‬քϘοΫεͷେ͖͞ʣΛద੾ʹઃఆ͢Δඞཁ͕͋Δɻρ−∇ V =
0γ0 β 0 × E ic(7.21)

ͱ‫͞ࢉܭ‬ΕΔɻ͜ͷి࣓৔Λ༻͍ͯɺཻ֤ࢠʹՃΘΔྗΛ‫͠ࢉܭ‬ɺ࣍ͷ࣌ؒεςοϓͰͷཻ֤ࢠͷҐஔͱ଎౓Λ‫͢ࢉܭ‬ΔɻҎ্͕ɺ੩ࢭ‫ܥ‬ͷ֨ࢠ্Ͱۭؒిՙޮ
ՌΛ‫͢ࢉܭ‬ΔྲྀΕͰ͋Δɻ

͜ͷిՙີ౓ ρ(r  ) ͔Β Poisson ํఔࣜ
2=͜ͷํ๏Ͱͷར఺͸ɺిՙີ౓Λ࿈ଓͱͯ͠ѻ͏ͨ

(7.18)

Ίʹɺ఺ిՙؒͷ૬‫༻࡞ޓ‬Λ௚઀‫͢ࢉܭ‬Δํ๏ʹൺ΂ͯߴ଎Ͱ͋Δͱ͍͏͜ͱ͕͋Δɻͨͩ͠ɺϏʔϜͷΛղ͍ͯɺ੩ࢭ‫ܥ‬ͷ੩ి৔Λ‫ٻ‬ΊΔ͜ͱʹͳΔɻ͜

ΤωϧΪʔ֦͕Γ͕େ͖͍৔߹ʹ͸ɺҰͭͷ੩ࢭ‫ܥ‬Λ͜Ͱ͸ɺPoisson ํఔࣜͷղ͖ํͷৄࡉʹ͍ͭͯ͸৮

ఆٛ͢Δ͜ͱ͕Ͱ͖ͳ͍ͨΊʹɺ஫ҙ͕ඞཁͰ͋ΔɻΕͳ͍ͷͰɺ਺஋‫ࢉܭ‬ͷจ‫Ͳͳݙ‬Λࢀরͯ͠ཉ͍͠ɻ

·ͨɺཻࢠີ౓͕૿େ͠ɺিಥతͳྗ͕ࢧ഑తʹͳΔ

͜͜Ͱ͸ɺPoisson ํఔࣜΛղ͘ࡍʹॏཁͱͳΔɺ‫ڥ‬ྖҬͰ͸ɺ͜͜Ͱͷํ๏ͷ‫׈‬Β͔ͳۭؒిՙྗ͚ͩͰ

ք৚݅ʹ͍ͭͯ঺հ͓ͯ͘͠ɻPoisson ํఔࣜ͸ɺ‫ڥ‬͸ɺਖ਼͍͠෺ཧΛऔΓೖΕͨ͜ͱʹ͸ͳΒͳ͍ɻ͜ͷքϘοΫε Ω ಺ʹઃஔ͞Εͨ֨ࢠ্Ͱղ͔ΕΔ͕ɺ

ͨΊɺର৅ͱ͢ΔϏʔϜͷ Debye ௕Λ‫͠ࢉܭ‬ɺۭؒ͜ͷ Ω ͷऔΓํͱ Ω ্Ͱͷ‫ڥ‬ք৚݅ͷऔΓํ͕ॏཁతʹ‫׈‬Β͔ͳۭؒిՙྗ͕ࢧ഑త͔ɺ͋Δ͍͸িಥత

ͱͳΔɻ‫ڥ‬ք৚݅ʹ͸ɺDirichlet ‫ڥ‬ք৚݅ (∂Ω ্Ͱͳྗ΋औΓೖΕͳ͚Ε͹ͳΒͳ͍͔Λ‫ۃݟ‬ΊɺۭؒిV = 0)ɺOpen ‫ڥ‬ք৚݅ (limr→∞ V = 0)ɺNeumann‫ڥ‬ք৚݅ (∂Ω ্Ͱ ∂V /∂r = 0) ͕͋Δɻ͜͜Ͱɺ∂Ωՙͷ‫ํࢉܭ‬๏Λબ୒͢Δඞཁ͕͋Δɻ͸‫ڥ‬քΛғ͏‫ڥ‬ք໘Λද͢ɻDirichlet ‫ڥ‬ք৚݅͸ɺ‫ڥ‬ք্Ͱ V = 0 ͱͳΓಋମද໘Λද͢͜ͱʹͳΔɻϏʔϜ͕ۚଐ੡ͷਅۭνΣϯόʔ಺Λਐߦ͢Δ৔߹ʹ͸ɺDirichlet ‫ڥ‬ք৚݅Λ࢖༻͠ɺ‫ڥ‬քϘοΫεΛਅۭνΣϯόʔͱಉαΠζʹऔΔ͜ͱʹΑΓɺಋମද໘ͷӨ‫ڹ‬Λද‫͢ݱ‬Δ͜ͱ͕Ͱ͖Δɻͨͩ͠ɺ‫ڥ‬քϘοΫεΛେ͖͘औΓɺ͞Βʹ֨ࢠ਺Λ૿΍͢ͱɺ௕͍‫͕ؒ࣌ࢉܭ‬ඞཁͱͳΔɻਅۭνΣϯόʔද໘ͰͷಋମͷӨ‫ڹ‬Λແࢹ͢Δ৔߹ʹ͸ɺOpen ‫ڥ‬ք৚݅Λ࢖͏͜ͱʹΑΓද‫͖Ͱݱ‬Δɻ͜ͷΑ͏ʹɺର৅ͱ͢Δ෺ཧ‫ݱ‬৅ʹ߹Θͤͯ‫ڥ‬ք৚݅ͱ‫ڥ‬քϘοΫεͷαΠζΛબ୒͢Δඞཁ͕͋Δɻ·ͨɺ‫ڥ‬ք৚݅͸ Poisson ํఔࣜΛ਺஋తʹղ͘৔߹ͷ҆ఆੑʹ΋Ө‫͢ڹ‬ΔͷͰɺ஫ҙ͢Δඞཁ͕
͋Δɻ

Poisson ํఔࣜΛղ͍ͯಘΒΕͨ੩ిϙςϯγϟϧ
V ͔Βɺ੩ࢭ‫Ͱܥ‬ͷ੩ి৔͸ɺE  = −∇V

(7.19)

ͱ‫͞ࢉܭ‬ΕΔɻE  ͸֨ࢠ఺্Ͱͷ V ͔Β‫͞ࢉܭ‬ΕΔ
ͨΊɺ֤ϚΫϩཻࢠ্Ͱͷ੩ి৔ E  (r i ) = E i ͸ɺr i

㸲㸫 

ిࢠॐ

8

͜͜Ͱ͸ɺcERL Ͱͷ࣮ࡍͷϏʔϜӡసʹΑΔ݁Ռ
Λ঺հ͠ͳ͕Βɺ࣮‫݁ݧ‬ՌΛ࠶‫͢ݱ‬ΔͨΊͷిࢠॐͷղੳϞσϧิਖ਼๏ͱɺిࢠॐۙ๣ͷۭؒిՙޮՌʹ͍ͭͯ঺հ͢Δɻ

8.1

ిࢠॐͷղੳϞσϧͷิਖ਼

8.1.1

ిࢠॐʹΑΔऩଋྗਤ 47: cERL ೖࣹ‫ۙث‬๣ͷϨΠΞ΢τɻిࢠॐɺ2 ୆ͷιϨϊΠυి࣓ੴɺόϯνϟʔۭಎɺ2 ୆ͷεΫηΫγϣϯ 2.3 ͷਤ 13 ʹࣔͨ͠Α͏ʹɺిࢠॐϦʔϯϞχλ (MS1, MS2) ʹΑͬͯߏ੒͞ΕΔɻ

ͷ੩ి৔෼෍ʹΑͬͯిࢠϏʔϜʹऩଋɾൃࢄྗ͕༩
͑ΒΕΔɻ͜ͷऩଋɾൃࢄྗ͸ɺి‫ܗۃ‬ঢ়ʹΑͬͯి৔ͷ‫౓ڧ‬෼෍͕มԽ͢Δ͜ͱʹΑͬͯੜ͍ͯ͡ΔɻΧιʔυ໘ۙ๣Ͱ͸ΤωϧΪʔ͕௿͍ͨΊʹɺ͜ͷӨ‫ڹ‬ΛΑΓ‫͘ڧ‬ड͚Δ͜ͱʹͳΔɻ͜ͷͨΊɺ࣮ࡍͷిࢠॐͰ͸ϏʔϜΛ࢖ͬͯిࢠॐͷऩଋྗΛධՁ͠ɺిࢠ
ॐͷղੳϞσϧͷิਖ਼Λߦ͏ඞཁ͕͋Δɻ͜͜Ͱ͸ɺਤ 48: ిࢠॐͷऩଋྗଌఆ๏ͷ໛ࣜਤɻྭ‫ى‬ϨʔβʔϏʔϜΛ࢖ͬͨऩଋྗͷଌఆ๏ͱղੳϞσϧͷิਖ਼ʹΛরࣹ͢ΔҐஔΛม͑ͳ͕ΒɺεΫϦʔϯϞχλ MS1͍ͭͯ঺հ͢Δɻ্ͰͷϏʔϜͷॏ৺ҐஔͷมԽΛଌఆ͢Δɻ

8.1.2

ͨΕ͍ͯΔ͜ͱΛ͍ࣔͯ͠Δɻ͔͠͠ͳ͕Βɺଌఆ݁ిࢠॐͷऩଋྗͷଌఆํ๏ՌͱղੳϞσϧʹΑΔγϛϡϨʔγϣϯ݁ՌͰ͸Ԡ౴cERL Ͱ͸ޫӄ‫ ۃ‬DC ిࢠॐ͕࢖ΘΕ͓ͯΓɺిࢠϏʔϜΛੜ੒͢Δࡍʹ͸ޫӄ‫ʹۃ‬ϨʔβʔΛরࣹ͠ͷͣΕ͕ΈΒΕͨɻ༌ૹ࿏্Ͱ֎ཚͱͳΔ΋ͷ͸΄΅͍ͯΔɻΧιʔυ໘্ͷϨʔβʔরࣹҐஔ͸ɺϛϥʔҼ͢Δ΋ͷͱਪଌ͞ΕΔɻͦ͜ͰɺղੳϞσϧͷిࢠΛಈ͔͢͜ͱʹΑ੍ͬͯ‫͢ޚ‬Δ͜ͱ͕Ͱ͖ΔɻরࣹҐॐి‫ܗۃ‬ঢ়Λ‫ʹ͔ۇ‬มԽͤͯ͞ɺ࠷΋ྑ࣮͘‫݁ݧ‬ՌΛஔΛม͑ͳ͕ΒԼྲྀʹ͋ΔεΫϦʔϯϞχλ্ͷϏʔ࠶‫͢ݱ‬Δ‫ܗ‬ঢ়Λ୳ࡧ͢Δ͜ͱͱͨ͠ɻ

ഉআ͞Ε͍ͯΔͨΊɺ͜ͷͣΕ͸ిࢠॐͷऩଋྗʹ‫ى‬ϜҐஔͷมԽΛଌఆ͢Δ͜ͱʹΑͬͯɺిࢠॐͷऩଋྗΛධՁ͢ΔɻϏʔϜҐஔͷଌఆʹར༻ͨ͠εΫϦʔϯϞχλ MS1 ͷ഑ஔΛਤ 47 ʹࣔ͢ɻిࢠॐͱεΫ

8.1.3

ిࢠॐղੳϞσϧͷิਖ਼ϦʔϯϞχλͷؒʹ͸ɺιϨϊΠυి࣓ੴ SL1 ͱόਤ 50 ʹΧιʔυ‫ܗ‬ঢ়ͷ֦େਤΛࣔ͢ɻి‫ۃ‬த৺෇ϯνϟʔۭಎ͕͋Δ͕ɺऩଋྗଌఆͷࡍʹ͸͜ΕΒ͸ۙʹԜΈ͕͋Γɺ͜ͷ෦෼͕ిࢠॐۙ๣ͷऩଋྗʹӨ͢΂ͯΦϑʹ͞Ε͍ͯΔɻ͜ͷͨΊɺిࢠॐ͔ΒεΫ‫ڹ‬Λ༩͑ΔɻGaAs ޫӄ‫ۃ‬Λอ࣋͢ΔΧιʔυύοΫϦʔϯϞχλؒΛࣗ༝ۭؒ (υϦϑτۭؒ) ͱΈͳ͢͸औΓ֎͕͠Մೳͳߏ଄ͱͳ͍ͬͯΔͨΊɺ͜ͷԜΈ͜ͱ͕Ͱ͖Δɻ͕ઃ‫ܭ‬ΑΓ‫ͣʹ͔ۇ‬ΕΔՄೳੑ͕͋Δɻઃ‫Ͱܭ‬͸͜ͷిࢠॐऩଋྗଌఆ๏ͷ໛ࣜਤΛਤ 48 ʹࣔ͢ɻΧιʔԜΈ͸ 0.5 mm ͱͳ͍ͬͯΔɻਤ 49 ͷଌఆ݁ՌΛ‫ݟ‬υ໘্ͷϨʔβʔরࣹҐஔΛม͑ͳ͕ΒɺMS1 ্Ͱ

Δͱɺ࣮‫݁ݧ‬Ռͷํ͕ϞσϧΑΓ΋ൃࢄྗ͕ऑ͍͜ͱ

ͷϏʔϜͷॏ৺ҐஔͷมԽΛଌఆ͢ΔɻଌఆͰ͸ɺۭ͕Θ͔Δɻ͜Ε͸ɺΧιʔυத৺෇ۙͷԜΈྔ͕ઃ‫ܭ‬ؒిՙޮՌʹΑΔӨ‫ڹ‬Λආ͚ΔͨΊʹόϯνిՙ͸ΑΓ΋େ͖͘ͳ͍ͬͯΔ͜ͱΛ͍ࣔͯ͠ΔɻԜΈྔ͕

10 fC ͱඇৗʹ௿͍஋ʹઃఆ͞ΕͨɻϨʔβʔরࣹҐஔΛม͑ͨͱ͖ͷϏʔϜҐஔͷมԽͷଌఆ݁ՌͱγϛϡϨʔγϣϯ݁ՌΛਤ 49 ʹࣔ͢ɻେ͖͍৔߹͸ɺΧιʔυ໘্ͷ Ez0 ্͕ঢ (ઈର஋ਫฏํ޲Ґஔͱਨ௚ํ޲Ґஔͷଌఆ݁Ռ͸΄΅ಉ͡Ԡಇ͘͜ͱʹͳΓɺిࢠॐશମͱͯ͠͸ൃࢄྗ͕ऑΊΒ౴ͱͳ͓ͬͯΓɺ͜Ε͸ిࢠॐऩଋྗͷ࣠ରশੑ͕อ

ΕΔ͜ͱʹͳΔɻͰ͸‫ݮ‬গ) ͢Δ͜ͱʹͳΓɺ∂Ez0 /∂z Λ‫ݮ‬গͤ͞Δ͜ͱʹͳΔɻैͬͯɺΧιʔυ໘্ͰΑΓ‫͍ڧ‬ऩଋྗ͕㸲㸫 390 kV

20
Measurement, x
Measurement, yCalculation

15
10

12
10
8
6
4
2
0
−2
−4
−6
−8
−10
−12
−6

5
0
−5

−10
−15
−20
−6

−4

−2
0
2
4
Laser offset on cathode (mm)390 kV

Position on MS1 (mm)Position on MS1 (mm)Measurement: x
Measurement: yCalculation: recess 0.9 mm
Calculation: recess 0.5 mm

6ਤ 49: ిࢠॐऩଋྗͷଌఆ݁Ռɻిࢠॐిѹ͸ 390

kVɻਫฏํ޲ͱਨ௚ํ޲ͷଌఆ݁ՌͱϞσϧ‫ʹࢉܭ‬
ΑΔ݁Ռ (ԁ౵ରশੑ͕͋ΔͨΊҰͭͷΈ) Λࣔ͢ɻ

−4

−2
0
2
4
Laser offset on cathode (mm)6

ਤ 51: ిࢠॐి‫ܗۃ‬ঢ়ͷղੳϞσϧͷิਖ਼ɻਤ 49 ͷଌఆ݁ՌΛ࠶‫͢ݱ‬ΔΑ͏ʹɺి‫ۃ‬த৺ͷԜΈΛิਖ਼
ͨ͠ɻ

8.1.4

ϏʔϜαΠζͷଌఆΧιʔυ͔ΒऔΓग़͞ΕΔిࢠϏʔϜͷੑ࣭͸ɺϨʔβʔͷ৚݅ (রࣹ௚‫ܘ‬ɺ೾௕ɺ࣌ؒ෯ɺग़ྗ) ʹ‫͘ڧ‬ґଘ͢ΔɻcERL ͷϏʔϜӡసͰ͸ɺ೾௕ 532
ਤ 50: Χιʔυ‫ܗ‬ঢ়ͷ֦େਤɻి‫ۃ‬த৺෇ۙʹԜΈ͕͋Γɺ͜ͷ෦෼͕ిࢠॐۙ๣ͷऩଋྗʹӨ‫ڹ‬Λ༩͑nm ͕༻͍ΒΕ͍ͯΔɻ͜ͷͱ͖ॳ‫ظ‬ΤϛολϯεΛ
ܾΊΔ kB T ͸ GaAs ޫӄ‫ۃ‬ͷ৔߹ɺ120 meV Ͱ͋ΔɻΧιʔυύοΫ͸औΓ֎͕͠Մೳͳߏ଄ͱͳͬͯ

Δ [16]ɻϨʔβʔͷ࣌ؒߏ଄͸ɺ3 ps rms ͷΨ΢ε෼͍ΔͨΊɺ͜ͷԜΈ͕ઃ‫ܭ‬ΑΓ‫ͣʹ͔ۇ‬ΕΔՄೳੑ͕෍Ͱ͋Δɻԣํ޲ͷ‫౓ڧ‬෼෍͸ϐϯϗʔϧͰ੾Γग़͋͠Δɻͯ΄΅Ұ༷ԁ෼෍ͱͳ͍ͬͯΔɻిࢠϏʔϜͷԣํ޲ϏʔϜμΠφϛΫεΛߟ͑Δͱ͖ɺ͜ͷॳ‫ظ‬Ϩʔβʔ࣮‫݁ݧ‬ՌΛղੳϞσϧʹΑΔγϛϡϨʔγϣϯͰ࠶௚‫͕ܘ‬ॏཁͱͳΔɻ͜ΕΛ࣮‫ݧ‬తʹ‫͢ূݕ‬ΔͨΊʹɺ‫͢ݱ‬ΔͨΊʹɺԜΈྔΛ 0.9, 1.0, 1.2 mm ͱͨ͠ిۭؒిՙޮՌΛແࢹͰ͖Δ 10 fC ͷόϯνిՙͷϏʔ‫ܗۃ‬ঢ়Λ࡞੒͠ɺγϛϡϨʔγϣϯΛߦͬͨɻਤ 51Ϝʹରͯ͠ɺMS1 ্ͰϏʔϜαΠζͷଌఆΛߦͬͨɻʹࣔ͞ΕΔΑ͏ʹɺԜΈྔ 0.9 mm ͷͱ͖ʹ࠷΋ྑਤ 52 ʹ SL1 Λม͑ͨͱ͖ͷ MS1 ্ͷϏʔϜαΠ࣮͘‫݁ݧ‬ՌΛ࠶‫͢ݱ‬Δ͜ͱͱͳͬͨɻ͜ΕΒͷ݁ՌΑζͷԠ౴ͷଌఆ݁ՌͱɺγϛϡϨʔγϣϯ݁Ռ (ิਖ਼ΓɺղੳϞσϧΛԜΈྔ 0.5 mm ͔Β 0.9 mm ʹิ͞Εͨి‫ܗۃ‬ঢ়Λ࢖༻) Λࣔ͢ɻγϛϡϨʔγϣϯͰ

ਖ਼͢Δ͜ͱʹͨ͠ɻͷॳ‫ظ‬Ϩʔβʔ௚‫ܘ‬͸ઃ‫ܭ‬஋Ͱ͋Δ 1.2 mm ͱͨ͠ɻ͜͜·Ͱ͸ɺϏʔϜͷॏ৺ҐஔͷΈΛண໨ͨͨ͠

ਤ 52 ΑΓɺଌఆ݁ՌͱϞσϧ‫ࢉܭ‬͸͔ͳΓ͍͕ۙɺΊɺϏʔϜΛ୯ཻࢠͱͯ͠औΓѻͬͨ͜ͱʹͳΔɻ࣮

‫ͣʹ͔ۇ‬Ε͕࢒͍ͬͯΔ͜ͱ͕Θ͔Δɻ࣮‫݁ݧ‬ՌΛΑࡍʹ͸ɺϏʔϜ͸༗‫͕֦ͳݶ‬ΓΛ΋ͭͨΊɺ࣍ͷஈ֊Γྑ͘࠶‫͢ݱ‬ΔͨΊʹɺϞσϧ‫ࢉܭ‬ͷϨʔβʔ௚‫ܘ‬Λͱͯ͠ϏʔϜαΠζʹର͢ΔӨ‫ڹ‬Λௐ΂Δ͜ͱͱ͠ม͑ͨ‫ࢉܭ‬Λߦͬͨɻਤ 52 ʹࣔ͢Α͏ʹɺϨʔβʔͨɻ͜ͷͨΊʹɺιϨϊΠυ SL1 ͷ࣓৔Λม͑ͨͱ௚‫ ܘ‬1.1 mm ͷͱ͖࠷΋ྑ࣮͘‫݁ݧ‬ՌΛ࠶‫͢ݱ‬Δ͜ͱ

͖ͷɺMS1 ্ͰͷϏʔϜαΠζͷԠ౴Λଌఆͨ͠ɻ

ͱͳͬͨɻ͜ͷ݁ՌΛ֬ೝ͢ΔͨΊʹɺ௚઀Ϩʔβʔ௚‫ܘ‬Λଌఆͨ͠ͱ͜Ζɺਫฏํ޲ 1.02 mmɺਨ௚ํ޲ 1.09 mmɺฏ‫ ۉ‬1.06 mm ͱͳΓɺϏʔϜͰଌఆͨ͠ 1.1 mm ͱ΄΅Ұக͢Δ͜ͱ͕֬ೝ͞ΕͨɻҎ্㸲㸫 

kBT = 120 meV

0.5 pC/bunch

Measurement: x
Measurement: x
Calculation: d = 1.0 mm
Calculation: d = 1.1 mm
Calculation: d = 1.2 mm

2σx, σy on MS1 (mm)σx, σy on MS1 (mm)

3

1

2

1

0

0
0

10
ISL1 (A)measurement x
measurement ysimulation, 0.5 pC
simulation, without SC

20

5
10
Excitation current of SL1 (A)15ਤ 53: ۭؒిՙޮՌ͕‫ݱ‬Ε͸͡ΊΔόϯνిՙ 0.5
ਤ 52: ۭؒిՙޮՌͷޮ͔ͳ͍όϯνిՙͰɺιϨϊΠυి࣓ੴ SL1 Λม͑ͨͱ͖ͷεΫϦʔϯϞχλpC ʹରͯ͠ͷιϨϊΠυి࣓ੴ SL1 ͷεΩϟϯ݁Ռɻۭؒిՙͳ͠ͷ৔߹ͷ‫݁ࢉܭ‬Ռͱൺ΂Δͱɺۭؒ

MS1 ্ͰͷϏʔϜαΠζͷԠ౴Λଌఆɻ࣮‫݁ݧ‬ՌΛిՙޮՌʹΑͬͯϏʔϜαΠζͷ֦͕Γ͕େ͖͘ͳͬ

࠶‫͢ݱ‬ΔΑ͏ʹɺղੳϞσϧͷྭ‫ى‬Ϩʔβʔ௚‫ܘ‬Λम͍ͯΔ͜ͱ͕Θ͔Δɻి‫ܗۃ‬ঢ়Λิਖ਼ͨ͠ղੳϞσϧਖ਼ͨ͠ɻd = 1.1 mm ͷͱ͖ʹ࠷΋ྑ࣮͘‫݁ݧ‬ՌΛ࠶Ͱ࣮‫݁ݧ‬ՌΛ࠶‫͍͖ͯͰݱ‬Δɻ

‫͢ݱ‬Δɻ
͖ͷԣํ޲ɺॎํ޲ͷϏʔϜԠ౴Λଌఆ͠ɺॳ‫ظ‬Ϩʔͷ݁ՌΑΓɺϞσϧͷϨʔβʔ௚‫ܘ‬Λ 1.1 mm ʹิਖ਼βʔύϧε௕Λ͖ͪΜͱ೺Ѳ͓ͯ͘͜͠ͱ͕ॏཁͰ

͢Δ͜ͱͱͨ͠ɻ

͋Δɻ

8.2

ۭؒిՙޮՌͷ‫ূݕ‬ղੳϞσϧͷి‫ܗۃ‬ঢ়ͱॳ‫ظ‬Ϩʔβʔ௚‫ܘ‬ͷิਖ਼͕ऴΘͬͨ͋ͱʹɺిࢠॐۙ๣ͰͷۭؒిՙޮՌͷଌఆΛߦͬͨɻଌఆͨ͠όϯνిՙ͸ɺ0.5 pC ͱ 7.38 pC
Ͱ͋Δɻਤ 53 ʹόϯνిՙ 0.5 pC ͷͱ͖ͷ SL1 εΩϟϯ݁ՌΛࣔ͢ɻۭؒిՙͳ͠ͷ৔߹ͷ‫݁ࢉܭ‬Ռͱൺ΂ΔͱɺۭؒిՙޮՌʹΑͬͯϏʔϜαΠζͷ֦͕Γ͕େ͖͘ͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻ·ͨɺి‫ܗۃ‬ঢ়Λิਖ਼ͨ͠ղੳϞσϧͰ࣮‫݁ݧ‬ՌΛൺֱ͢Δͱɺ΄΅߹͍ͬͯΔ͜ͱ͕֬ೝ͞Εͨɻ
ਤ 54 ʹɺۭؒిՙޮՌ͕ࢧ഑తʹͳΔόϯνిՙ7.38 pC ʹରͯ͠ͷιϨϊΠυి࣓ੴ SL1 ͷεΩϟϯ݁ՌΛࣔ͢ɻόϯνిՙΛ্͛ͯ΋ɺ௿όϯνిՙͰి‫ܗۃ‬ঢ়Λิਖ਼ͨ͠ղੳϞσϧΛ༻͍࣮ͯ‫݁ݧ‬ՌΛ࠶‫͖Ͱݱ‬Δ͜ͱ͕֬ೝ͞Εͨɻ͞ΒʹόϯνిՙΛ্ۭ͛ͯؒిՙޮՌ͕‫ͬͳ͘ڧ‬ͨͱ͖ͷϏʔϜ༌ૹͰ͸ɺਤ 55 ʹࣔ͢Α͏ʹྭ‫ى‬ϨʔβʔύϧεΛελοΫͯ͠ॳ‫ࢠిظ‬όϯν௕Λ৳͹ͯ͠ిՙີ౓௿‫ݮ‬ΛਤΔඞཁ͕͋Δɻ͜ͷ৔߹ɺॳ‫ظ‬Ϩʔβʔύϧε௕Λਖ਼֬ʹ੍‫͢ޚ‬Δ͜ͱ͕ཁ‫͞ٻ‬Ε
Δ͜ͱʹͳΔɻ࣮ࡍͷϏʔϜӡసͰύϧεελοΩϯ
άΛߦ͏৔߹ʹ͸ɺελοΩϯά͢ΔִؒΛม͑ͨͱ

㸲㸫 

4

σx, σy on MS1 (mm)7.38 pC/bunch3

measurement x
measurement ysimulation, 7.38 pC

2

1

0

8

9
10
11
Excitation current of SL1 (A)12ਤ 54: ۭؒిՙޮՌ͕ࢧ഑తʹͳΔόϯνిՙ 7.38

pC ʹରͯ͠ͷιϨϊΠυి࣓ੴ SL1 ͷεΩϟϯ݁ՌɻόϯνిՙΛ্͛ͯ΋ి‫ܗۃ‬ঢ়Λิਖ਼ͨ͠ղੳϞσϧͰ࣮‫݁ݧ‬ՌΛ࠶‫͍͖ͯͰݱ‬Δɻਤ 55: ύϧεελοΩϯάʹΑΔɺ௕͍ྭ‫ى‬Ϩʔβʔύϧεͷੜ੒ɻ3 ps rms Ψ΢ε෼෍ͷϨʔβʔ෼෍Λ 8 ‫ݸ‬ελοΩϯάͯ͠ɺ௕͍ϑϥοττοϓΛ΋
ͭϨʔβʔύϧεΛ࡞ΔɻిࢠϏʔϜͷ࣌ؒํ޲ʹ௕͘ͳΓɺόϯν಺ͷిՙີ౓͕௿Լ͠ɺిࢠॐ௚‫Ͱޙ‬ͷۭؒిՙྗΛऑΊΒΕΔɻ㸲㸫 ೖࣹ‫ث‬

9͜͜Ͱ͸ɺERL ೖࣹ‫ث‬ͷ࠷దԽ‫ࢉܭ‬ͷྫʹ͍ͭͯ
঺հ͢Δɻόϯνిՙ্͕͕ͬͯ͘ΔͱۭؒిՙޮՌ
͕ࢧ഑తʹͳΔͨΊɺϏʔϜͷӡಈΛԣํ޲ͱॎํ޲
Ͱಠཱʹѻ͏͜ͱ͕Ͱ͖ͣɺ6 ࣍‫ݩ‬Ґ૬ۭؒ෼෍Λ૯߹తʹ੍‫͢ޚ‬Δ͜ͱ͕ඞཁͱͳΔɻ·ͨɺઢ‫ܗ‬Ճ଎‫ث‬ͷಛ௃Ͱ͋Δ্ྲྀ͔ΒԼྲྀ΁Ө‫఻͕ڹ‬೻͢Δͱ͍͏ਤ 56: ༌ૹ৚݅Λ࠷దԽ͢ΔͨΊͷ ERL ೖࣹ‫ث‬഑ੑ࣭͕͋ΔͨΊɺిࢠॐ͔Βೖࣹ‫Ͱ·ث‬Ұମͱͯ͠ߟ

ஔྫɻ߹ྲྀ෦ग़‫Ͱޱ‬ɺ‫֨ن‬ԽΤϛολϯεͱόϯν༻͍͑ͯ͘ඞཁ͕͋Δɻೖࣹ‫ث‬ϏʔϜϥΠϯͷઃ‫͓ʹܭ‬

ͷ 2 ͭΛಉ࣌ʹ࠷খ͔͢Δ͜ͱΛߟ͑ΔɻDC ͕ి͍ͯ΋ɺ࠷খΤϛολϯεΛૂ͏Α͏ͳઃ‫ܭ‬Λߦ͏৔ࢠॐɺSL ͕ιϨϊΠυి࣓ੴɺBC ͕όϯνϟʔۭ߹ʹ͸ɺ࠷্ྲྀͷిࢠॐ͔Βೖࣹ‫ث‬ग़‫Ͱ·ޱ‬Λ௨ͨ͠

ಎɺSC ͕ 2 ηϧͷೖࣹ‫ث‬௒఻ಋۭಎɺQM ͕࢛‫ిۃ‬ϏʔϜ༌ૹ৚݅

12ͷ࠷దԽ͕ඞཁͱͳΔɻ͜͜Ͱ͸ɺ

࣓ੴɺBM ͕ภ޲ి࣓ੴΛද͢ɻ਺஋‫ࢉܭ‬Λ༻͍ͨ ERL ೖࣹ‫ث‬༌ૹ৚݅ͷ࠷దԽྫΛ
঺հ͢Δɻ਺஋‫ࢉܭ‬ίʔυͱͯ͠ɺGeneral Particle

9.1.1

Tracer (GPT) Λ༻͍͍ͯΔɻ

MOGA Λ༻͍ͨ࠷దԽMOGA ͷ֓ཁΛ؆୯ʹ঺հ͢ΔɻϚϧνΦϒδΣ
ΫςΟϒ๏Ͱ͸ɺෳ਺ͷྔΛಉ࣌ʹ࠷దԽ (࠷খԽ͋9.1

Δ͍͸࠷େԽ) ͢Δ͜ͱΛߟ͑Δɻೖࣹ‫ث‬ͷ৔߹ʹ͸ɺೖࣹ‫ث‬༌ૹ৚݅ͷ࠷దԽిࢠॐ͔Βೖࣹ‫ث‬Λ௨ͬͯ߹ྲྀ෦ग़‫Ͱޱ‬Τϛολϯ
εΛ࠷খԽ͢Δ͜ͱΛߟ͑Δɻ͜ͷͱ͖ͷೖࣹ‫ث‬ͷ഑ஔΛਤ 56 ʹࣔ͢ɻ500 kV ޫӄ‫ ۃ‬DC ిࢠॐͷ‫ޙ‬ʹɺ1 ൪໨ͷιϨϊΠυɺόϯνѹॖ༻ͷόϯνϟʔۭಎɺ2 ൪໨ͷιϨϊΠυ͕ઃஔ͞Εɺͦͷ‫ ʹޙ‬2 ηϧͷ௒఻ಋۭಎ͕ 3 ୆ଓ͘ɻ͜ͷϏʔϜϥΠϯͰௐ੔Մೳͳύϥϝλ͸ɺॳ‫ظ‬෼෍ిࢠ෼෍ 13 (σx , σy , σt )ɺ
ిࢠॐిѹɺιϨϊΠυ 1 ͷ‫͞ڧ‬ɺιϨϊΠυ 2 ͷ‫͞ڧ‬ɺόϯνϟʔͷՃ଎ిѹͱҐ૬ɺ௒఻ಋՃ଎ۭಎ

3 ୆ʹ͍ͭͯͷҐ૬ͱՃ଎ిѹͷ‫ ܭ‬14 ‫͋Ͱݸ‬Δɻ͜ΕΛද 1 ʹࣔ͢ɻ߹ྲྀ෦ग़‫Ͱ·ޱ‬Λߟ͑Δͱɺ͜ΕΒͷଞʹϚονϯά෦ͷ 4 ‫࣓ిۃ‬ੴͷ‫͞ڧ‬΋ՃΘΓɺଟ͘ͷࣗ༝౓͕͋Δ͜ͱʹͳΔɻೖࣹ‫ث‬ͷઃ‫Ͱܭ‬͸ɺ͜ΕΒͷύϥϝλΛ࠷దԽ͢Δඞཁ͕͋Δ͕ɺύϥϝλ਺͕ଟ͍ͱ͍͏͜ͱ͕͜ΕΛ೉͍ͯ͘͠͠Δɻ‫ݱ‬࣮తͳ࣌ؒͰ࠷దԽΛߦ͏ͨΊʹ͸ɺޮ཰ͷྑ͍୳ࡧํ๏͕ඞཁͱͳΔɻೖࣹ‫ث‬ͷ࠷దԽํ๏ͱͯ͠͸ɺ͜
Ε·ͰʹҨ఻తΞϧΰϦζϜΛ༻͍ͨํ๏ͳͲ༗ྗ
ͳํ๏͕‫͔ͭز‬ఏҊ͞Ε͓ͯΓɺೖࣹ‫ث‬ͷઃ‫࢖ʹܭ‬ΘΕ͍ͯΔ [17, 36]ɻ͜͜Ͱ͸ɺϚϧνΦϒδΣΫςΟ
ϒ๏ͱҨ఻తΞϧΰϦζϜ (Multi Objective Genetic

Algorithm, MOGA) [17] Λ༻͍ͨϏʔϜ༌ૹ৚݅ͷ࠷దԽ݁Ռʹ͍ͭͯ঺հ͢Δɻग़‫Ͱޱ‬ͷ‫֨ن‬ԽΤϛολϯεͱόϯν௕Λಉ࣌ʹ࠷খ
Խ͢Δ͜ͱ͕ଟ͍ɻ‫֨ن‬ԽΤϛολϯεͱόϯν௕Λಉ࣌ʹ࠷খԽ͠Α͏ͱߟ͑ͨͱ͖ɺ͜ͷ 2 ͭͷྔ͸ۭؒిՙޮՌΛ௨ͯ͠૬൓͢Δؔ܎ʹͳΔɻ͜Ε͸ɺόϯν௕Λ୹͘͢Δͱిՙີ౓্͕ঢۭؒ͠ిՙྗ͕
‫ͳ͘ڧ‬ΓɺΤϛολϯε͕ѱԽ͢Δͱ͍͏͜ͱΛද͠
͍ͯΔɻ͜ͷΑ͏ͳ૬൓͢Δ 2 ͭͷྔΛಉ࣌ʹ࠷খԽ
͢Δ͜ͱ͸ɺ2 ͭͷંΓ߹͍Λ͚ͭΔΑ͏ͳ৚݅Λ୳͢͜ͱʹ૬౰͢ΔɻϚϧνΦϒδΣΫςΟϒ๏Ͱ͸ɺෳ਺ͷ࠷దԽ͍ͨ͠ྔΛಉ࣌ʹ࠷దԽ͢ΔͨΊͷํ๏Ͱ͋Δɻೖࣹ‫ث‬ϏʔϜϥΠϯͰ͸ɺ‫֨ن‬ԽΤϛολϯεͱόϯν௕͸ϏʔϜ༌ૹ࿏৚݅ʹΑܾͬͯ·Δɻೖࣹ‫ۭث‬ಎग़‫Ͱ·ޱ‬Λߟ͑ͨͱ͖ɺϏʔϜ༌ૹ࿏৚݅Λ੍‫͢ޚ‬Δͷ͸ද 1 ʹࣔͨ͠ 14 ‫ݸ‬ͷม਺ͱͳΔɻ͜͜Ͱ͸ɺ͜ͷม਺ͷηοτΛ༌ૹ৚݅ηοτͱ‫͢ʹͱ͜Ϳݺ‬ΔɻҰͭͷ༌ૹ৚݅ηοτ͸͋Δ 14 ‫ݸ‬ͷม਺ͷ஋Λ΋ͪɺ͜ͷ஋Λ΋ͱʹγϛϡϨʔγϣϯΛ࣮ࢪ͢Δ͜ͱͰɺҰͭͷ༌ૹ৚݅ηοτʹର͢Δ‫֨ن‬ԽΤϛολϯεͱόϯν௕ΛಘΔ͜ͱ͕Ͱ͖ΔɻϚϧνΦϒδΣ
ΫςΟϒ๏Ͱ͸ɺಉ࣌ʹෳ਺ͷ༌ૹ৚݅ηοτΛอ࣋ͯ͠ɺͦΕͧΕʹରͯ͠‫֨ن‬ԽΤϛολϯεͱόϯν௕Λ‫͢ࢉܭ‬Δɻ͜ͷͱ͖ɺ༌ૹ৚݅ηοτͷ਺͚ͩɺ‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷ૊͕Ͱ͖Δ͜ͱʹͳ

12 ϏʔϜ༌ૹ৚݅͸ి࣓ੴͷऩଋྗ΍ɺՃ଎ۭಎͷՃ଎ޯ഑ɾՃ
଎Ґ૬ʹΑܾͬͯ·ΔͨΊɺ͜ΕΒͷύϥϝλΛҰవΊʹͯ͠ϏʔϜ༌ૹ৚݅ͱ‫Ϳݺ‬ɻ
13 ॳ‫ࢠిظ‬෼෍͸ྭ‫ى‬Ϩʔβʔ෼෍ɺ೾௕ɺΧιʔυࡐ࣭ʹΑܾͬͯ·Δɻ

Δɻ͜ͷ‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷ૊ΛͦΕͧΕਫฏ࣠ɺਨ௚࣠ͱͯ͠ϓϩοτ͢Δͱɺ࠷খԽ͠Α͏ͱͨ͠ͱ͖ʹ૬൓͢Δ෺ཧྔͰ͋ΔͨΊɺ൓ൺྫ‫ۂ‬

㸲㸫 

1
2

ిࢠॐిѹ

ϯά‫ࢉܭ‬Λߦ͍‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷϨʔβʔॳ‫ظ‬෼෍ σx0

૊ΛಘΔ

3
4
5

Ϩʔβʔॳ‫ظ‬෼෍ σy0

6
7ιϨϊΠυ 2 ͷ‫͞ڧ‬8910όϯνϟʔۭಎ (Ґ૬)
SC ۭಎ 1ʢి৔)SC ۭಎ 1ʢҐ૬)

11
12SC ۭಎ 2ʢి৔)SC ۭಎ 2ʢҐ૬)

13
14SC ۭಎ 3ʢి৔)SC ۭಎ 3ʢҐ૬)

3. 50 ‫ݸ‬ͷ‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷ૊͔ΒɺϨʔβʔॳ‫ظ‬෼෍ σt0༌ૹ৚݅ηοτͷධՁΛߦ͏ιϨϊΠυ 1 ͷ‫͞ڧ‬

4. Ҩ఻తΞϧΰϦζϜͰ࣍ͷੈ୅ͷ 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτͷੜ੒

όϯνϟʔۭಎ (ి৔)

5. ઃఆͨ͠ੈ୅ (‫܁‬Γฦ͠ճ਺) ʹ౸ୡ͢Δ·Ͱɺ
2. - 4. ͷૢ࡞Λ‫܁‬Γฦ͢͜ͷΑ͏ͳྲྀΕͱͳΔ͕ɺ1 ੈ୅ຖʹ 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτʹରۭͯؒ͠ిՙޮՌΛ‫ؚ‬ΜཻͩࢠτϥοΩϯά‫ࢉܭ‬Λߦ͏͜ͱʹͳΓɺ‫ؒ࣌ࢉܭ‬͸๲େͳ΋ͷͱ
ͳΔɻ‫ؒ࣌ࢉܭ‬Λ୹ॖԽ͢ΔͨΊʹɺ্‫ه‬ͷ 2. ͷखද 1: ERL ೖࣹ‫Ͱث‬ௐ੔Մೳͳ߲໨ɻ࠷దԽ‫Ͱࢉܭ‬͸͜ΕΒͷ߲໨Λࣗ༝ม਺ͱͯ͠ௐ੔͢Δ͜ͱʹͳΔɻ
ͨͩ͠ɺిࢠॐͷిѹ͸‫ج‬ຊతʹߴ͚Ε͹ߴ͍΄Ͳྑ͍ͷͰɺ௨ৗͷ࠷దԽ‫Ͱࢉܭ‬͸ҹՃՄೳͳ࠷େిѹʹॱ͸ฒྻԽͯ͠‫͢ࢉܭ‬Δ͜ͱ͕ଟ͍ɻͨͩ͜͠ͷ৔߹ʹ͸ɺCPU ͷ਺͕ඞཁͱͳΔɻ͜ͷΑ͏ʹ MOGAΛ࣮ࡍʹߦ͏৔߹ʹ͸ɺ‫͕ڥ؀ػࢉܭͳྗڧ‬ඞཁͱͳΔɻ‫ݻ‬ఆ͢Δ͜ͱ͕ଟ͍ɻ

9.1.2
ઢͷΑ͏ͳઢ (࠷લઢ) ͕‫ܗ‬੒͞ΕΔ͜ͱʹͳΔɻϚϧνΦϒδΣΫςΟϒ๏Ͱ͸͜ͷΑ͏ʹͯ͠‫࡞ܗ‬ΒΕͨ࠷લઢΛͳΔ΂͘‫͚ͮۙʹ఺ݪ‬ΔΑ͏ͳ (྆ํΛಉ࣌ʹ࠷খԽ͢ΔΑ͏ͳ) ༌ૹ৚݅ηοτΛ୳ࡧ͢Δ͜ͱΛߦ͍ͬͯΔɻྫ͑͹ɺ͜͜Ͱ঺հ͢Δ࠷దԽྫͰ
͸ɺಉ࣌ʹ 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτΛอ࣋͢ΔΑ͏ʹ͍ͯ͠Δɻ50 ‫ݸ‬ͷ༌ૹ৚͔݅Β‫ٻ‬Ίͨ‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷ૊ʹରͯ͠ɺ‫͔఺ݪ‬Β͍ۙ৚݅Λ༩
͑Δ༌ૹ৚݅ηοτΛΑΓదԠ౓͕ߴ͍ͱධՁ͢Δɻ͜͜·Ͱ͸ɺҟͳΔ 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτʹରͯ͠ɺ50 ‫ݸ‬ͷ‫֨ن‬ԽΤϛολϯεͱόϯν௕ͷ૊Λ‫ࢉܭ‬ͯ͠ɺͦͷ݁Ռ͔ΒͳΔ΂͘‫ͮۙ͘ʹ఺ݪ‬Α͏ͳ࠷લઢΛಘΔͱ͍͏͜ͱߟ͖͕͑ͯͨɺ࣍ʹඞཁͳͷ͸ɺ͜ͷ༌ૹ৚݅ΛͲͷΑ͏ʹมԽ͍͔ͤͯ͘͞ͱ͍͏͜
ͱͰ͋ΔɻMOGA Ͱ༻͍Δͷ͸Ҩ఻తΞϧΰϦζϜ
Ͱ͋ΔɻҨ఻తΞϧΰϦζϜͷৄࡉ͸͜͜Ͱ͸লུ͢
Δ͕ɺ50 ‫ݸ‬ͷ༌ૹ৚݅ʹରͯ͠ɺબ୒ɺަࠥɺಥવ
มҟͱ͍͏ૢ࡞Λߦ͍ɺ࣍ͷੈ୅ͷ 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτΛ࡞Δ͜ͱʹͳΔɻ͜ΕΛ‫܁‬Γฦͯ͠ɺదԠ౓
ͷߴ͍༌ૹ৚݅ηοτΛ୳ࡧ͍ͯ͘͜͠ͱʹͳΔɻ۩
ମతͳ‫ࢉܭ‬ͷྲྀΕ͸࣍ͷΑ͏ʹͳΔɻ

ೖࣹ‫࠷ث‬దԽ‫ࢉܭ‬ͷྫ

ਤ 56 ʹࣔ͢ೖࣹ‫ʹث‬ରͯ͠ɺిࢠॐ͔Β߹ྲྀ෦ग़‫Ͱ·ޱ‬ͷ༌ૹ৚݅ͷ࠷దԽΛߦͬͨྫΛ঺հ͢Δɻ༌ૹ৚݅ηοτ͸ɺද 1 ͷ͏ͪͷిࢠॐిѹΛআ͍ͨม਺ʹɺೖࣹ‫ۭث‬ಎग़‫͔ޱ‬Β߹ྲྀ෦ೖ‫Ͱ·ޱ‬ͷؒʹ഑ஔ͞Εͨ 5 ୆ͷ࢛‫࣓ిۃ‬ੴͷ‫͞ڧ‬ΛՃ͑ͨ΋ͷͱͳΔɻ
࠷దԽ‫ࢉܭ‬͸࣍ͷΑ͏ͳ৚݅ͷԼͰߦΘΕͨɻॳ‫ظ‬෼෍ͷੜ੒ʹ͸ɺ४ཚ਺ [37] Λ࢖༻ͨ͠ɻ·ͨɺτϥοΩϯά‫ʹࢉܭ‬͸ GPT [33] Λ࢖༻͠ɺۭؒిՙޮ
Ռͷ‫ʹࢉܭ‬͸ 3 ࣍‫ݩ‬ϝογϡ๏ (sc3dmesh) Λ༻͍ͨɻॳ‫ࢠిظ‬෼෍͸ϏΞ؈෼෍ (ਤ 10) ͱͨ͠ɻόϯνిՙ͸ −80 pC ͱͨ͠ɻ·ͨɺ࠷దԽ‫࣌ࢉܭ‬ͷిࢠϏʔϜΛߏ੒͢ΔϚΫϩཻࢠͷ਺͸ 5 k ‫ͨ͠ͱݸ‬ɻ

MOGA Λ༻͍ͯ༌ૹ৚݅Λ࠷దԽͨ݁͠ՌΛਤ 57
ʹࣔ͢ɻਤ 57 ͷΑ͏ʹɺ‫֨ن‬ԽΤϛολϯεͱόϯν௕͸૬൓͢Δؔ܎Ͱ͋Δ͜ͱ͕Θ͔Δɻද tab:paramʹ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ηοτͷม਺ͷ஋Λࣔ͢ɻ࠷దԽ͞Εͨ༌ૹ৚݅ʹର͢ΔϚΫϩཻࢠ਺ͷӨ‫ڹ‬Λਤ 58 ʹࣔ͢ɻਤ 58 ͸ɺϚΫϩཻࢠ਺Λม͑ͨͱ͖
ͷ‫֨ن‬ԽΤϛολϯεͷมԽ͕ࣔ͞Ε͍ͯΔɻϚΫϩཻࢠ਺͕গͳ͍ͱ‫݁ࢉܭ‬Ռ͕มԽ͢Δ͜ͱ͕Θ͔Δɻ

1. 50 ‫ݸ‬ͷॳ‫ظ‬༌ૹ৚݅ηοτͷઃఆ (ཚ਺͔Βε࠷దԽ‫Ͱࢉܭ‬͸๲େͳ਺ͷ‫ࢉܭ‬Λߦ͏ͨΊɺ࠷దԽͷॳ‫ظ‬ஈ֊Ͱ͸‫ؒ࣌ࢉܭ‬Λઅ໿͢ΔͨΊʹগͳ͍ϚΫϩλʔτ͢Δ৔߹͕ଟ͍)ཻࢠ਺Ͱ‫ࢉܭ‬Λߦ͍ͬͯΔɻ࠷ऴతͳ݁ՌΛಘΔʹ͸

2. 50 ‫ݸ‬ͷ༌ૹ৚݅ηοτʹରͯ͠ɺཻࢠτϥοΩ㸲㸫 

ද 2: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ηοτParameterValueਤ 57: ࠷దԽ͞Εͨ༌ૹ৚݅ͷԼͰͷόϯν௕ͱΤϛολϯεͷؔ܎ɻόϯν௕ͱΤϛολϯε͸߹ྲྀ෦ग़‫͞ࢉܭͰޱ‬Εͨɻ࠷దԽ‫Ͱࢉܭ‬͸ 50 ‫ݸ‬ͷҟͳΔϏʔϜ༌ૹ৚݅ͷηοτ͕࡞ΒΕɺϚϧνΦϒδΣΫ
ςΟϒ๏Λ༻͍ͨҨ఻తΞϧΰϦζϜ (MOGA) ʹΑͬͯ࠷దԽ͞Ε͍ͯΔɻਤͷதͷ఺͸ͦΕͧΕҟͳΔϏʔϜ༌ૹ৚݅Λ΋ͭɻόϯν௕ͱΤϛολϯε͸
૬൓͢Δؔ܎Ͱ͋Δ͜ͱ͕Θ͔Δɻ
ཻࢠ਺Λ૿΍ͨ͠‫ࢉܭ‬Λߦ͍ɺ݁Ռ͕҆ఆ͍ͯ͠Δ͔
Λ‫ݟ‬Δඞཁ͕͋Δɻ

σx0
σt0

0.34 (mm)17.6 (ps)

BSLA01
BSLA020.037 (T)0.024 (T)VBCA01
Eacc,SCA01
Eacc,SCA02

96.0 (kV)
7.6 (MV/m)
13.1 (MV/m)

Eacc,SCA03
φSCA0115.0 (MV/m)−20.9◦

φSCA02
φSCA03
K1,QM A01

−20.8◦
10.0◦−27.4 (m−2 )K1,QM A02
K1,QM A03

−4.6 (m−2 )29.8 (m−2 )K1,QM A04
K1,QM A05
zSLA01−2.7 (m−2 )−24.3 (m−2 )0.38 (m)zSLA02zBCA010.71 (m)1.14 (m)ϚΫϩཻࢠ਺ 5 k Ͱ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ηοτʹରͯ͠ɺཻࢠ਺Λ 200 k ʹͨ͠‫݁ࢉܭ‬ՌΛࣔ
͢ɻਤ 59 ʹ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ηοτʹର͢Δ‫֨ن‬ԽΤϛολϯε εnx , εny ͷ࣌ؒൃలΛࣔ͢ɻ
Normalized rms emittance, εnx, εny (mm mrad)·ͨɺਤ 60ɺਤ 61 ʹ͸ɺͦΕͧΕԣํ޲ϏʔϜαΠζ σx , σy ͱɺόϯν௕ σz ͷ࣌ؒൃలɺӡಈΤωϧΪʔ T ͱɺΤωϧΪʔ޿͕Γ σE ͷ࣌ؒൃలΛࣔ
͢ɻਤ 62 ʹɺ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹର͢Δ
ਫฏํ޲Ґ૬ۭؒ෼෍Λࣔ͢ɻ͜ͷΑ͏ʹɺMOGAΛ༻͍ͨ࠷దԽ‫ʹࢉܭ‬Αͬͯɺ߹ྲྀ෦ग़‫֨نͰޱ‬ԽΤϛολϯεͱόϯν௕ͷ྆ํΛಉ࣌ʹ࠷దԽ͢Δ͜ͱ
͕Ͱ͖Δɻ

0.8

cERL injector with rectangular magnets, −80 pC

εnx
εny

0.7

0.6

0.5
0

1

2
3
Number of particles

4

5

+5

[×10 ]

ਤ 58: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹରͯ͠ɺ਺஋
‫ͨ͠༻࢖Ͱࢉܭ‬ϚΫϩཻࢠ਺Λม͑ͨͱ͖ͷ‫֨ن‬ԽΤϛολϯεͷมԽɻγϛϡϨʔγϣϯͰ͸ϚΫϩཻࢠͰϏʔϜ෼෍Λۙࣅͯ͠‫͢ࢉܭ‬Δ͕ɺͦͷ਺͕গͳ͍
ͱ‫݁ࢉܭ‬Ռ͕มΘΔ͜ͱʹͳΔɻ㸲㸫 Normalized rms emittance (mm mrad)cERL injector with rectangular magnets, −80 pC, 200 k particles

εnx
εny

2

1

0

0

2
4
6
8
Longitudinal distance from cathode, s (m)ਤ 59: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹର͢Δ‫֨ن‬ԽΤϛολϯε εnx , εny ͷ࣌ؒൃలɻ

cERL injector with rectangular magnets, −80 pC, 200 k particles
cERL injector with rectangular magnets, −80 pC, 200 k particles

4
3
2
1
0

0

ϏʔϜαΠζ σx , σy ͱɺόϯν௕ σz ͷ࣌ؒൃలɻ

Rms beam size (mm)20
Kinetic energy
σE

10
0

2
4
6
8
Longitudinal distance from cathode, s (m)ਤ 62: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹର͢Δਫฏํ

cERL injector with rectangular magnets, −80 pC, 200 k particles

σx
σy
σz

3
2
1
0

0

30

޲Ґ૬ۭؒ෼෍ɻ

6

4

40

0

2
4
6
8
Longitudinal distance from cathode, s (m)ਤ 60: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹର͢Δԣํ޲

5

50

9
8
7
6
5
4
3
2
1
0

Energy spread, σE (keV)

σx
σy
σz

5

Kinetic energy (MeV)

Rms beam size (mm)

6

2
4
6
8
Longitudinal distance from cathode, s (m)ਤ 61: ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ʹର͢ΔӡಈΤωϧΪʔ T ͱɺΤωϧΪʔ޿͕Γ σE ͷ࣌ؒൃలɻ

㸲㸫 

10

͜͜Ͱɺs Λཧ૝‫ي‬ಓ্ͰͷϏʔϜਐߦํ޲ͷిࢠό߹ྲྀ෦ͷ෺ཧ

ϯνͷ࠲ඪɺs0 Λภ޲ి࣓ੴೖ‫ޱ‬ͷ࠲ඪͱͨ͠ɻҎԼ

ERL ͷ߹ྲྀ෦͸ɺೖࣹ‫͋Ͱث‬Δఔ౓ͷΤωϧΪʔ·ͰՃ଎͞ΕͨϏʔϜΛɺओՃ଎ۭಎͷ͋Δपճ෦ʹͷٞ࿦Ͱ͸ɺΤωϧΪʔ֦͕Γͱ͍͏ͱ͖ʹɺϏʔϜ‫͙ͨܨ‬Ίͷ෦෼Ͱ͋Δɻ͜ͷ෦෼Ͱ͸ɺภ޲ి࣓ੴΛ΋ͷͱ͢Δɻ͜ͷΑ͏ͳԾఆͷ΋ͱͰ͸ɺઢ‫ʹࣅۙܗ‬༻͍ͯೖࣹ‫͔ث‬ΒདྷͨϏʔϜΛ࣓৔ʹΑΓ‫ͯ͛ۂ‬ɺप

ΑΔిࢠͷਫฏํ޲ͷӡಈ͸ɺภ޲ి࣓ੴͷ‫཰ۂ‬൒‫ܘ‬ճ෦ͷ‫ي‬ಓʹ߹ྲྀ͢ΔΑ͏ʹ͍ͯ͠Δɻ߹ྲྀ෦ͷ࠷‫ޙ‬Λ ρɺx Λཧ૝‫ي‬ಓ͔Βͷਫฏํ޲ͷͣΕͱ͢Δͱɺ

ͷΤωϧΪʔ E0 Ͱ‫֨ن‬Խ͞Εͨ δ = δE /E0 Λࢦ͢Ͱ͸ϏʔϜͷपճ෦Λ௨͖ͬͯͨߴΤωϧΪʔϏʔϜ1xx = − 2 + (δ0 + δSC + κ(s − s0 ))ρρͱೖࣹ‫͔ث‬Βདྷͨ௿ΤωϧΪʔϏʔϜ͕߹ྲྀ͢Δ͜ͱʹͳΔɻ

(10.1)

߹ྲྀ෦खલ·Ͱ͸ɺೖࣹ‫ث‬Λߏ੒͢Δཁૉ͸௚ઢঢ়ͱ‫ه‬ड़͞ΕΔɻ͜͜Ͱɺx = d2 x/ds2 Ͱ͋ΔɻδSCʹ഑ஔ͞ΕΔͨΊɺϏʔϜͷ‫ي‬ಓ͕‫͕ۂ‬Δ͜ͱ͸ͳ͸ɺ߹ྲྀ෦ೖ‫Ͱ·ޱ‬ͷ্ྲྀ෦Ͱॎํ޲ۭؒిՙྗʹ͍ɻ͜ͷͨΊɺϏʔϜʹΤωϧΪʔ֦͕Γ͕ੜͨ͡৔

Αͬͯੜͨ͡ΤωϧΪʔ֦͕ΓͰ͋Δɻδ0 ͸ s = 0߹ʹ΋ɺϏʔϜ͸௚ਐ͍ͯ͠ΔͨΊʹɺΤωϧΪʔ֦ͰͷΤωϧΪʔ֦͕Γɺκ ͸ཁૉதͰͷॎํ޲ۭؒి

͕Γ͕ԣํ޲ͷӡಈʹӨ‫ڹ‬Λ༩͑Δ͜ͱ͸ͳ͔ͬͨɻ

ՙྗʹΑΔ‫֨ن‬Խ͞ΕͨϙςϯγϟϧͰ͋Δɻ͜ͷͱ͔͠͠ɺ߹ྲྀ෦Ͱ͸ภ޲ి࣓ੴʹΑͬͯϏʔϜ͕‫͛ۂ‬͖ɺ߹ྲྀ෦தͰͷਫฏํ޲ͷӡಈ͸ɺΒΕΔͨΊɺΤωϧΪʔ֦͕Γ͕෼ࢄؔ਺Λ௨ͯ͠ԣx ≡ (x, x , δ0 , δSC , κ)ํ޲ͷӡಈʹӨ‫ڹ‬Λ༩͑Δ͜ͱʹͳΔɻಛʹɺۭؒి(10.2)

ՙޮՌʹΑͬͯɺϏʔϜ͕߹ྲྀ෦Λ௨ա͢Δؒʹ΋ΤʹΑͬͯ‫ه‬ड़͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷ 5 ࣍‫ݩ‬ͷϕΫτωϧΪʔ֦͕Γ͕มԽ͢Δ৔߹ʹ͸ɺ௨ৗͷ෼ࢄؔ਺ϧͷ s0 → s1 ʹ͍ͭͯͷ࣌ؒൃల͸ɺ1 ࣍ͷసૹߦྻΛௐ੔͢Δ͚ͩͰ͸ɺԣํ޲ͷӡಈʹର͢ΔӨ‫ڹ‬ΛආR(s1 |s0 ) Λ༻͍ͯ

͚Δ͜ͱ͕ग़དྷͳ͍ɻͦͷͨΊɺ߹ྲྀ෦ͰͷϏʔϜͷx(s1 ) = R(s1 |s0 )tx(s0 )t෺ཧΛߟ͑Δࡍʹ͸ɺۭؒిՙޮՌͷӨ‫ڹ‬Λ‫ؚ‬Μͩ෼ࢄؔ਺Λߟ͑Δඞཁ͕͋Δɻ
·ͨɺ߹ྲྀ෦Ͱ͸΋͏Ұͭॏཁͳ෺ཧ‫ݱ‬৅͕͋Δɻ
ͦΕ͸ CSR ʹΑΔΤωϧΪʔ֦͕ΓͷมԽͰ͋ΔɻERL Ͱ͸୹͍όϯν௕ͷϏʔϜ͕ཁ‫͞ٻ‬ΕΔͨΊɺϏʔϜͷΤωϧΪʔ͕௿͍৔߹ʹ΋ɺCSR ͷӨ‫ڹ‬Λߟྀ͢Δඞཁ͕͋Δɻ͜͜Ͱ͸ɺۭؒిՙޮՌʹΑΔΤωϧΪʔ֦͕Γ͕ϏʔϜ͕૸ͬͨ‫ʹ཭ڑ‬ઢ‫ʹܗ‬ґଘ͢ΔͱԾఆͯ͠ɺԣ

ͱ‫ه‬ड़Ͱ͖Δɻ͜ͷࣜ͸ɺΤωϧΪʔ֦͕Γ͕͋Δ৔߹ͷӡಈΛ‫ه‬ड़͢Δɺ3 × 3 ͷߦྻΛ֦ுͨ͠΋ͷͰ͋Γɺ࣍ͷΑ͏ʹॻ͘͜ͱ͕Ͱ͖Δɻ
⎞⎛R11 R12 R13 R14 R15⎟⎜⎜ R21 R22 R23 R24 R25 ⎟
⎟
⎜
R=⎜
0
0 ⎟
⎟
⎜ R31 R32 R33
⎟
⎜
0
0
1
L ⎠
⎝ 0

ํ޲ͷࣹӨΤϛολϯεʹ༩͑ΔӨ‫ڹ‬ΛߟΔɻ·ͨɺ߹ྲྀ෦ͷΑ͏ͳϏʔϜΤωϧΪʔͷ௿͍ (∼ 10 MeV)৔߹Ͱͷ CSR ͷ‫ྫࢉܭ‬Λ঺հ͢Δɻ

(10.3)

00

0

0(10.4)

1

L ͸ཁૉͷ௕͞ɺRij (i, j = 1, 2, 3) ͸௨ৗͷ 3 × 3 ͷసૹߦྻͰ͋Δ 14 ɻ‫཰ۂ‬൒‫ ܘ‬ρɺ‫ ֯͛ۂ‬θ Λ΋ͭηΫλʔ‫ܕ‬ภ޲ి࣓ੴͷ৔߹ɺR11 = cos θɺR12 = ρ sin θɺ

10.1ॎํ޲ۭؒిՙྗʹΑΔΤϛολϯ
ε૿େͱͦͷิঈ͜͜Ͱ͸ɺӋౡࢯʹΑͬͯఏҊ͞Εͨ߹ྲྀ෦Ͱͷॎํ޲ۭؒిՙྗͷઢ‫ܗ‬ղੳ [21] ʹ͍ͭͯ঺հ͢Δɻ
·ͣɺߟ͍͑ͯΔిࢠόϯν͕ɺ߹ྲྀ෦಺Ͱͦͷԣํ޲ͱॎํ޲ͷ෼෍Λ΄ͱΜͲม͑ͳ͍ͱԾఆ͢Δ
ͱɺॎํ޲ۭؒిՙϙςϯγϟϧ͸ҰఆʹอͨΕΔͱ

R13 = ρ(1−cos θ)ɺR14 = ρ(1−cos θ)ɺR15 = ρ2 (θ −sin θ)ɺR21 = −1/ρ·cos θɺR22 = cos θɺR23 = sin θɺ
R24 = sin θɺR25 = ρ(1 − cos θ) ͱͳΔɻ·ͨɺଞͷཁૉ (ࣗ༝ۭؒɺ4 ‫࣓ిۃ‬ੴͳͲ) ʹ͍ͭͯ͸ɺ֤ཁૉͷ Rij (i, j = 1, 2, 3) ͱ R14 = R15 = R24 = R25 = 0Ͱ‫ه‬ड़͞ΕΔɻ͜ͷ৽ͨͳ 5 × 5 ͷసૹߦྻΛ༻͍Δ
ͱɺ࣍ͷΑ͏ʹۭؒిՙ෼ࢄؔ਺ (ζx , ζx ) Λఆٛ͢Δ

ߟ͑Δ͜ͱ͕Ͱ͖Δɻ͜ͷΑ͏ͳԾఆͷ΋ͱͰ͸ɺॎํ޲ۭؒిՙྗʹΑͬͯੜ͡ΔΤωϧΪʔ֦͕Γ͸ɺిࢠόϯν͕ਐΜͩ‫ ཭ڑ‬s − s0 ʹൺྫ͢ΔΈͳͤΔɻ

14 ֤ཁૉͰͷసૹߦྻʹ͍ͭͯ͸ɺࢀߟจ‫[ ݙ‬38] ͳͲΛࢀর͠
ͯཉ͍͠ɻ

㸲㸫 

͜ͱ͕Ͱ͖Δɻ⎛⎛⎞⎞ζx (s0 )ζx (s1 )
⎜ 
⎜ 
⎟
⎟
⎜ ζx (s0 ) ⎟⎜ ζx (s1 ) ⎟
⎜
⎜
⎟
⎟
⎜
⎟ = R(s1 |s0 ) ⎜
⎟
0
0
⎜
⎜
⎟
⎟
⎜
⎜
⎟
⎟
⎝ L(s0 ) ⎠⎝ L(s1 ) ⎠1(10.5)

1

͜͜ͰɺL(s) = s ͸‫ي‬ಓ௕Λද͢ɻ͜ͷۭؒిՙ෼ࢄؔ਺ (ζx , ζx ) Λ༻͍Δͱɺॎํ޲ۭ

ؒిՙྗʹΑΔҐ૬ۭؒ (x, x ) தͰͷҐஔͷͣΕ͸ɺ

d = (κζx , κζx )ਤ 63: ߹ྲྀ෦Ͱͷॎํ޲ۭؒిՙྗʹΑΔࣹӨΤϛολϯεͷ૿େͱͦͷิঈɻ

(10.6)
࣮‫͢ݱ‬ΔΑ͏ʹ߹ྲྀ෦खલͷϚονϯά෦Λௐ੔ඞཁͱͳΓɺۭؒిՙ෼ࢄؔ਺ͱۭؒిՙϙςϯγϟϧͷ

͕͋ΔɻੵʹΑͬͯ‫ه‬ड़͞ΕΔɻ͜ͷͱ͖ɺ߹ྲྀ෦ग़‫Ͱޱ‬ͷ֤όϯνεϥΠε͸ɺҐ૬ۭؒதͷ ζx x− ζx x = 0 ͷ௚ઢঢ়ʹ੔ྻ͢Δɻ͜ͷ༷ࢠΛਤ 63(a) ʹࣔ͢ɻ·ͨɺCSR ʹΑΔΤωϧΪʔ֦͕ΓʹΑΔޮՌ΋ಉ༷ͳઢ‫ʹࣅۙܗ‬Αͬͯղੳ͢Δ͜ͱ͕Ͱ͖ɺCSR
෼ࢄؔ਺Λఆٛ͢Δ͜ͱ͕Ͱ͖Δ [22]ɻ‫࣮ݱ‬ͷ߹ྲྀ෦߹ྲྀ෦ग़‫͚͓ʹޱ‬ΔΤϛολϯε͸ɺͰ͸ɺ͜ΕΒͷॎํ޲ۭؒిՙ෼ࢄؔ਺ͱ CSR ෼ࢄ

ε2 = (ε0 βx +D)2 (ε0 γx +D2 )−(ε0 αx +DD )2 (10.7)

ؔ਺ͷޮՌΛಉ࣌ʹ࠷খʹ͢ΔΑ͏ʹ͠ͳ͚Ε͹ͳΒͳ͍ɻͰධՁ͢Δ͜ͱ͕Ͱ͖Δɻ͜͜Ͱɺ(αx , βx , γx ) ͸߹ྲྀ෦ग़‫͚͓ʹޱ‬Δ Courant-Snyder ύϥϝλɺε0 ͸߹ྲྀ෦ೖ‫Ͱޱ‬ͷॳ‫ظ‬ΤϛολϯεͰ͋Δɻ(D, D  ) ͸Ґ

10.2

͜͜Ͱ͸ɺERL ೖࣹ‫ྲྀ߹ث‬෦ͷΑ͏ͳΤωϧΪʔͷ૬ۭؒதͰͷόϯνεϥΠεͷ rms ֦͕ΓͰ͋Γɺ

(D, D ) ≡ κrms (ζx , ζx )߹ྲྀ෦Ͱͷ CSR ͷӨ‫ڹ‬௿͍ྖҬͰͷ CSR ʹ͍ͭͯɺSagan ʹΑΔํ๏ [39](10.8)

Λ༻͍ͯ‫݁ͨ͠ࢉܭ‬ՌΛ঺հ͢Δ [40]ɻCSR ͷৄࡉʹ͍ͭͯ͸ɺౡాࢯͷςΩετ΍ࢀߟจ‫[ ݙ‬41] ͳͲ͔Β‫͞ࢉܭ‬ΕΔɻ͜ͷΑ͏ʹɺҐ૬ۭؒதͰͷ෼෍ͷ܏͖͕ɺۭؒి

Λࢀরͯ͠ཉ͍͠ɻ

ՙ෼ࢄؔ਺ʹΑͬͯόϯνεϥΠε͕੔ྻ͢Δํ޲ͱ
ͣΕ͍ͯΔͱɺਤ 63(a) ʹࣔ͢Α͏ʹΤϛολϯεͷ

10.2.1૿େ͕ੜ͡ΔɻͰ͸ɺ͜ΕΛճආ͢Δʹ͸Ͳ͏͢Ε͹Α͍͔ʁ߹ྲྀ෦Ͱ͸ϏʔϜޫֶؔ਺Λௐ੔Ͱ͖Δࣗ༝
౓͕ͳ͍ͨΊɺ߹ྲྀ෦खલͷϚονϯά෦ͷ 4 ‫࣓ిۃ‬ੴΛௐ੔ͯ͠ɺ߹ྲྀ෦ग़‫Ͱޱ‬ͷϏʔϜޫֶؔ਺Λௐ੔ͯ͠΍Ε͹Α͍ɻ߹ྲྀ෦ग़‫Ͱޱ‬ͷҐ૬ۭؒͷ෼෍ͷ܏
͖͸ɺ

ξ=

1
arctan22αxγx − βx(10.9)GPT/CSR ͷ։ൃ௨ৗΑ͘࢖༻͞ΕΔ CSR ͷ 1 ࣍‫ ݩ‬wake ‫ࢉܭ‬Λ‫ؚ‬Μͩίʔυ [43] Ͱ͸ɺ௒૬ର࿦తͳ৚͕݅Ծఆ͞
Ε͍ͯΔͨΊɺERL ೖࣹ‫ث‬ͷΑ͏ͳ௿ΤωϧΪʔྖҬͷϏʔϜμΠφϛΫεʹద༻͢Δ͜ͱ͕Ͱ͖ͳ͍ɻ
·ͨɺۭؒిՙޮՌͱ CSR ͷޮՌΛશͯ‫ؚ‬ΊͯΑΓ‫͢ࢉܭʹີݫ‬Δʹ͸ɺTREDI [44] ΍ TraFic4 [45] ͷΑ͏ʹɺLienard-Wiechert ϙςϯγϟϧΛηϧϑί

Ͱ༩͑ΒΕΔɻҰํɺॎํ޲ۭؒిՙྗʹΑͬͯόϯϯγεςϯτʹղ͘͜ͱ͕ඞཁͱͳΔ͕ɺ͜Ε͸๲େνεϥΠε͕੔ྻ͞ΕΔํ޲͸ɺͳ‫ؒ࣌ࢉܭ‬Λඞཁͱ͠ɺ‫܁‬Γฦ͠‫ࢉܭ‬Λߦ͏ύϥϝλ

ξLSC = arctan

ζxζx࠷దԽʹ͸޲͔ͳ͍ɻ͜ΕΒͷ୅ΘΓʹɺ߹ྲྀ෦Λ‫ؚ‬(10.10)Ή ERL ೖࣹ‫ث‬શମΛγϛϡϨʔτ͢ΔͨΊʹɺۭؒిՙ‫ࢉܭ‬Λ‫ؚ‬Μཻͩࢠ௥੻ίʔυͰ͋Δ GPT [33] ༻Ͱ‫͞ࢉܭ‬ΕΔɻਤ 63(b) ͷΑ͏ʹɺξ = ξLSC ͷ৔߹ʹɺ৽ͨͳ CSR ‫ࢉܭ‬ϧʔνϯɺGPT/CSR Λ։ൃ͠ʹ͸ɺ෼෍ͷ௕࣠ͱεϥΠε͕੔ྻ͞ΕΔํ޲͕Ұக

ͨɻGPT/CSR Ͱ࢖༻͍ͯ͠ΔΞϧΰϦζϜ [42] Ͱ͠ɺΤϛολϯεͷ૿େ͕࠷খʹ཈͑ΒΕΔɻ͜ΕΛ͸ɺ௒૬ର࿦తͳ৚݅͸Ծఆ͞Ε͓ͯΒͣɺೖࣹ‫ث‬ͷ㸲㸫 Α͏ͳ௿ΤωϧΪʔྖҬͷ߹ྲྀ෦΍γέΠϯͰͷϏʔ

+20

[×10

ϜμΠφϛΫεͷ‫ʹࢉܭ‬΋ద༻͢Δ͜ͱ͕Ͱ͖Δɻ
CSR power, dε /dt (eV/s)‫يܗ‬ಓ্Ͱͷ CSR ʹΑΔΤωϧΪʔଛࣦ͓ΑͼΤωϧΪʔ֦͕ΓΛ‫͠ࢉܭ‬ɺelegant ʹΑΔ݁Ռͱղੳ
తͳ݁ՌͱΛൺֱ͢Δɻ·ͨɺ3 ୆ͷภ޲ి࣓ੴ͔Β
ͳΔ߹ྲྀ෦ʹରͯ͠ɺCSR ʹΑΔΤϛολϯε૿େΛ‫݁ͨ͠ࢉܭ‬Ռʹ͍ͭͯ΋঺հ͢Δ

−4
−6
−8
0

1) Λ༻͍͍ͯͳ
͍ͨΊɺERL ೖࣹ‫ث‬ͷΑ͏ͳ 10 MeV ఔ౓ͷ௿͍Τ௒૬ର࿦తͰ͋Δͱ͍͏ۙࣅ (γGPT/CSRelegant
Theory−2

GPT/CSR ʹΑΔ‫݁ࢉܭ‬Ռ

GPT/CSR ͸ Sagan [42] ͷํ๏Λ༻͍ͯɺ1 ࣍‫ݩ‬
CSR wake Λ‫͢ࢉܭ‬Δɻ͜ͷํ๏͸ɺిࢠϏʔϜ͕10+5[×10 ]RMS energy spread, dσδ /dt (1/s)͋Γɺ҆ఆঢ়ଶ͚ͩͰͳ͘ա౉ঢ়ଶ΋ѻ͏͜ͱ͕Ͱ͖
Δɻ͞ΒʹɺGPT/CSR ͸ɺ‫ి૾ڸ‬ՙΛಋೖ͢Δ͜ͱʹΑΓɺਅۭνΣϯόʔʹΑΔ CSR ͷःณޮՌͷӨ‫ڹ‬΋औΓೖΕ͍ͯΔɻ

ΤωϧΪʔଛࣦͱΤωϧΪʔ֦͕Γσs = 0.6 mm

5GPT/CSRelegant(1/3)
Theory for γ >> (R/σs)

4
3
2
1
0

0

10

GPT/CSRɺelegantɺղੳࣜΛ༻͍ͯɺ൒‫ ܘ‬ρ =ͨɻ·ͨɺॳ‫ظ‬ͷཻࢠ෼෍͸ਐߦํ޲ɺԣํ޲ͱ΋ʹ Gaussian ෼෍ͱͨ͠ɻਤ 64 ʹɺΤωϧΪʔଛࣦdε/dtɺਤ 65 ʹɺΤωϧΪʔ֦͕Γ dσδ /dt ͷ‫݁ࢉܭ‬ՌΛࣔ͢ɻਤ 64 தͰͷ੺ઢ͸ɺC. Mayes [46] ʹΑͬͯಋग़͞ΕͨղੳࣜʹΑΔ‫݁ࢉܭ‬ՌͰ͋Δɻdε2 (re me c2 )cβ 4 γ 4N (1 + (N − 1)T (a)) ,
=−
dt
3
ρ2
(10.11)͜͜Ͱɺa = 3/2 · γ 3σs /(βρ) Ͱ͋ΓɺT (a) =

9 1
32π a31e (8a2 )

√

πK5/6

1
8a2

− 2πa ,

(10.12)
K5/6 (x) ͸ม‫ܗ‬ϕοηϧؔ਺ɺN ͸όϯν಺Ͱͷిࢠ਺ɺme ͸ిࢠͷ࣭ྔɺre ͸ిࢠͷ‫ݹ‬య൒‫ܘ‬ɺc ͸ਅۭதͰͷޫ଎ɺγ ͸ϩʔϨϯπҼࢠͰɺβ = (1−1/γ 2 )1/2

50

ϏʔϜΤωϧΪʔґଘੑɻ೚ҙͷϏʔϜ‫ي‬ಓʹରͯ͠ɺCSR wake Λ‫ࢉܭ‬ՄೳͰ

1.0 m ͷԁ‫يܗ‬ಓʹରͯ͠ɺ҆ఆঢ়ଶͷΤωϧΪʔଛࣦͱΤωϧΪʔ֦͕ΓΛ‫ͨ͠ࢉܭ‬ɻ͜ͷ‫Ͱࢉܭ‬͸ɺόϯν௕ σs = 0.6 mmɺόϯνిՙ −80 pC ͱ͠

20
30
40
Beam energy (MeV)ਤ 64: ҆ఆঢ়ଶͰͷ CSR ʹΑΔΤωϧΪʔଛࣦͷωϧΪʔྖҬͰ΋༗ޮͰ͋Δɻ·ͨɺGPT/CSR ͸ɺ

10.2.3σs = 0.6 mm

0͜͜Ͱ͸ɺGPT/CSR ͷ༗ޮੑΛࣔͨ͢Ίʹɺԁ

10.2.2]20
30
40Beam energy (MeV)50

ਤ 65: ҆ఆঢ়ଶͰͷ CSR ʹΑΔΤωϧΪʔ֦͕ΓͷϏʔϜΤωϧΪʔґଘੑɻ
க͢ΔɻҰํɺਤ 64 ʹࣔ͞ΕΔΑ͏ʹɺGPT/CSR͸શͯͷΤωϧΪʔྖҬʹ͓͍ͯղੳ஋ͱҰக͍ͯ͠Δɻ

(ρ/σs )1/3 ͷ৔߹ͷۙࣅࣜ [47,48] ͔ΒಘΒΕͨΤωϧΪʔ֦͕ΓͰ͋Δɻ
ਤ 65 ͷ੺ઢ͸ɺγdσδ
re N cβ.
≈ 0.224/3
dtγρ2/3 σsߴ ͍ Τ ω ϧ Ϊ ʔ ྖ Ҭ (E0>(10.13)40MeV) Ͱ ͸ ɺ

GPT/CSRɺelegant ͱ΋ʹۙࣅࣜʹΑΔ݁ՌΛ࠶‫ݱ‬͍ͯ͠Δɻ͔͠͠ɺ௿ΤωϧΪʔྖҬͷ‫( ݶۃ‬E0 → 0)Ͱ͸ɺۙࣅࣜͱ elegant Ͱ͸ͱ΋ʹൃࢄͯ͠͠·͏ɻ͜Εʹରͯ͠ɺGPT/CSR Ͱ͸ 0 ʹ͖ۙͮɺ༧૝௨Γͷ݁ՌΛࣔͨ͠ɻҎ্ͷ݁Ռ͕ࣔ͢Α͏ʹɺGPT/CSR͸޿͍ΤωϧΪʔྖҬͰ༗ޮͰ͋ΔɻͱͳΔɻelegant ͸௒૬ର࿦తͳۙࣅ͕༻͍ΒΕ͍ͯΔͨΊ [43]ɺE0 > 40 MeV ͷ৔߹ʹ‫ݶ‬Γղੳ஋ͱҰ㸲㸫 ρ = 10 m, E0 = 128 MeV, no shielding

1000

−2000
−3000
−4000
−1.5

−1

−0.5

CSR power, dε/dt (eV/s)

−1000

]

R = 10 m, Q = −77 pC, σs = 1 mm

8

0.1 m
0.2 m
0.3 m
0.4 m
0.5 m
0.6 m
0.7 m
0.8 m
0.9 m
1.0 m

0
dε/dt (eV/m)

+19

[×10

0
s (mm)

0.5

1

1.5

10 MeV
20 MeV
40 MeV
80 MeV
160 MeV
1 GeV

6
4
2
0
−2
−4

no shielding

−6
−8
0

0.05

0.1
0.15
Shield height, h (m)0.2ਤ 66: ःณͳ͠ͷ৔߹ͷภ޲ి࣓ੴग़‫ޱ‬Ҏ߱ͷ CSR
ਤ 68: CSR ʹΑΔΤωϧΪʔଛࣦʹର͢Δःณͷߴwake ͷมԽɻॳ‫ظ‬ͷόϯν௕͸ɺ0.3 mmɻ
ρ = 10 m, E0 = 128 MeV, h = 2 cm

2000

0.1 m
0.2 m
0.3 m
0.4 m
0.5 m
0.6 m
0.7 m
0.8 m
0.9 m
1.0 m

1000
dε/dt (eV/m)͞ͷӨ‫ڹ‬ɻCSR ͸҆ఆঢ়ଶɻ‫ి૾ڸ‬ՙͷ਺͸ 32ɻ

0
−1000

10.2.5ਅۭνΣϯόʔʹΑΔ CSR ͷःณ

GPT/CSR Λ༻͍ͯɺCSR ͷःณΛ ρ = 10 m ͷԁ‫يܗ‬ಓ্Ͱ‫ͨ͠ࢉܭ‬ɻ͜ͷ‫Ͱࢉܭ‬͸ɺCSR ͸҆ఆঢ়ଶʹ͋Δͱͨ͠ɻਤ 68 ͸ɺΤωϧΪʔଛࣦͷਅۭνΣϯόʔͷߴ͞ʹର͢ΔґଘੑΛࣔ͢ɻ‫Ͱࢉܭ‬͸ɺόϯν௕Λ 1.0 mmɺόϯν಺ͷిՙΛ −77 pCɺ‫ڸ‬

−2000
−1.5

−1

−0.5

0
s (mm)

0.51૾ిՙͷ਺Λ 32 ͱͨ͠ɻ·ͨɺॳ‫ࢠཻظ‬෼෍͸ɺਐ

1.5ߦํ޲ɺԣํ޲ͱ΋ʹ Gaussian ෼෍ͱͨ͠ɻਤ 68 ͷதͰͷ௚ઢ͸ɺࣜ (10.11) ͔Β‫͞ࢉܭ‬Εͨःณͳ͠ͷਤ 67: ःณ͋Γͷ৔߹ͷภ޲ి࣓ੴग़‫ޱ‬Ҏ߱ͷ CSR

৔߹ͷղੳ஋Λࣔ͢ɻःณͷߴ͕͞૿Ճ͢Δʹै͍ɺwake ͷมԽɻਅۭνΣϯόʔͷߴ͞͸ 2 cmɻΤωϧΪʔଛࣦ͕ղੳ஋ʹۙͮ͘͜ͱ͕Θ͔Δɻ

10.2.4ա౉ঢ়ଶͰͷ CSR wake10.2.6
ภ޲ి࣓ੴͷೖ‫ޱ‬΍ग़‫ޱ‬௚‫ޙ‬ͷա౉ঢ়ଶʹ͋Δ CSR߹ྲྀ෦Ͱͷ CSR ͷӨ‫ڹ‬

GPT/CSR Λ࢖ͬͨ‫ͯ͠ͱྫࢉܭ‬ɺίʔωϧେֶͷӨ‫ڹ‬ͷ‫݁ࢉܭ‬ՌΛࣔͨ͢Ίʹɺภ޲ి࣓ੴग़‫͔ޱ‬Βࣗ༝ۭؒΛਐΉϏʔϜʹ͍ͭͯɺGPT/CSR Λ༻͍

ERL ‫ܭ‬ըͷ߹ྲྀ෦Ͱͷ CSR ʹΑΔΤϛολϯε૿ͯ CSR wake ؔ਺ͷมԽΛ‫ͨ͠ࢉܭ‬ɻ‫Ͱࢉܭ‬͸ɺϏʔେΛ‫͠ࢉܭ‬ɺelegant ʹΑΔ‫݁ࢉܭ‬Ռͱൺֱͨ͠ɻ‫ܭ‬ϜΤωϧΪʔ 128 MeVɺॳ‫ظ‬ͷόϯν௕ 0.3 mmɺόࢉͰ͸ɺϏʔϜͷӡಈྔΛɺ(a) p0 = 10 MeV/cɺ (b)ϯν಺ͷిՙ −80 pC ͱͨ͠ɻ·ͨɺॳ‫ظ‬ͷཻࢠ෼෍͸ਐߦํ޲ɺԣํ޲ͱ΋ Gaussian ෼෍ͱͨ͠ɻਤ 66

p0 = 500 MeV/c ͱͨ͠ɻ·ͨɺCSR ʹΑΔޮՌͷΈΛ‫ݟ‬ΔͨΊʹɺۭؒిՙޮՌ͸औΓೖΕ͍ͯͳ͍ɻ

͸ःณޮՌͷͳ͍৔߹ɺਤ 67 ͸ःณޮՌͷ͋Δ৔߹ਤ 30 ʹ߹ྲྀ෦ͷϨΠΞ΢τΛࣔ͢ɻ߹ྲྀ෦ʹ͸ɺภͷա౉ঢ়ଶͷ CSR wake ͷ‫݁ࢉܭ‬ՌΛࣔ͢ɻਤ 67 Ͱ޲ి࣓ੴ 3 ୆ͷଞʹ 4 ‫࣓ిۃ‬ੴ 2 ୆͕഑ஔ͞Ε B3 ͷ͸ɺਅۭνΣϯόʔͷߴ͞Λ 2 cm, ‫ి૾ڸ‬ՙͷ਺Λग़‫Ͱޱ‬͸෼ࢄؔ਺͕ 0 ʹͳΔΑ͏ʹ‫͕͞ڧ‬ௐ੔͞Ε32 ͱͨ͠ɻ‫Ͱࢉܭ‬͸ɺ൒‫ ܘ‬ρ = 10 m Λ΋ͭภ޲ి࣓ੴͷग़‫ޱ‬Λ Δs = 0 m ͱͯ͠ɺ͜ͷ఺Ͱ CSR ͸҆Δɻ·ͨɺ̎ͭͷӡಈྔʹରͯ͠ಉ͡ϏʔϜޫֶؔఆঢ়ଶʹ͋Δͱ͠ɺ͔ͦ͜Βࣗ༝ۭؒΛਐΉ͏ͪʹͲ

Δɻਤ 69 ʹਫฏํ޲ͷ RMS ϏʔϜαΠζͷมԽΛɺ਺Λ༩͑ΔΑ͏ʹɺΤωϧΪʔʹΑͬͯεέʔϧ͞ΕͷΑ͏ʹ CSR wake ͕มԽ͢Δ͔Λ‫ٻ‬Ίͨɻਤ 66ɺ

ਤ 70 ʹਫฏํ޲‫֨ن‬Խ RMS ΤϛολϯεͷมԽΛ

67 ͕ࣔ͢Α͏ʹɺCSR wake ͸ग़‫͔ޱ‬Β཭ΕΔͱ‫ݮ‬গ͍ͯ͘͠ɻ

ࣔ͢ɻ͜ΕΒͷਤͰ͸ɺԣ࣠͸߹ྲྀ෦ೖ‫͔ޱ‬Βͷਐߦํ޲ͷ‫཭ڑ‬Λද͢ɻ‫Ͱࢉܭ‬͸ɺόϯν௕Λ 0.3 mmɺόϯν಺ͷిՙΛ −80 pCɺॳ‫ࢠཻظ‬෼෍Λ Gaussian෼෍ͱͨ͠ɻਤ 69ɺ 70 ΑΓɺ(a) p0 = 10 MeV/c㸲㸫 

(a) p0 = 10 MeV/c, CSR

0.04

elegant
GPT/CSR

Horizontal beam size, σx (mm)

0.03
0.02
0.01
0
−4
[1×10 ](b) p0 = 500 MeV/c, CSR

6
4
2
0

0

1

2

3

s (m)

ਤ 69: ߹ྲྀ෦ (ਤ 30) Ͱͷ CSR ʹΑΔਫฏํ޲ RMS
ϏʔϜαΠζͷมԽɻ−3[1×10 ]
Normalized emittance, εnx (mm mrad)

4(a) p0 = 10 MeV/c, CSR
elegant
GPT/CSR2

0−5[1×10 ]
8(b) p0 = 500 MeV/c, CSR

4

0

0

1

2

3

s (m)

ਤ 70: ߹ྲྀ෦ (ਤ 30) Ͱͷ CSR ʹΑΔ‫֨ن‬Խਫฏํ޲ RMS ΤϛολϯεͷมԽɻͷΤωϧΪʔ͕௿͍৔߹͸ɺelegant ͱ GPT/CSRͷ݁ՌͰҧ͍͕‫ݟ‬ΒΕͨɻ͜Ε͸ɺelegant ͷ CSR‫͕ࢉܭ‬௒૬ର࿦తͳۙࣅΛ‫ؚ‬ΉͨΊɺ௿ΤωϧΪʔྖҬͰ༗ޮͰͳ͍͜ͱʹ‫ى‬Ҽ͢Δͱਪଌ͞ΕΔɻҰํɺ(b) p0 = 500 MeV/c ͷ௒૬ର࿦తͳۙࣅ͕੒Γཱͭ৔߹ʹ͸ɺ̎ͭͷίʔυʹΑΔ‫݁ࢉܭ‬Ռ͸ྑ͍ҰகΛࣔͨ͠ɻ͜ΕΒͷ͜ͱ͔ΒɺGPT/CSR ͸޿͍ΤωϧΪʔൣғͰ༗ޮͰ͋Δ͜ͱ͕֬ೝͰ͖ͨɻ

㸲㸫 

11

ೖࣹ‫͔ث‬Βओۭಎ΁ͷϏʔϜ༌ૹ͜͜Ͱ͸ɺೖࣹ‫͔ث‬Βओۭಎ·Ͱ௿ΤωϧΪʔͷೖࣹϏʔϜΛ༌ૹ͢Δ͜ͱΛߟ͑ΔɻલͷηΫγϣϯͰ͸ɺ௿Τϛολϯε༌ૹͷͨΊʹೖࣹ‫ث‬ͷ༌ૹ৚݅Λ
࠷దԽ͢Δख๏Λ֤۠ؒຖ (ೖࣹ‫ث‬ɺ߹ྲྀ෦) ʹ঺հਤ 71: ిࢠॐ͔Βओۭಎग़‫Ͱ·ޱ‬ͷ࠷దԽखॱɻ࣍

ͨ͠ɻ࣮ࡍͷϏʔϜ༌ૹͰ͸ɺ௿ΤωϧΪʔϏʔϜΛ

ͷ 3 ͭͷํ๏͕͋Δɻํ๏ 1: ిࢠॐ͔Βओۭಎग़‫ޱ‬ओۭಎͰՃ଎͞ΕΔ·ͰҰ؏ͯ͠ѻ͏ඞཁ͕͋Δɻ··ͰҰ౓ʹ࠷దԽ (1 ஈ֊๏)ɻํ๏ 2: ೖࣹ‫ۭث‬ಎग़ͣɺೖࣹ‫͔ث‬Βओۭಎ·Ͱͷ༌ૹ৚݅ͷઃ‫ܭ‬खॱʹͭ‫࠷Ͱ·ޱ‬దԽͨ͋͠ͱɺओۭಎग़‫࠷Ͱ·ޱ‬దԽ (2 ஈ͍ͯ঺հ͢Δɻͦͷ͋ͱɺcERL Ͱ࣮ࡍͷϏʔϜΛઃ֊๏)ɻํ๏ 3: ೖࣹ‫ۭث‬ಎग़‫࠷Ͱ·ޱ‬దԽͨ͋͠ͱɺ

‫ܭ‬৚݅ʹ͚͍ۙͮͯͨ͘Ίͷௐ੔ํ๏ (ϏʔϜޫֶؔ߹ྲྀ෦ग़‫Ͱ·ޱ‬Λ࠷దԽ͠ɺ࠷‫ʹޙ‬ओۭಎग़‫࠷Ͱ·ޱ‬਺Ϛονϯά) ʹ͍ͭͯ঺հ͢ΔɻదԽ (3 ஈ֊๏)ɻओۭಎग़‫Ͱޱ‬͸ɺपճ෦΁ͷϏʔϜޫֶؔ਺ϚονϯάͷͨΊʹɺϏʔϜޫֶؔ਺͕͋Δ11.1

ൣғʹ੍‫͞ݶ‬ΕΔɻϏʔϜ༌ૹ৚݅ͷઃ‫ܭ‬खॱೖࣹ‫ۭث‬ಎ͔Βओۭಎ΁ͷϏʔϜ༌ૹ৚݅ͷઃ‫ܭ‬Ͱ͸ɺిࢠॐ͔Βओۭಎग़‫Ͱ·ޱ‬Ұ؏ͨ͠γϛϡϨʔγϣϯΛߦ͏ʹ͸ɺ్தͷ࢛‫࣓ిۃ‬ੴͷௐ੔΋ߦ͏͜ͱ͕‫·ؚ‬ΕΔɻ͔͠͠ͳ͕Βɺ͜ͷ৔߹ʹ͸ௐ੔Մೳͳ

1. ΤϛολϯεѱԽΛආ͚Δ͜ͱ߲໨਺ (ࣗ༝౓) ͕૿͑Δ͜ͱʹͳΓ (cERL ͷ৔߹

2. όϯν௕͋Δఔ౓୹͘ (1 ∼ 3 ps)ʹ͸ɺ5 ୆ + 8 ୆ͷ࢛‫࣓ిۃ‬ੴ)ɺ࠷దԽ‫͔ͳ͕ࢉܭ‬3. पճ෦ʹ༌ૹ͢ΔͨΊͷϏʔϜޫֶؔ਺ͷௐ੔ग़‫Ͱ·ޱ‬ͷҰ؏ͨ͠γϛϡϨʔγϣϯʹΑΔ༌ૹ৚͕݅ཁ‫͞ٻ‬ΕΔɻ3 ͭ໨ͷओۭಎग़‫͚͓ʹޱ‬ΔϏʔϜޫͷ࠷దԽ͸ɺ࠷ྑͷ༌ૹ৚݅Λ‫͚ͭݟ‬ΔՄೳੑ΋͋Δ

ֶؔ਺ͷ‫ڐ‬༰ൣғ͸ɺपճ෦ͷϏʔϜޫֶؔ਺ઃ‫ʹܭ‬͕ɺࣗ༝౓͕૿͑͗͢Δ͜ͱʹͳΓɺͦ͜ʹ౸ୡ͢Δ

Αܾͬͯ·ΔͨΊɺೖࣹ‫͔ث‬ΒདྷΔϏʔϜͷ༌ૹ৚݅ͷ͕೉͍͠έʔε΋ଟ͍ɻҎ্ͷ͜ͱ͔Βߟ͑ΔͱɺΛ͜ͷൣғ಺ʹ߹ΘͤࠐΉ͜ͱ͕ඞཁͱͳΔɻ

༌ૹ৚݅ͷ࠷దԽʹ͸࣍ͷ 3 ͭͷํ๏͕ߟ͑ΒΕΔɻ

ͳ͔ऩଋ͠ͳ͍ͱ͍͏ࣄଶʹؕΔɻిࢠॐ͔Βओۭಎೖࣹ‫ث‬ͷϏʔϜ༌ૹ৚݅ͷઃ‫Ͱܭ‬͸ɺ਺஋‫ࢉܭ‬Λར

1. ిࢠॐ͔Βओۭಎग़‫Ͱ·ޱ‬Ұ౓ʹ࠷దԽ (1 ஈ༻ͯ͠࠷దԽΞϧΰϦζϜʹΑͬͯόϯν௕ͱΤϛο֊๏)λϯεͷ 2 ͭΛಉ࣌ʹ࠷খԽ͖ͯͨ͠ɻͨͩ͠ɺ͜

2. ೖࣹ‫ۭث‬ಎग़‫࠷Ͱ·ޱ‬దԽͨ͋͠ͱɺओۭಎग़‫ޱ‬ͷ࠷খԽ͸͋Δ 1 ఺ͷΈͰϏʔϜ඼࣭Λ‫͍ͯݟ‬Δ͜

·Ͱ࠷దԽ (2 ஈ֊๏)ͱʹͳΓɺ͞ΒʹԼྲྀʹ༌ૹ͢Δ৔߹ʹ͸ɺ࠷దԽ͠
ͨ఺͔ΒԼྲྀͷ༌ૹ৚݅Λௐ੔͍ͯ͘͜͠ͱ͕ඞཁͱ

3. ೖࣹ‫ۭث‬ಎग़‫࠷Ͱ·ޱ‬దԽͨ͋͠ͱɺ߹ྲྀ෦ग़‫ޱ‬·ͰΛ࠷దԽ͠, ࠷‫ʹޙ‬ओۭಎग़‫࠷Ͱ·ޱ‬దԽ (3ͳΔɻͨͩ͠ɺۭؒిՙޮՌ͕ࢧ഑తͳ৔߹ʹ͸ɺ͋
Δ఺Ͱ࠷దԽͯ͠΋ͦͷԼྲྀʹ༌ૹ͢Δ్தͰۭؒి
ՙޮՌʹΑΔΤϛολϯεѱԽ͕ੜ͡Δ৔߹͕͋Δɻ͜ͷͨΊɺిࢠॐ͔Βओۭಎग़‫Ͱ·ޱ‬Ұ؏ͨ͠γϛϡϨʔγϣϯ͕ඞཁͱͳΔɻೖࣹ‫ث‬ग़‫͔ޱ‬Βओۭಎ·Ͱͷ۠ؒΛߟ͑ͯΈΔͱɺओʹ 2 ͭͷ໨త͕͋ΔɻҰͭ͸ೖࣹ‫͔ث‬Β߹ྲྀ෦ͷؒʹ഑ஔ͞Ε࢛ͨ‫࣓ిۃ‬ੴʹΑͬͯ߹ྲྀ෦ͰͷΤϛολ
ϯεѱԽΛආ͚Δͱ͍͏͜ͱͰ͋Δɻ΋͏Ұͭ͸߹ྲྀஈ֊๏)͜ΕΛ໛ࣜతʹࣔͨ͠ͷ͕ɺਤ 71 Ͱ͋Δɻిࢠॐ͔
Βओۭಎग़‫Ͱ·ޱ‬ͷҰ؏ͨ͠γϛϡϨʔγϣϯ͸ɺํ๏ 1 ʹରԠ͢Δɻ͜͜Ͱ͸ɺ3 ͭͷํ๏Ͱ࠷దԽͨ݁͠Ռʹ͍ͭͯ঺հ͢Δɻ·ͣɺओۭಎग़‫ ఺( ޱ‬A) ͰͷϏʔϜޫֶؔ਺ͷ‫ڐ‬༰ൣғ͸ɺ෦͔Βओۭಎ·Ͱͷؒʹ഑ஔ͞Ε࢛ͨ‫࣓ిۃ‬ੴʹΑͬͯɺपճ෦ʹ޲͚ͨ༌ૹ৚݅Λ੔͑Δͱ͍͏͜ͱͰ͋
Δɻ͜ͷΑ͏ʹ༌ૹ৚݅Λௐ੔͢ΔͨΊͷ࢛‫࣓ిۃ‬ੴ• 1 < βx < 5 m• 6 < βy < 18 m
• −0.3 < αx < −0.1ʹ͸ɺͦΕͧΕͷ໾ׂ͕͋Δ͜ͱ͕Θ͔Δɻ㸲㸫 ਤ 74: ߹ྲྀ෦ग़‫( ޱ‬point C) Ͱ࠷దԽ͞ΕͨϏʔϜ༌ૹ৚݅ͷͱ͖ͷҐ૬ۭؒ෼෍ɻೖࣹ‫ۭث‬ಎ͔Β߹ྲྀ෦ͷؒʹ͋Δ 5 ୆ͷ࢛‫࣓ిۃ‬ੴͷ‫࠷͕͞ڧ‬దԽ͞Εͨɻ͜ͷ৚݅Λॳ‫ظ‬஋ͱͯ͠ओۭಎग़‫Ͱ·ޱ‬ͷ࠷దԽਤ 72: ೖࣹ‫ۭث‬ಎग़‫( ޱ‬point B) ͰϏʔϜ༌ૹ৚݅Λߦ͏ɻΛ࠷దԽͨ݁͠Ռɻσz = 0.59 mm ෇ۙͷϏʔϜ༌ૹ࣓ੴ͕ௐ੔͞ΕΔ͜ͱʹͳΔɻΤϛολϯεΛѱԽ͞৚݅Λॳ‫ظ‬஋ͱͯ͠ɺԼྲྀͷ࠷దԽΛߦ͏ɻͤͣʹɺ఺ A ͷϏʔϜޫֶؔ਺͕‫ڐ‬༰ൣғʹऩ·ΔΑ͏ʹ࠷దԽ‫ߦ͕ࢉܭ‬ΘΕΔɻ3 ஈ֊๏Ͱ͸ɺ఺ C·Ͱ 5 ୆ͷ࢛‫࣓ిۃ‬ੴ͕ௐ੔͞ΕΔɻ͜ͷ৔߹͸ओʹ߹ྲྀ෦ͰͷΤϛολϯεѱԽΛ࠷খʹ཈͑ΔΑ͏ͳௐ੔Λ͢Δ͜ͱʹͳΔɻਤ 74 ʹ 3 ஈ֊๏Ͱ࣮ࢪͨ͠
఺ C ʹ͓͚ΔҐ૬ۭؒ෼෍Λࣔ͢ɻ͜ͷ৚݅Λॳ‫ظ‬஋ͱͯ͠ ఺ A ·Ͱͷ࠷దԽ (8 ୆ͷ࢛‫࣓ిۃ‬ੴͷௐ੔ɺ໨త͸पճ෦ʹ޲͚ͨϏʔϜޫֶؔ਺ͷௐ੔) ͕ਤ 73: ೖࣹ‫ۭث‬ಎग़‫( ޱ‬point B) Ͱ࠷దԽ͞ΕͨϏʔߦΘΕΔɻϜ༌ૹ৚݅ͷͱ͖ͷҐ૬ۭؒ෼෍ɻ͜ͷ৚݅Λॳ‫ظ‬஋ͱͯ͠Լྲྀͷ࠷దԽΛߦ͏ɻ

ਤ 75 ʹ 3 ͭͷํ๏Ͱ࠷దԽͨ͠ͱ͖ͷిࢠॐ͔Βओۭಎग़‫Ͱ·ޱ‬ͷ‫֨ن‬ԽΤϛολϯεͷ࣌ؒൃలΛࣔ͢ɻ·ͨɺਤ 76 ʹϏʔϜޫֶؔ਺ βx , βy ͷ࣌ؒ• 2.5 < αy < 3.5ൃలΛࣔ͢ɻ1 ஈ֊๏Ͱ͸఺ A ͷϏʔϜޫֶؔ਺৚

ͱͳΔɻೖࣹ‫͔ث‬Βओۭಎ΁༌ૹ͢Δ৔߹ɺ࠷ऴతʹ݅Λຬ͕ͨ͢Τϛολϯε͕େ͖͍ɺ2 ஈ֊๏Ͱ͸Τ͜ͷ৚݅Λຬͨ͞ͳ͚Ε͹ͳΒͳ͍ɻΤϛολϯεͱϛολϯε͸ͦͦ͜͜খ͍͕͞ϏʔϜޫֶؔ਺৚݅Λόϯν௕ʹ͍ͭͯ͸ɺ࠷దԽ‫Ͱࢉܭ‬͸ͱ΋ʹ࠷খԽ͢ຬͨ͞ͳ͍ɺ3 ஈ֊๏Ͱ͸Τϛολϯε͕࠷΋খ͘͞Δ͜ͱΛ৚݅ͱͨ͠ɻ͜͜Ͱ঺հ͢Δ‫Ͱࢉܭ‬͸ɺೖࣹ͔ͭޫֶؔ਺৚݅ʹ͔ͳΓ͍ۙͱ͍͏݁Ռͱͳͬͨɻ

ϏʔϜͷΤωϧΪʔ 5 MeVɺओۭಎʹΑΔՃ଎‫ޙ‬ͷΤωϧΪʔΛ 35 MeV ͱ͍ͯ͠Δɻਤ 71 ʹࣔ͢Α͏

1 ஈ֊๏ͷ৔߹͕࠷΋ྑ͍༌ૹ৚݅Λ୳͠౰ͯΔ͜ͱ
͕Մೳͳ͸ͣͰ͋Δ͕ɺࣗ༝ಈ͕ଟ͍ͨΊʹ࠷ྑͷղʹɺ2 ஈ֊๏Ͱ͸ೖࣹ‫ۭث‬ಎग़‫ ఺( ޱ‬B) Λ۠੾Γ఺ʹ୧Γண͚ͳ͍݁Ռͱͳͬͨɻ
Ҏ্ͷ݁ՌΛ౿·͑ͯɺcERL ͷ༌ૹ৚݅ͷઃ‫Ͱܭ‬

ͱ͠ɺ3 ஈ֊๏Ͱ͸͜Εʹ߹ྲྀ෦ग़‫ ఺( ޱ‬C) Λ۠੾͸ɺओʹ 3 ஈ֊๏Λ༻͍͍ͯΔɻͨͩ͠ɺ3 ஈ֊๏ͰΓ఺ͱͯ͠௥Ճ͢Δɻ
ਤ 72 ʹɺ఺ B ʹ͓͚ΔϏʔϜ༌ૹ৚݅Λ࠷దԽ͠ಘΒΕͨ༌ૹ৚݅Λॳ‫ظ‬஋ͱͯ͠ɺ1 ஈ֊๏ʹΑΔ༌

ͨ݁ՌΛࣔ͢ɻ͜͜·Ͱͷ࠷దԽͰ͸ɺྭ‫ى‬Ϩʔβʔૹ৚݅ͷ࠷దԽΛ͞Βʹ௥Ճͯ͠ɺΑΓΤϛολϯε৚݅ (εϙοτ‫ܘ‬ɺ࣌ؒύϧε෯)ɺιϨϊΠυి࣓ΛԼ͛Δํ๏΋ࢼΈ͍ͯΔɻ

ੴɺόϯνϟʔۭಎ (Ճ଎ిѹ)ɺೖࣹ‫ث‬௒఻ಋۭಎ

(Ճ଎ޯ഑ͱҐ૬) ͕ௐ੔Մೳͳ߲໨ͱͳ͍ͬͯΔɻ2
ஈ֊๏ͱ 3 ஈ֊๏Ͱ͸ɺਤ 72 ͷ σz = 0.59 mm ෇

11.2

ۙͷϏʔϜ༌ૹ৚݅Λॳ‫ظ‬஋ͱͯ͠ɺԼྲྀͷ࠷దԽΛߦͬͨɻ͜ͷͱ͖ͷԣํ޲ͱॎํ޲ͷҐ૬ۭؒ෼෍Λ15ϏʔϜ༌ૹ࿏ͷௐ੔ํ๏cERL Ͱ͸ޫӄ‫ۃ‬Λ༻͍ͨ DC ిࢠॐ͔Β 390 keVͷిࢠϏʔϜ͕ੜ੒͞Εɺೖࣹ‫ث‬௒఻ಋۭಎͰ 2.4

ਤ 73 ʹࣔ͢ɻ

2 ஈ֊๏Ͱ͸ɺ఺ B ͔Β఺ A ·Ͱ 13 ୆ͷ࢛‫ిۃ‬㸲㸫 

15 ͜͜Ͱ͸ɺӡಈΤωϧΪʔ T = (γ − 1)mc2 Λ༻͍Δɻpoint B

1.5

Single−step, original layout
Two−step, original layout
Three−step, new layout

point A1

point A40

0.5

0.42 mm mrad0.26 mm mrad
point C0
1.5

Single−step, original layout
Two−step, original layout
Three−step, new layout

1

point C

20

0
50

0.5
0
0

30

10

40

βy (m)

εny (mm mrad)

point B50

βx (m)εnx (mm mrad)1.36 mm mrad

10
20
Longutidilan distance from cathode, z (m)

30
20
10

ਤ 75: 3 ͭͷํ๏Ͱ࠷దԽͨ͠ϏʔϜ༌ૹ৚݅ɻిࢠ

00ॐ͔Βओۭಎग़‫Ͱ·ޱ‬ͷ‫֨ن‬ԽΤϛολϯεͷ࣌ؒൃలΛࣔ͢ɻMeV ·ͰՃ଎͞ΕΔɻೖࣹ‫Ͱث‬Ճ଎͞ΕͨిࢠϏʔϜ͸ɺ߹ྲྀ෦Ͱपճ෦‫ي‬ಓʹೖΓɺओۭಎͰ 19.4 MeV·ͰՃ଎͞ΕΔɻ19.4 MeV ͷిࢠϏʔϜ͸पճ෦Λ

1020Longutidilan distance from cathode, z (m)ਤ 76: 3 ͭͷํ๏Ͱ࠷దԽͨ͠ϏʔϜ༌ૹ৚݅ɻిࢠ
ॐ͔Βओۭಎग़‫Ͱ·ޱ‬ͷϏʔϜޫֶؔ਺ βx , βy ͷ࣌ؒൃలΛࣔ͢ɻ

௨ͬͯ࠶ͼओۭಎʹೖΓɺΤωϧΪʔճऩ͞Εओμϯϓʹ༌ૹ͞ΕΔɻ‫ࡏݱ‬ͷయ‫ܕ‬తͳӡస৚݅͸ɺόϯν
ిՙ 0.5 pCɺ‫܁‬Γฦ͠प೾਺ 162.5 MHz Ͱฏ‫ۉ‬ϏʔϜిྲྀ 80 μA ͷ CW ӡసͱͳ͍ͬͯΔɻόϯνి
ՙ 0.5 pC ͸ɺ‫܁‬Γฦ͠प೾਺ 1.3 GHz ͷ CW ӡసʹ͓͚ΔϐʔΫిྲྀ 0.65 mA ʹ૬౰͠ɺۭؒిՙ
ޮՌ͕‫ݱ‬Εͯ͘Δిՙྔͱͳ͍ͬͯΔɻಛʹɺిࢠॐ͔Βओۭಎ·Ͱͷ۠ؒͰ͸ɺΤωϧΪʔ͕ 390 keV∼ 2.4 MeV ͱ௿͍ͨΊɺۭؒిՙޮՌͷ੍‫ͱͱޚ‬΋ʹɺ࢒ཹ࣓৔΍‫࣓ڥ؀‬৔౳ʹΑΔӨ‫ڹ‬΋ิਖ਼͍ͯ͘͠

ਤ 77: cERL ʹ͓͚ΔϏʔϜޫֶؔ਺ϚονϯάՕ
ॴɻMP1 ∼ MP7 ͷ 7 ՕॴͰϚονϯάΛߦ͏ɻ

11.2.1ԣํ޲ͷϏʔϜޫֶؔ਺Λઃ‫ܭ‬৚݅ʹ߹ΘͤΔͨ

ඞཁ͕͋Δɻ͜ͷྖҬͰ͸ɺ4 ࣍‫ݩ‬ͷԣํ޲Ґ૬ۭؒϏʔϜޫֶؔ਺ͷ࠷దԽख๏

(x, x , y, y ) ͱɺਐߦํ޲Ґ૬ۭؒ (z, δ) ͕ۭؒిՙ

Ίͷख๏ͱͯ͠ɺ࢛‫࣓ిۃ‬ੴͷ k ஋Λม͑ͨͱ͖ͷޮՌΛ௨ͯ݁͠߹͢ΔͨΊɺ6 ࣍‫ݩ‬Ґ૬ۭؒΛ੍‫͢ޚ‬ϏʔϜαΠζͷมԽ (Q-scan ͷԠ౴) ΛԼྲྀͷεΫΔ͜ͱ͕ඞཁͱͳΔɻͨͩ͠ɺ࣮ࡍͷϏʔϜௐ੔Ͱ͸ɺϦʔϯϞχλͰଌఆ͠ɺͦͷԠ౴͕ઃ‫ܭ‬৚݅ͱಉ͡ʹԣํ޲ͱॎํ޲ͷӡಈͷ݁߹Λආ͚ͯɺۭؒిՙޮՌ

ͳΔΑ͏ʹௐ੔͢Δͱ͍͏ํ๏Λ༻͍Δɻଌఆ͞ΕͨͷແࢹͰ͖Δඇৗʹ௿͍όϯνిՙ (਺ 10 fC) ͔Βௐ੔Λ։͍࢝ͯ͠Δɻԣํ޲ͱॎํ޲ΛͦΕͧΕಠཱ

Q-scan ͷԠ౴͔ΒϏʔϜޫֶؔ਺ α, β ͱΤϛολϯ
ε ε Λ‫ٻ‬Ίͯɺ͜ΕΒΛิਖ਼͢Δ͜ͱ΋Ͱ͖Δ͕ɺۭ

ʹௐ੔ͯ͠ิਖ਼ํ๏Λཱ֬ͨ͠ͷͪʹɺόϯνిՙΛؒిՙޮՌͷࢧ഑తͳΤωϧΪʔྖҬʹద༻͢ΔͨΊ૿‫ిۭؒͯ͠ڧ‬ՙޮՌΛ‫ؚ‬Ίͨ 6 ࣍‫ݩ‬Ґ૬੍ۭؒ‫ޚ‬ΛʹɺϏʔϜޫֶؔ਺ͷ୅ΘΓʹϏʔϜαΠζͷԠ౴Λߦ͍ͬͯΔɻ͜͜Ͱ͸ɺԣํ޲ͷϏʔϜޫֶؔ਺ͷิଌఆͱઃ‫ܭ‬৚݅Ͱൺֱ͢Δํ๏Λ༻͍Δ͜ͱͱ͢Δɻ

ਖ਼ํ๏։ൃͱͦͷࢼ‫݁ݧ‬Ռʹ͍ͭͯ঺հͨ͋͠ͱɺॎ

ిࢠϏʔϜ͕Ճ଎͞ΕۭͯؒిՙޮՌΛແࢹͰ͖ΔΤํ޲ͷϏʔϜ༌ૹ৚݅ͷࢼ‫݁ݧ‬Ռʹ͍ͭͯ঺հ͢Δɻ

ωϧΪʔྖҬͰ͸ɺϏʔϜαΠζͷԠ౴͔ΒϏʔϜޫֶؔ਺Λ‫ٻ‬ΊͯɺͦΕΛิਖ਼͢ΔΑ͏ʹ֦ு͢Δ͜ͱ͸༰қͰ͋Δɻ·ͣɺԣํ޲ͷϏʔϜޫֶؔ਺ͷઃ‫ܭ‬৚݅Λ߹Θͤ㸲㸫 ΕͨԠ౴ؔ਺ͷࠩ ΔM = (M − T ) = (Δσxm (k1 )Δσym (k1 ) . . . Δσxm (kn ) Δσym (kn ) )T Λ‫ٻ‬ΊΔɻ͜ͷ͕ࠩઃ‫ܭ‬৚݅ͱͷͣΕΛද͢ɻΔख๏ʹ͍ͭͯ঺հ͢Δɻਤ 78 ʹࣔ͢Α͏ʹɺ͜͜Ͱ঺հ͢ΔϏʔϜޫֶؔ਺ͷ࠷దԽख๏Ͱ͸ɺ5 ୆ͷ࢛‫࣓ిۃ‬ੴͱ 1 ୆ͷεΫϦʔϯϞχλʔΛ༻͍Δɻิ

(Q5) ͷऩଋྗ (k ஋) Λม͑ͳ͕ΒɺͦͷԼྲྀʹ͋

3 ൪໨ͷεςοϓͱͯ͠ɺ͜ΕΛิਖ਼͢ΔͨΊͷԠ౴ߦྻΛଌఆ͢Δɻ͜ͷͨΊʹɺQ1 ͷ k ஋Λઃ‫ܭ‬৚ΔεΫϦʔϯϞχλʔͰ rms ϏʔϜαΠζΛଌఆ͠ɺ͔݅Β ΔK ͚ͩͣΒͨ͠ঢ়ଶͰɺQ5 ͷԠ౴ؔ਺ΛQ5 ʹର͢ΔϏʔϜαΠζͷԠ౴ؔ਺ΛಘΔɻ͜ͷଌఆ͞ΕͨԠ౴ؔ਺Λઃ‫ܭ‬৚݅ͷԠ౴ؔ਺ʢλʔήοτʣ

ଌఆ͢Δɻ͜ΕΛ MQ1 = (σxm,Q1 (k1 ) σym,Q1 (k1 )ਖ਼ͷྲྀΕ͸ҎԼͷΑ͏ʹͳΔɻ࠷Լྲྀͷ࢛‫࣓ిۃ‬ੴ‫࣓ిۃ‬ੴ (Q1, Q2, Q3, Q4) ͷ k ஋Λิਖ਼͢Δɻ
e

Q1

Q2

Q3

Q4

ߦ͍ɺΔMQ2 , ΔMQ3 , ΔMQ4 Λ‫ٻ‬ΊΔɻ͜ΕΒͷଌ

Screen monitorఆ݁Ռ͔Βิਖ਼ྔΛ‫͢ࢉܭ‬ΔͨΊͷԠ౴ߦྻΛ‫ٻ‬ΊΔ

To measure Q−scan response͜ͱ͕Ͱ͖ΔɻԠ౴ߦྻ͸ɺ

R = ΔMQ1 ΔMQ2 ΔMQ3

ਤ 78: 5 ୆ͷ࢛‫࣓ిۃ‬ੴͱϏʔϜαΠζଌఆ༻ͷεΫϦʔϯϞχλΛ༻͍ͨϏʔϜޫֶؔ਺Ϛονϯά๏ςοϓͱͯ͠ɺλʔήοτͱͳΔԠ౴ؔ਺Λ‫ٻ‬ΊΔɻ

Q5 ͷऩଋྗͷ஋Λ k ͱ͠ɺ͜ΕΛ Δk ͣͭมԽ͞
ͤɺͦͷͱ͖ͷεΫϦʔϯϞχλʔ্Ͱͷਫฏɾਨ௚
ํ޲ͷ rms ϏʔϜαΠζ σxt , σyt Λ‫͢ࢉܭ‬Δɻ۩ମతʹ͸ɺi = 1, 2, ..., n ͱͯ͠ɺn ‫ݸ‬ͷऩଋྗɺΔk,,

͢ΔͱɺQ1, Q2, Q3, Q4 ͷͦΕͧΕͷ࢛‫࣓ిۃ‬ੴͷ࣍ʹɺ۩ମతͳ࠷దԽखॱΛࣔ͢ɻ·ͣ࠷ॳͷε

n−1
2


ΔMQ4

(11.2)
ͱͳΔɻΔK ͕খ͍͞ͱͯ͠Ԡ౴͕ઢ‫͋Ͱܗ‬ΔͱԾఆͷ໛ࣜਤɻki = k0 + i − 1 −σym,Q1 (kn ) )T ͱ͢ΔɻΔK Λ

Δσym,Q1 (kn ) )T ͸ɺQ1 ʹ ΔK ͚ͩՃ͑ͨͱ͖ͷԠ౴ؔ਺ͷมԽྔΛද͢͜ͱʹͳΔɻQ1 ͷ k ஋Λ‫ݩ‬ͷ஋ʹ໭ͨ͋͠ͱɺಉ༷ͷଌఆΛ Q2, Q3, Q4 ʹରͯ͠Q5

To adjust optics

σxm,Q1 (kn )Ճ͑Δલͷଌఆ݁Ռͱͷࠩ ΔMQ1 = (MQ1 − M )= (Δσxm,Q1 (k1 ) Δσym,Q1 (k1 ) . . . Δσxm,Q1 (kn )ͱൺֱ͠ɺ͜ͷ͕ࠩখ͘͞ͳΔΑ͏ʹ্ྲྀͷ࢛୆ͷ࢛

−...

(11.1)k ஋Λ (ΔKQ1 ΔKQ2 ΔKQ3 ΔKQ4 )T ͚ͩมԽ͞
ͤͨͱ͖ͷԠ౴ؔ਺ͷมԽྔ͸ɺ
⎞
⎛
ΔKQ1
⎟
⎜
⎟
R ⎜
⎜ ΔKQ2 ⎟ ,(11.3)⎜ΔK ⎝ ΔKQ3 ⎟⎠ΔKQ4ͱͳΔɻ͜ΕΑΓɺ(ΔKQ1 ΔKQ2 ΔKQ3 ΔKQ4ʹରͯ͠ϏʔϜαΠζΛ‫͢ࢉܭ‬Δɻ͜͜Ͱɺn ͸‫਺ح‬)T ͕ɺλʔήοτԠ౴ؔ਺ͱଌఆ͞ΕͨԠ౴ؔ਺ͷͱ͠ɺk0 ͸ઃ‫ܭ‬৚݅ͷ k ஋Ͱ͋Δͱ͢Δɻ͜ͷͱ͖ɺ

ࠩ ΔM Λੜͤ͡͞Δͱߟ͑Δͱɺ
⎞
⎛
ΔKQ1
⎟
⎜
ΔKQ2 ⎟
R ⎜
⎟,
⎜
ΔM =
⎟
ΔK ⎜
⎝ ΔKQ3 ⎠ΔKQ4λʔήοτͷԠ౴ؔ਺͸ɺT =
σxt (k1 )σyt (k1 )...

σxt (kn )σyt (kn )T,(11.4)

ͱ‫·ٻ‬ΔɻۭؒిՙޮՌͷࢧ഑తͳ৔߹ʹ͸ɺۭؒి

ͱ͢Δ͜ͱ͕Ͱ͖Δɻ͜ͷͱ͖ɺR ͷ‫ ྻߦٯ‬R−1 ΛՙޮՌΛ‫ؚ‬ΜཻͩࢠτϥοΩϯάίʔυΛ࢖༻ͯ͠ɺ

‫ٻ‬ΊΔ͜ͱͰɺλʔήοτԠ౴ؔ਺ͱଌఆ͞ΕͨԠ౴λʔήοτͷԠ౴ؔ਺ T Λ‫ٻ‬ΊΔɻۭؒిՙޮՌΛؔ਺ͷࠩ ΔM ͔Β Q1, Q2, Q3, Q4 ͷิਖ਼ྔΛɺ
⎛
⎞
ΔKQ1
⎜
⎟
⎜ ΔKQ2 ⎟⎜⎟ = ΔKR−1 ΔM,(11.5)
⎜ ΔK
⎟
Q3 ⎠
⎝
ΔKQ4

ແࢹͰ͖Δ৔߹ʹ͸ɺQ5 ೖ‫ޱ‬ͷϏʔϜޫֶؔ਺ α, βͱΤϛολϯε ε Λॳ‫ظ‬஋ͱͯ͠ɺQ5 ͱεΫϦʔϯ
Ϟχλ·ͰͷసૹߦྻΛ༻͍ͯɺT Λ‫͢ࢉܭ‬Δ͜ͱ΋Ͱ͖Δɻ

2 ൪໨ͷεςοϓͱͯ͠ɺn ‫ݸ‬ͷ ki ʹରͯ͠ɺ࣮ࡍͷՃ଎‫Ͱث‬ϏʔϜαΠζ σxm , σym Λଌఆ͢Δɻଌఆ͞ͱ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖ΔɻҎ্ʹࣔͨ͠ํ๏ͰɺQ5 ͷΕͨԠ౴ؔ਺Λ M = (σxm (k1 ) σym (k1 ) . . . σxm (kn )Ԡ౴ؔ਺Λઃ‫߹ͱܭ‬ΘͤΔ͜ͱ͕Ͱ͖ΔɻखॱΛ·ͱσym (kn ) )Tͱ͢ΔɻλʔήοτԠ౴ؔ਺ͱଌఆ͞

ΊΔͱ࣍ͷΑ͏ʹͳΔɻ㸲㸫 

1. λʔήοτԠ౴ؔ਺ T ͷ‫ࢉܭ‬(a) before optics matching
2

2

TargetMeasurement

σy (mm)σx (mm)2. Q5 Ԡ౴ؔ਺ M ͷଌఆ
1

3. Q1, Q2, Q3, Q4 ͷԠ౴ߦྻ R ͷଌఆ

1

4. Q1, Q2, Q3, Q4 ͷิਖ਼ྔͷ‫ͱࢉܭ‬ઃఆͷࠩ ΔM ͕খ͘͞ͳΔ·Ͱ‫܁‬Γฦ͢ɻ
2͜ͷํ๏Ͱ͸ɺԠ౴͸ઢ‫͋Ͱܗ‬Δͱ͍͏͜ͱҎ֎͸ͯ௚઀Ճ଎‫ث‬ύϥϝλͷ࠷దԽΛ࣮ࢪ͍ͯ͠Δ͜ͱʹ

1

ରԠ͢Δɻ͜ͷͨΊɺՃ଎‫ث‬ͷϞσϧʹґଘͤͣʹิ
ਖ਼Ͱ͖Δͱ͜Ζ͕௕ॴͱͳΔɻͨͩ͠ɺઢ‫ͳܗ‬Ԡ౴Λ0−5 0Ծఆ͍ͯ͠ΔͨΊɺઃ‫ܭ‬৚݅ͱ࣮ࡍͷ৚͕݅཭Ε͍ͯΔ৔߹ʹ͸ɺઃ‫ܭ‬৚݅ʹऩଋ͠ͳ͍Մೳੑ΋͋Δɻ͜
ͷํ๏Ͱ͸ɺ͋Δఔ౓ઃ‫ܭ‬৚݅ʹ͍ۙ৚͔݅Β։࢝ͯ͠ɺ‫܁‬Γฦ͠ิਖ਼Λ࣮ࢪ͢Δ͜ͱ͕ඞཁͱͳΔɻ࣍ͷηΫγϣϯͰ͸ɺ͜ͷํ๏Λ cERL Ճ଎‫ʹث‬ద༻ͯ͠ɺԣํ޲ͷϏʔϜ༌ૹ࿏ޫֶؔ਺ͷิਖ਼Λߦͬͨྫ

TargetMeasurement

σx (mm)Կ΋ԾఆΛಋೖ͍ͯ͠ͳ͍ͨΊɺ‫࣮ݱ‬ͷԠ౴Λར༻͠0
5 10 15 20 25−5 0 5 10 15 20 25
22
k (1/m )k (1/m )(b) after optics matching2

σy (mm)0−5 0͜ͷखॱΛλʔήοτԠ౴ؔ਺ͱଌఆ͞ΕͨԠ౴ؔ਺

5 10 15 20 25
2
k (1/m )

1

0
−5 0

5 10 15 20 25
2
k (1/m )ਤ 79: ߹ྲྀ෦ೖ‫ޱ‬ͷϚονϯά఺ MP1 ʹ͓͚ΔϏʔϜޫֶؔ਺ͷϚονϯά݁ՌɻϚονϯάલͱ‫ޙ‬ͷ Q5
ͷ‫͞ڧ‬Λม͑ͨͱ͖ͷϏʔϜαΠζͷԠ౴Λࣔ͢ɻΛ঺հ͢Δɻ
·ͨɺॎํ޲ͷϏʔϜ༌ૹ৚݅ͷิਖ਼ʹ͍ͭͯ΋ɺ‫ج‬ຊతʹ͸ԣํ޲ͷ৔߹ͱಉ͡ɺՃ଎‫ث‬ཁૉͷύϥϝλΛม͑ͨͱ͖ͷϏʔϜ৚݅ͷԠ౴Λଌఆɾิਖ਼͢Δ
ํ๏Λ༻͍Δ͜ͱ͕Ͱ͖Δɻॎํ޲ͷ৔߹ʹ͸ɺ࢛‫ۃ‬ి࣓ੴͷ k ஋ͷ୅ΘΓʹɺߴप೾ۭಎͷՃ଎ిѹɺՃ(a) before optics matching(b) after optics matching଎Ґ૬Λม͑ͨͱ͖ͷόϯν௕ɺΤωϧΪʔ֦͕ΓͷԠ౴Λଌఆ͠ɺ͜ΕΛઃ‫ܭ‬৚݅ͱ߹ΘͤΔΑ͏ʹิਖ਼ਤ 80: ߹ྲྀ෦ೖ‫ޱ‬ͷϚονϯά఺ MP1 ʹ͓͚Δԣ

͢Δ͜ͱʹͳΔɻํ޲ϏʔϜϓϩϑΝΠϧͷଌఆ݁ՌɻϚονϯάલͱ‫ޙ‬ͷϓϩϑΝΠϧͷมԽΛࣔ͢ɻ

11.2.2cERL ʹ͓͚Δԣํ޲༌ૹ৚݅ͷ࠷దԽόϯνిՙ (਺ 10 fC) ͰऩଋɾൃࢄྗͷଌఆΛߦ͍ɺcERL Ͱ͸ɺલͷηΫγϣϯͰ঺հͨ͠ํ๏Λ༻͍ͯɺ্ྲྀ͔Βԣํ޲ͷϏʔϜ༌ૹ࿏ޫֶ‫ܥ‬ͷิਖ਼Λ࣮‫݁ݧ‬ՌΛϞσϧͰ࠶‫͖Ͱݱ‬ΔΑ͏ʹ͍ͯ͠Δ [49]ɻ

ߦ͍ͬͯΔɻॎํ޲ʹ͍ͭͯ͸ɺ‫ج‬ຊతʹΤωϧΪʔੴʹΑΔԣํ޲ͷऩଋྗΛड͚ͨ͋ͱɺόϯνϟʔۭͱΤωϧΪʔ֦͕ΓΛଌఆ͠ͳ͕ΒՃ଎ۭಎͷՃ଎ిಎ [50] Ͱόϯνѹॖ͞Εɺೖࣹ‫ۭث‬ಎͰ 2.4 MeV ·ѹͱҐ૬ͷௐ੔Λߦ͍ͬͯΔɻۭؒిՙޮՌ͕‫ݦ‬ஶʹͰՃ଎͞ΕΔɻೖࣹ‫ۭث‬ಎʹΑΔՃ଎Ͱ͸ɺిࢠϏʔͳΔόϯνిՙͰ͸ɺΑΓਫ਼ີͳՃ଎ిѹͱҐ૬ͷௐϜΛۭಎͷத৺ʹ߹Θͤͯ௨͢͜ͱ͕ɺ௿Τϛολϯ

੔͕ඞཁͱͳΔͨΊɺόϯν௕ʹର͢ΔԠ౴Λଌఆ͢εϏʔϜΛอ࣋͢Δ͏͑ͰॏཁͱͳΔɻ͜ͷͨΊʹɺΔํ๏Λࢼ‫ͨ͠ݧ‬ɻೖࣹ‫ۭث‬ಎͷߴ࣍Ϟʔυ৴߸Λར༻ͨ͠ϏʔϜҐஔϞ

390 keV ʹՃ଎͞ΕͨిࢠϏʔϜ͸ɺιϨϊΠυి࣓ԣํ޲ͷ༌ૹ৚݅ͷ࠷దԽʹ͋ͨͬͯ͸ɺ࠷্ྲྀͷχλͷ։ൃΛߦ͍ɺ͜ΕΛ༻͍ͯ‫ي‬ಓௐ੔ࢼ‫ݧ‬Λߦͬిࢠॐ͔Βิਖ਼Λߦ͍ͬͯΔɻcERL ͷిࢠॐ͸ DC

͍ͯΔ [51]ɻ͜ͷΑ͏ʹೖࣹ‫ۭث‬ಎ্ྲྀͰ͸ɺిࢠޫӄ‫ࢠిۃ‬ॐΛ༻͍͓ͯΓɺ160 mm ͷΪϟοϓͷؒॐɺιϨϊΠυి࣓ੴɺόϯνϟʔۭಎͱೖࣹ‫ۭث‬ಎʹ 390 kV ΛҾՃͯ͠ిࢠϏʔϜΛੜ੒ɾՃ଎͍ͯ͠

ʹΑΔӨ‫ڹ‬Λ૯߹తʹௐ੔ͯ͠ɺ༌ૹ৚݅ͷ࠷దԽΛΔɻ͜ͷ۠ؒͰ͸ɺి‫ܗۃ‬ঢ়ʹΑΔԣํ޲ͷऩଋɾൃ

ߦ͍ͬͯΔɻ

ࢄྗ͕ಇ͖ɺ͜ΕΛϞσϧͰਖ਼֬ʹද‫͢ݱ‬Δ͜ͱ͕ॏೖࣹ‫ث‬ΑΓԼྲྀͰ͸ɺ༌ૹ৚݅ͷௐ੔ͷͨΊʹ࢛‫ۃ‬ཁͱͳΔɻ·ͣɺۭؒిՙޮՌͷޮ͔ͳ͍‫ۃ‬Ίͯ௿͍ి࣓ੴΛ༻͍ΔɻओۭಎͰՃ଎͞Εͨ͋ͱ΋ϏʔϜ㸲㸫 ΤωϧΪʔ͸ 19.4 MeV ͱ௿͍ͨΊɺి࣓ੴࣗ਎ʹ(a) before optics matching
62ΑΔ࢒ཹ࣓৔΍ɺਅۭήʔδͷ࣓ੴ͕࡞Δ‫࣓ڥ؀‬৔౳

TargetMeasurement݅ʹ͚ۙͮΔʹ͸ɺ্ྲྀ͔Β࢛‫࣓ిۃ‬ੴͷิਖ਼Λߦͬσy (mm)σx (mm)ͷӨ‫ڹ‬Λ‫͘ڧ‬ड͚ͯ͠·͏ɻ͜ͷͨΊɺઃ‫ܭ‬ͷ༌ૹ৚

1͍ͯ͘ඞཁ͕͋Δɻ͜ΕΛ࣮ࢪ͢ΔͨΊʹɺલͷηΫγϣϯͰ঺հͨ͠ɺ5 ୆ͷ࢛‫࣓ిۃ‬ੴΛҰηοτʹ͠

04ͨ༌ૹ৚݅ิਖ਼๏Λ༻͍͍ͯΔɻपճӡసͷͱ͖ʹ

6

͸ɺਤ 77 ʹࣔ͢ 7 ՕॴͷϚονϯάϙΠϯτ (MP)2‫ٻ‬Ί͍ͯΔɻ࣮ࡍͷิਖ਼Ͱ͸ɺଌఆ‫ࠩޡ‬ͷӨ‫ڹ‬Λߟྀ

1

ͯ͠ 4 ‫ݸ‬ͷ͏ͪͷ 2 ‫ݸ‬ͷ‫ݻ‬༗஋Λ༻͍ͯٙࣅ‫ྻߦٯ‬

12

TargetMeasurement

σx (mm)ࡍʹ͸ɺಛҟ஋෼ղ (SVD ๏) ʹΑͬͯٙࣅ‫ྻߦٯ‬Λ2

0
8 10 12
4
6
8 10
2
2
k (1/m )k (1/m )(b) after optics matching6

σy (mm)Ͱิਖ਼Λߦ͍ͬͯΔɻԠ౴ߦྻ R ͷ‫ྻߦٯ‬Λ‫ٻ‬ΊΔ

442Λ‫͍ͯ͠ࢉܭ‬Δɻ
04

͜͜Ͱ͸ɺόϯνిՙ 0.5 pC ͷӡస৚݅࣌ͷϚονϯά݁Ռʹ͍ͭͯ঺հ͢Δɻਤ 79 ʹ߹ྲྀ෦ʹ͋Δ

6

8 10
2
k (1/m )

12

0
4

6

8 10
2
k (1/m )12ϚονϯάϙΠϯτ MP1 ʹ͓͚Δ༌ૹ৚݅ิਖ਼ͷ݁ՌΛࣔ͢ɻԠ౴ؔ਺ͷλʔήοτ஋͸ɺۭؒిՙޮՌΛਤ 81: ओۭಎೖ‫ޱ‬ͷϚονϯά఺ MP2 ʹ͓͚ΔϏʔ‫ؚ‬ΉཻࢠτϥοΩϯάίʔυͰ͋Δ General Particle

Ϝޫֶؔ਺ͷϚονϯά݁ՌɻϚονϯάલͱ‫ޙ‬ͷ Q5

Tracer (GPT) [33] Λ༻͍ͯ‫͞ࢉܭ‬Εͨɻਤ 79 ͷ (a)͸ิਖ਼લͷ Q5 ͷԠ౴ؔ਺Λ͍ࣔͯ͠Δɻิਖ਼લ͸ɺͷ‫͞ڧ‬Λม͑ͨͱ͖ͷϏʔϜαΠζͷԠ౴Λࣔ͢ɻਨ௚ํ޲ͷԠ౴ؔ਺ͷଌఆ஋͕λʔήοτ͔Βେ͖͘

11.2.3ॎํ޲༌ૹ৚݅ͷԠ౴ଌఆͣΕ͍ͯΔ͜ͱ͕Θ͔Δɻ͜ͷঢ়ଶ͔ΒલͷηΫγϣ

ॎํ޲ͷ༌ૹ৚݅ͷ࠷దԽʹ͸ɺߴप೾Ճ଎ۭಎͷϯͰࣔͨ͠༌ૹ৚݅ͷิਖ਼Λ࣮ࢪͨ͠ɻิਖ਼Λ 3 ճ࣮Ճ଎ిѹͱҐ૬Λิਖ਼͢Δ͜ͱ͕ඞཁͱͳΔɻ͜Εࢪͨ͋͠ͱͷԠ౴ؔ਺Λਤ 79 ͷ (b) ʹࣔ͢ɻ‫׬‬શʹ·ͰͷۭؒిՙޮՌ͕ൺֱతऑ͍όϯνిՙͰ͸ɺ֤Ұக͍ͯ͠ΔΘ͚Ͱ͸ͳ͍͕ɺ͔ͳΓλʔήοτԠ౴ۭಎԼྲྀͷΤωϧΪʔ͕ઃ‫ܭ‬৚݅ʹͳΔΑ͏ʹௐ੔ؔ਺ʹۙ͘ͳ͍ͬͯΔ͜ͱ͕Θ͔Δɻਤ 80 ʹิਖ਼લΛ࣮ࢪ͢Δ͜ͱͰɺ༌ૹ৚݅Λิਖ਼͖ͯͨ͠ɻ͔͠͠

‫ޙ‬ͷϏʔϜϓϩϑΝΠϧΛࣔ͢ɻԠ౴ߦྻͷଌఆ࣌ؒͳ͕ΒɺόϯνిՙΛ૿‫ిۭؒͯ͠ڧ‬ՙޮՌ͕ࢧ഑

͸ɺ1 ճ͋ͨΓ 5 ෼ఔ౓Ͱ͋Γɺ͜͜Ͱͷิਖ਼ʹ͸తʹͳͬͯ͘ΔͱɺΑΓৄࡉͳՃ଎ిѹͱҐ૬ͷௐ੔

20 ෼ఔ౓Λཁ͍ͯ͠ΔɻMP1 Ͱ͸ΤωϧΪʔ͕ 2.9
MeV Ͱ͋ΓɺۭؒిՙޮՌ͕·ͩ࢒ΔྖҬͰ͋Δ͕ɺ

͕ඞཁͱͳͬͯ͘Δɻ࠷ऴతʹ͸ɺόϯν௕ͱΤωϧ͜ͷํ๏ʹΑۭͬͯؒిՙޮՌΛ‫ؚ‬Ίͯ༌ૹ৚݅Λิͯิਖ਼͢Δ͜ͱ͕໨ඪͰ͋Γɺͦͷ४උͱͯ͠ɺόϯ

ਖ਼Ͱ͖Δ͜ͱ͕֬ೝ͞ΕͨɻνϟʔۭಎͷՃ଎ిѹͱόϯνిՙΛม͑ͨͱ͖ͷόΪʔ֦͕Γʹର͢ΔɺՃ଎ిѹͱҐ૬ͷԠ౴Λଌఆ͠

ਤ 81 ʹ߹ྲྀ෦ͱओۭಎͷؒʹ͋ΔϚονϯάϙΠ

ϯν௕ͷԠ౴ؔ਺ଌఆΛߦͬͨɻόϯν௕͸ɺਤ 77

ϯτ MP2 ʹ͓͚Δ༌ૹ৚݅ͷิਖ਼݁ՌΛࣔ͢ɻͪ͜

ʹࣔ͢Α͏ʹɺೖࣹ‫ث‬਍அϥΠϯʹઃஔ͞Εͨภ޲ۭΒ΋ 3 ճิਖ਼Λ࣮ࢪ͠ɺ΄΅ઃ‫ܭ‬৚݅ʹิਖ਼Ͱ͖ͯ

ಎ [52] Λ༻͍ͯଌఆ͞Εͨɻ

͍Δ͜ͱ͕Θ͔Δɻ͜ͷΑ͏ʹɺ5 ୆ͷ࢛‫࣓ిۃ‬ੴΛଌఆ࣌ͷόϯνిՙ͸ 7.7 pC Ͱ͋Γɺ͜Ε͸ 1.3

Ұηοτʹͯ͠ɺଌఆ͞ΕͨԠ౴ߦྻΛ༻͍Δํ๏GHz Ͱฏ‫ۉ‬ϏʔϜిྲྀ 10 mA ʹ૬౰͢ΔɻόϯνͰɺԣํ޲ͷ༌ૹ৚݅Λิਖ਼Ͱ͖Δ͜ͱ͕֬ೝ͞Εిՙ 7.7 pC Ͱ͸ۭؒిՙޮՌ͕ࢧ഑తͰ͋Γɺॎํ

ͨɻcERL ͷӡసͰ͸ɺ͜ΕΒͷิਖ਼͸ɺຖ೔ߦ͏Θ޲ͱԣํ޲ͷ༌ૹ৚͕݅݁߹͢Δɻ·ͣɺԣํ޲ͷ༌͚Ͱ͸ͳ͘ɺ༌ૹ৚݅Λେ͖͘ม͑ͨͱ͖ʹ࣮ࢪͯ͠ૹ৚݅ͷௐ੔Λ࣮ࢪͨ͋͠ͱʹɺόϯν௕ͷԠ౴ଌఆ

͍ΔɻओۭಎԼྲྀͷΤωϧΪʔ 19.4 MeV ͷྖҬ΋Λߦͬͨɻਤ 82 ʹόϯνϟʔۭಎͷՃ଎ిѹΛม͑ಉ༷ͷํ๏ͰϚονϯάΛ࣮ࢪ͠ɺϏʔϜαΠζΛখͨͱ͖ͷόϯν௕ͷԠ౴Λࣔ͢ɻγϛϡϨʔγϣϯͷ͘͞཈͑ͯ༌ૹͰ͖Δ͜ͱ͕֬ೝ͞Ε͍ͯΔɻ

Ԡ౴͸ GPT ʹΑͬͯ‫͞ࢉܭ‬ΕͨɻΤωϧΪʔ߹ΘͤʹΑΔҐ૬ௐ੔ͷΈΛ࣮ࢪͨ͋͠ͱͷଌఆ݁ՌͰ͋Δ
͕ɺόϯνϟʔిѹʹର͢Δόϯν௕ͷԠ౴͸ɺϞσ

㸲㸫 

11
Measurement
Simulation

Bunch length, σt (ps)

10
9
8
7
6
5
4
30

40

50
60
70
80
Buncher voltage, Vb (kV)90

ਤ 82: όϯνϟʔۭಎͷՃ଎ిѹΛεΩϟϯͨ͠ͱ͖ͷόϯν௕ͷมԽͷଌఆ݁Ռɻόϯν௕͸ਤ 77 ͷೖࣹ‫ث‬਍அϥΠϯʹ͋Δภ޲ۭಎΛ༻͍ͯଌఆ͞Εͨɻ

Bunch length, σt (ps)

6
5
4
3
2
Measurement
Simulation

1
0
0

2

4
6
8
Bunch charge, Q (pC)10

ਤ 83: όϯνిՙΛม͑ͨͱ͖ͷόϯν௕ͷมԽͷଌఆ݁Ռɻόϯν௕͸ਤ 77 ͷೖࣹ‫ث‬਍அϥΠϯʹ͋
Δภ޲ۭಎΛ༻͍ͯଌఆ͞Εͨɻϧ‫͔ࢉܭ‬Βେ͖ͣ͘Ε͍ͯͳ͍͜ͱ͕֬ೝ͞Εͨɻ
ਤ 83 ʹόϯνిѹΛม͑ͨͱ͖ͷόϯν௕ͷԠ౴Λࣔ͢ɻόϯνిՙ͸ۭؒిՙޮՌͷޮ͔ͳ͍ඍখి
ՙ͔Βɺ7.7 pC ·Ͱ૿‫͞ڧ‬ΕͨɻγϛϡϨʔγϣϯͷԠ౴ͱଌఆ݁ՌΛൺֱ͢Δͱɺ͔ͳΓ͍ۙԠ౴ʹͳͬ
͍ͯΔ͜ͱ͕֬ೝ͞Εͨɻ͜ͷ͜ͱΑΓɺΤωϧΪʔ߹ΘͤʹΑΔҐ૬ௐ੔ͷΈͰɺ͋Δఔ౓ઃ‫ܭ‬৚݅ʹ͚ۙͮΒΕΔ͜ͱ͕֬ೝ͞Εͨɻόϯνѹॖ৚݅͸ɺόϯνϟʔۭಎ͚ͩͰͳ͘ೖࣹ
‫ۭث‬ಎͷՃ଎ిѹͱҐ૬ʹ΋ґଘ͢ΔͨΊɺ࣍ͷஈ֊ͱͯ͠ɺͦΕΒͷԠ౴ߦྻΛଌఆ͠ɺͦΕΛ༻͍ͨิਖ਼๏ͷࢼ‫ݧ‬Λߦ͏༧ఆͰ͋Δɻ·ͨɺۭؒిՙޮՌΛհͯ͠ɺԣํ޲ͷӡಈ͕ॎํ޲ͷӡಈʹ΋Ө‫ڹ‬Λ༩͑
ΔͨΊɺ6 ࣍‫ݩ‬Ґ૬ۭؒͷԠ౴Λ߹ΘͤΔΑ͏ʹิਖ਼๏Λ֦ு͍ͯ͘͠༧ఆͰ͋Δɻ

㸲㸫 

12

Ճ଎ɾ‫ݮ‬଎ϏʔϜͷΦϓςΟΫ
εઃ‫ܭ‬

12.1

ߴௐ೾ʹΑΔ Beam Break Up(HOM BBU) [53–56]ϏʔϜ͕ۭಎͷத৺͔ΒͣΕͨ৔ॴ (x = 0) Λ௨աਤ 84: ۭಎʹΑΔΩοΫͱपճϧʔϓͷ‫ي‬ಓͷ໛ࣜਤɻ

͢Δͱɺԣํ޲ʹΩοΫΛ͢Δߴௐ೾ʢHigher-Order

Mode, HOMʣ͕ൃੜ͢Δɻ͜ͷΑ͏ʹɺϏʔϜʹӨ‫ڹ‬Λ͋ͨ͑ΔΑ͏ͳి࣓৔Λ wake ﬁeld ͱ‫ͼݺ‬ɺΤϛολϯε͕ѱԽͨ͠ΓɺϏʔϜΛࣦͬͨΓ͢ΔϏʔϕΫτϧ (x, x , y, y  , z, p/p) ʹର͢Δసૹߦྻ R Λ
ߟ͑Δɻసૹߦྻ R12 ͷ੒෼Λ༻͍Δͱɺ࠶ͼۭಎͷೖΓ‫ʹޱ‬໭ͬͨ࣌ͷҐஔ x(t) ͸࣍ͷΑ͏ʹͳΔɻ

ϜϒϨʔΫΞοϓʢBeam Break Up, BBUʣ͕ൃੜx(t) = R12 x (t − tr ).

͢Δɻ͜Ε͸ɺՃ଎ۭಎ͕௕‫཭ڑ‬ฒͿେ‫ܕ‬ͷઢ‫ܗ‬Ճ଎

(12.4)‫Ͱث‬໰୊ʹͳ͓ͬͯΓɺϦχΞίϥΠμʔ (ILC) Ͱ͜͜Ͱɺtr ͸पճϧʔϓΛ௨ա͢ΔͨΊʹ͔͔Δ࣌΋‫ݒ‬೦͞Ε͍ͯΔɻҰํɺERL Ͱ͸पճͯ͠໭ͬͯ

ؒͰ͋Δɻࣜ (12.3) ͱ (12.4) ͔Βɺपճ‫ޙ‬ͷ x(t) ʹ͖ͨϏʔϜ͕ wake ﬁled Λ‫ڧ‬ΊΔ৔߹΋͋ΔɻՃ଎͍ۭͭͯಎ͕ 100 ୆Ҏ্ฒͿ GeV Ϋϥεͷ ERL Ͱਂࠁͳ໰୊ͱͳ͓ͬͯΓɺ҆ఆʹϏʔϜΛӡసͰ͖Δฏ‫ిۉ‬x(t) =

 t−teR12 −∞r W (t − tr − τ )I(τ )x(τ )dτpcྲྀͷ্‫ݶ‬ΛܾΊͯ͠·͏ཁҼͱͳ͍ͬͯΔɻӡಈྔ pͷిࢠ͕ۭಎೖΓ‫ Ͱޱ‬x = 0, x = 0 Ͱೖࣹͨ͠৔
߹Λߟ͑Δʢ࣮͸ΩοΫΛड͚Δଆͷిࢠ͕ x = 0,x = 0 Ͱ͋ͬͯ΋ෆ҆ఆੑʹ͸‫د‬༩͠ͳ͍ʣɻ͜͜
Ͱɺx ͸ x ͷ‫ي‬ಓ্ͷҐஔ s ͷඍ෼Ͱ͋Δ͕ɺਫฏํ޲ͷӡಈྔΛ px Ͱ࣍ͷΑ͏ʹॻ͘͜ͱ΋Ͱ͖Δɻ
x =

px
,
p

(12.1)

ۭಎͷ਺͸ҰͭͰɺՃ଎ྔ͸े෼খ͍͞΋ͷͱ͢Δɻ

(12.5)
ͱ͍͏͕ࣜಋ͔ΕΔɻR12 = 0 ͷͱ͖͸ɺϏʔϜ͕

x = 0 ͷҐஔʹ໭Γɺ৽ͨʹ wake ﬁeld Λ૿෯ͤ͞Δ͜ͱ͕ͳ͍ͷͰ҆ఆͰ͋Δɻ͔͠͠ɺ|R12 | ͕େ͖͍࣌ʹ͸ɺपճ‫ ʹޙ‬x ํ޲ͷҐஔͷΦϑηοτ͕֦େ͢Δ͜ͱ΋͋ΔɻHOM BBU ͕ൃੜ͢Δ͔Ͳ͏͔͸ɺV (t) ͕૿෯͢Δ͔‫ݮ‬ਰ͢Δ͔Ͱܾ·Δɻ۩ମతʹෆ҆ఆʹͳΔ৚݅Λ‫ٻ‬ΊΔͨΊʹ V (t) ʹ͍ͭͯॻ͖Լ͢ɻલํͷిࢠόϯν͕ۭಎͷத৺͔Β x ͚ͩͣΕͨ৔V (t) =

ॴΛ௨ա͢Δͱɺwake ﬁeld, V (t) Λۭಎͷதʹ࢒͢ɻ͜ͷ V (t) ͸࣌ؒͷؔ਺Ͱ͋Δɻ࣌ؒ ͷిྲྀ͕ੜ੒͠

,teR12 −∞ W (t − τ )I(t )V (τ − tr )dτpc.

(12.6)

ͨ wake ﬁeld Λ࣌ؒ t ͷిࢠ͕‫͡ײ‬Δ΋ͷͱ͢Δͱɺ͜ͷੵ෼ํఔࣜΛղͨ͘Ίʹ V (t) ∝ eiωt ͱ͢Δͱɺిྲྀ I(t) ͱ wake function, W (t) ͱ‫ݺ‬͹ΕΔۭಎಛ‫ݻ‬༗஋ํఔ͕ࣜಘΒΕΔɻIm(ω) ͷූ߸͕ਖ਼ͷ࣌ʹ͸༗ͷؔ਺Ͱද͢͜ͱ͕Ͱ͖Δɻ
 t
V (t) =W (t − τ )x(τ )I(τ )dτ.

ͱ͖͕ BBU ͕ൃੜ͠ͳ͍ฏ‫ྲྀిۉ‬ͷᮢ஋ Ith Ͱ͋Δ

V (t) ͸‫ݮ‬ਰ͠ɺෛͷͱ͖͸૿෯͢ΔɻIm(ω) = 0 ͷ(12.2)−∞

ͱΈͳ͢͜ͱ͕Ͱ͖Δɻ؆୯ͷͨΊʹ͋ΔಛఆͷϞʔυͷߴௐ೾प೾਺ m ͷΈΛߟྀ͢Δͱɺ࣍ͷΑ͏ͳ͜ͷͱ͖ʹ‫ํޙ‬ͷిࢠʹΩοΫ x (x  1) Λ༩͑Δɻ

͕ࣜಘΒΕΔɻ͢Δͱɺۭಎग़‫ޱ‬ͷిࢠͷ֯౓ x (t) ͸࣍ͷΑ͏ʹදIth = −

͢͜ͱ͕Ͱ͖Δɻx (t) =px (t)eV (t)=.
ppc(12.3)

e ωcm



Rm
Qm



2pc

.

(12.7)Qm R12 sin(ωm tr )͜͜ͰɺRm ͸γϟϯτΠϯϐʔμϯεɺQm ͸ৼಈͷ‫ܧ‬ଓ࣌ؒΛද͢ແ࣍‫Ͱྔݩ‬ɺۭಎ‫ݻ‬༗ͷύϥϝʔλͰ͋ΔɻͦͷಛఆͷϞʔυ m ͷ wake function ͱ࣍࣍ʹपճϧʔϓΛ௨ա͢Δ͜ͱʹͳΔɻਤ 84 ʹ໛ͷΑ͏ͳؔ܎͕͋ΔɻࣜਤΛࣔ͢ɻपճϧʔϓͷϏʔϜ༌ૹ͸γϯϓϧʹઢ‫ܗ‬ΦϓςΟΫεͷΈͰܾ·Δͱ͢Δɻ͜͜Ͱɺ6 ࣍‫ݩ‬㸲㸫 

W (t) =

Rm
Qm

2
ωm
e−(ωm /2Qm )t sin ωm t.
2c(12.8)

͜ͷ୯७Խͨ͠Ϟσϧ͔Βɺ͍͔ͭ͘ͷ͜ͱ͕Θ͔Δɻ

ͱͳΔɻQm ͕Լ͕ΔΑ͏ͳޮՌɺ͢ͳΘͪৼಈͷ‫ܧ‬• Rm ͷେ͖͍Ϟʔυ͕ɺฏ‫ྲྀిۉ‬ͷᮢ஋ Ith ʹҰ൪Ө‫͕͋ڹ‬Δɻ

ଓ͕࣌ؒ‫ݮ‬গ͢ΔɻՃ଎प೾਺ 1.3GHz ͸νϡʔφʔ• R12 sin(ωm tr ) > 0 Ͱ͋Ε͹ৗʹ҆ఆʢ͔͠͠ɺ
τϥοΩϯάͷ݁ՌʹΑΔͱᮢ஋͕ଘࡏ [55]ʣɻ
• |R12 | ͷখ͍͞ΦϓςΟΫε͕͍͍ɻʢ͢ͳΘͪϕʔλؔ਺͕খ͍͞ʣ

Λ༻͍ͯਖ਼֬ʹ߹ΘͤΔ͕ɺߴௐ೾͸੡࡞ਫ਼౓ͷ͹Β͖ͭͳͲ͕‫ݪ‬ҼͰ߹Θͳ͍͜ͱ΋͋Δ΄͔ɺҙਤతʹ͹Β͖ͭΛ༩͑Δ͜ͱ΋ՄೳͰ͋Δɻ
ΦϓςΟΫεͷઃ‫Ͱܭ‬͸ɺՃ଎‫ث‬શൠʹΘͨͬͯ

|R12 | Λখ͘͞཈͑ͨΓɺӨ‫ڹ‬ͷେ͖͍ߴௐ೾ ωm ʹ
ରͯ͠ tr Λ࠷దԽͨ͠Γɺਫฏɾਨ௚ํ޲ͷΧοϓϦϯάΛௐ੔͢Δ͜ͱʹΑͬͯɺᮢ஋ిྲྀΛ্͛Δ޻

• ߴௐ೾ͷप೾਺ ωm ͱपճ࣌ؒ tr ͷؔ܎͕ॏཁɻ
͜ΕΒͷ݁Ռ͔Βɺۭಎͷ wake function ͱઢ‫ܗ‬Ճ଎෉Λ͢Δɻ࣍ͷষͰ͸ɺઢ‫ܗ‬Ճ଎‫ث‬ͷΦϓςΟΫεઃ‫͍ͯͭʹܭ‬આ໌͢Δɻ‫ͱث‬पճϧʔϓͷΦϓςΟΫεͷ࠷దԽ͕ඞཁෆՄܽ
Ͱ͋Δ͜ͱ͕Θ͔Δɻ

12.2

͜Ε·Ͱ͸ɺਫฏํ޲ͷΈΛߟ͖͕͑ͯͨɺಉ͡Α͏ʹਨ௚ํ޲ʹ΋ HOM BBU ͸ൃੜ͢Δɻ͜͜Ͱɺ

HOM BBU ͷిྲྀᮢ஋Λ্͛ΔͨΊʹ͸ɺ|R12 | Λਫฏɾਨ௚ํ޲ͷ֯౓ ͷΧοϓϦϯά͕͋Δ৔߹Λখ͘͞཈͑Δ͜ͱ͕ॏཁͰ͋Δ͜ͱΛઆ໌ͨ͠ɻ͜͜

ߟ͑Δɻͦͷͱ͖͸ɺిࢠͷҐஔΛ

x → x cos θ + y sin θ,ͰɺՃ଎‫ث‬ͷΦϓςΟΫεઃ‫ʹܭ‬Α͘࢖ΘΕΔϕʔλ(12.9)

ͱ͠ɺసૹߦྻͷ R12 ΛҎԼͷࣜͰஔ͖‫ͯ͑׵‬ٞ࿦
͢Δ͜ͱ͕Ͱ͖Δɻ∗R12= R12 cos2 θ

3 GeV ERL ͷઢ‫ܗ‬Ճ଎‫ث‬ͷΦϓςΟΫεσβΠϯ [1, 57–59]ؔ਺ ͱҐ૬ͷਐΈ Ͱॻ͖௚͢ɻۭಎͰՃ‫ݮ‬଎͢ΔͨΊɺϩʔϨϯπϑΝΫλʔΛ ͱ͠ɺΤωϧΪʔͷมԽ΋ߟྀ͢Δɻసૹߦྻ R ͷग़ൃ఺Λ iɺऴண఺Λ fͷԼ෇͖Ͱද͢ͱɺ

+(R14 + R32 ) sin θ cos θ+

R34 sin2 θ.

R12 (i → f ) = γi(12.10)βi βf
sin Δφ,γi γf(12.13)∗ΧοϓϦϯάʹΑͬͯ͸ R12Λখ͘͢͞Δ͜ͱ͕ͰͱͳΔɻՃ଎ۭಎ͕Ұ͔ͭ͠ͳ͍৔߹͸ɺΔφ Λθϩ͖ΔͨΊɺपճ෦ͷΦϓςΟΫεͷΧοϓϦϯάͷௐʹ͚ۙͮΕ͹͍͍͕ɺଟ਺ͷՃ଎ۭಎ͕͋Δ৔߹͸ඇ੔Ͱిྲྀᮢ஋Λ্͛Δ͜ͱ͕Ͱ͖Δɻৗʹࠔ೉Ͱ͋ΔɻͦΕ͸ɺͦΕͧΕͷۭಎΛ௨ա͢Δ

࣮ࡍʹ͸ɺwake function ͸ΤωϧΪʔʢͭ·Γ ʣ͕ҟͳΓɺผͷՃ଎ۭಎͰൃ
ੜͨ͠ wake ﬁeld ͕ྡ઀͢Δۭಎʹ఻Θͬͯ૿෯͢2
ωm
e−(ωm /2Qm )t sin ωm t,
2c
m(12.11)ͱɺϞʔυͷ଍͋͠Θͤʹͳ͓ͬͯΓෳࡶͰ͋Δɻ·
ͨɺ100 ୆Ҏ্ͷۭಎ͕ฒΜͰ௕େͳ্ʹɺࣜ (12.3)

ಎͷ૊Έ߹Θͤ͸΄΅ۭಎͷ਺ͷ 2 ৐͋ΔͨΊɺ͢Ͱ͸Ұఆͱͨ͠ӡಈྔ p ͕Ճ଎ʹΑͬͯঃʑʹมԽ͠ʹͳΔɻ

W (t) =Rm
Qm

Δ͜ͱ΋͋Δʢcoupling HOMʣͨΊͰ͋Δɻͦͷۭ΂ͯʹ͓͍ͯ Λθϩʹ͚ۙͮΔ͜ͱ͸ෆՄೳʹ͍ۙɻͦ͜Ͱɺϕʔλؔ਺Λશମతʹখ͘͢͞Δ͜ͱ͕ॏཁ

∗͍ͯ͘ɻͻͱͭͷۭಎͰ͋Ε͹ɺ R12΍पճ࣌ؒ tr௨ৗͷઢ‫ܗ‬Ճ଎‫͋Ͱث‬Ε͹ɺ࢛‫࣓ిۃ‬ੴʹΑΔऩଋ

ͷ࠷దͳղ͕؆୯ͳϞσϧͰΘ͔Δ͕ɺ਺ଟ͘ͷશͯྗΛ‫͘͢ڧ‬Ε͹ϕʔλؔ਺Λখ͘͢͞Δ͜ͱ͕Ͱ͖

ͷۭಎʹରͯ͠࠷దԽ͢Δʹ͸τϥοΩϯάʹΑΔγ

Δɻ͔͠͠ɺERL ͷ৔߹ɺՃ଎ɾ‫ݮ‬଎ϏʔϜͱಉ͡ϛϡϨʔγϣϯ͕ඞཁͰ͋ΔɻҰํͰɺۭಎ͕਺ଟ͘

ύεΛ௨ա͢Δɻऩଋྗ͸ϏʔϜͷӡಈྔʹ൓ൺྫ͢

͋Δͱ͖ʹɺޮՌతʹిྲྀᮢ஋Λ্͛Δํ๏͕͋Δɻ

ΔͨΊɺߴΤωϧΪʔͷϏʔϜʹରͯ͠‫͍ڧ‬ऩଋྗΛՃ଎ۭಎຖʹߴௐ೾ͷप೾਺Λม͑Δํ๏ͰɺHOM

༩͑Δ͜ͱ͕Ͱ͖ͣɺͦͷΑ͏ͳ৚݅ͷதͰϕʔλؔ

randomization ͱ‫ݺ‬͹ΕΔɻߴௐ೾ͷप೾਺͕ ωm Λத৺ʹ σωm ͷ͹Β͖͕ͭ͋Δͱɺࣜ (12.8) ͸ɺ਺Λখ͘͢͞Δඞཁ͕͋Δɻ͜͜Ͱɺ3 GeV ERL ͷ

W (t) =

Rm
Qm

2
2
2
ωm
e−(ωm /2Qm )t e−σωm t /2 sin ωm t,
2c(12.12)ઃ‫ํܭ‬๏Λ঺հ͢Δɻ
·ͣɺՃ଎ۭಎͱి࣓ੴͷϨΠΞ΢τΛਤ 85 ʹࡌͤΔɻՃ଎ۭಎ͸ΫϥΠΦελοτͷதʹ഑ஔ͞Εɺ֤ΫϥΠΦελοτͷ྆ଆʹ 3 ୆Ұ૊ͷ࢛‫࣓ిۃ‬ੴΛ㸲㸫 ਤ 85: 3-GeV ERL ͷՃ଎ۭಎͷ௚ઢ෦ͷϨΠΞ΢τɻ
഑ஔ͢Δɻ3 GeV ͷઢ‫ܗ‬Ճ଎‫ث‬͸͜ΕΛ਺ 100 ‫ݸ‬ฒ΂ͨ΋ͷͰ͋Δɻ͜ͷ 3 ୆Ұ૊ͷ࢛‫࣓ిۃ‬ੴΛ tripletͱ‫ͼݺ‬ɺਫฏ΋ਨ௚ํ޲΋ऩଋ͢Δ͜ͱ͕Ͱ͖Δ࠷খͷ૊Έ߹ΘͤͰ͋Δɻ͜͜ͰɺՃ଎ۭಎͷ௚ઢ෦ͷ֤ཁૉͰϏʔϜʹಇ͘సૹߦྻΛઆ໌͢Δɻ·ͣɺ‫཭ڑ‬L ͷυϦϑτεϖʔε͸ਤ 86: ͻͱ૊ͷΫϥΠΦελοτͱ triplet ͱͦͷυϦϑτεϖʔε :

1

L

0

1

,

(12.14)ϕʔλؔ਺ɻ

cos α −

ͱͳΔɻ࢛‫࣓ిۃ‬ੴͷऩଋྗ͸ K ஋ͱ͍͏ม਺Ͱද

e(∂B/∂x),p

2 sin α− √38 γγf sin α͞Εɺ࣓ੴ಺ͷ࣓৔ޯ഑Λ ∂B/∂ ͱ͢Δͱɺ

K=

√

1
γf
α = √ ln ,8 γi(12.15)ͱ͍͏ؔ܎͕͋Δɻ͜ͷసૹߦྻ͸√√√1 sin Ks
cos KsK,࢛‫࣓ిۃ‬ੴ :√√√cos Ks− K sin Ks(12.16)√ γi
8 γ  sin α
√


γi
cos
α + 2 sin αγf,(12.19)

ͱͳΔ [60]ɻ
Ճ଎ۭಎͷ௚ઢ෦ΦϓςΟΫεͷઃ‫ܭ‬͸ɺҎ্ͷసૹߦྻΛ΋ͱʹߦ͏ɻ·ͣɺͻͱͭͷΫϥΠΦε
λοτͱ triplet ͷ૊Έ߹ΘͤΛߟ͑ɺۭಎ಺ͰϕʔͱͳΔɻ͜͜ͰɺK ஋͕ਖ਼ͷͱ͖ਫฏํ޲ʹରͯ͠λؔ਺Λখ͘͢͞ΔΑ͏ʹ triplet Λ࠷దԽ͢Δɻࠨ

ऩଋ͢Δ͜ͱΛ͕ࣔ͢ɺ͜ͷఆٛ͸ਓʹΑͬͯҟͳΔ

ӈରশʹઃ‫͕ྫͨ͠ܭ‬ਤ 86 Ͱ͋Γɺࠨӈͷ (QF0,ͷͰ஫ҙ͕ඞཁͰ͋Δɻઢ‫ܗ‬Ճ଎‫ث‬ͷ৔߹ɺԣํ޲ͷQD0) ͱ (QF1,QD1) ͸ triplet ͷ൒෼Λ͍ࣔͯ͠Δɻӡಈྔ px ͸มΘΒ͵··Ճ଎͞Εɺશମͷ p ͕૿ՃதԝͰՃ଎΋͘͠͸‫ݮ‬଎͢ΔͨΊʹɺ(QF0,QD0) ͱ

͢Δ [57]ɻ͢Δͱɺࣜ (12.4) ʹࣔͨ֯͠౓ x ͸‫ݮ‬গ

(QF1,QD1) ͰΤωϧΪʔ͕ҟͳΔ͕ɺಉ͡ऩଋൃࢄ

͢Δɻ͜ΕΛஅ೤‫ݮ‬ਰͱ‫ͼݺ‬ɺసૹߦྻΛ༻͍ͯɺ࣍

ྗ͕ಇ͘Α͏ʹ͍ͯ͠Δɻ3 GeV ͷઢ‫ܗ‬Ճ଎‫ث‬͸͜ͷΑ͏ʹද͢͜ͱ͕Ͱ͖ΔɻΕΛͭͳ͍͛ͯͬͨ΋ͷͰɺਤ 87 ʹࣔ͢ɻ͜͜Ͱɺ

அ೤‫ݮ‬ਰ :

1
0

γf
γi
γ  ln γiγiγf஫ҙ͍ͨ͜͠ͱ͸ɺͻͱͭͷՃ଎ۭಎͰՃ‫ݮ‬଎Λߦ͏

.

(12.17)ͨΊɺΤωϧΪʔͷҟͳΔ 2 ͭͷϏʔϜ͕௨ա͢Δ͜
ͱͰ͋Δɻྫ͑͹ɺ3 GeV ·ͰՃ଎ͨ͠௚‫ޙ‬ͷ࢛‫ిۃ‬͜͜Ͱ ͸Ճ଎ޯ഑Λද͠ɺՃ଎ͷͱ͖ʹਖ਼Ͱ͋Δɻ࣓ੴʹ͸ 10 MeV ·Ͱ‫ݮ‬଎͞ΕͨϏʔϜ΋௨ա͢Δɻ

υϦϑτεϖʔεͷసૹߦྻͱྨࣅ͓ͯ͠ΓɺR12 ੒‫ݪ‬ଇతʹ࢛‫࣓ిۃ‬ੴ͸ΤωϧΪʔͷ௿͍ํʹ߹Θͤͯ෼Λ࣮ޮతͳ௕͞ͱΈͳ͢͜ͱ͕Ͱ͖Δɻ600 m ͷௐ੔͢ΔͨΊɺࣜ (12.15) ʹࣔ͢Α͏ʹɺΤωϧΪʔՃ଎ۭಎͰ 10 MeV ͔Β 3 GeV ·ͰՃ଎͢Δͱ͖͸ͷߴ͍ϏʔϜʹରͯ͠ऩଋྗ͕ಇ͔ͳ͍ɻͦͷΑ͏ͳ͓Αͦ 10 m Ͱ͋Δͷʹର͠ɺ‫ݮ‬଎͢Δͱ͖͸͓Αͦ৚݅ͷதͰɺΦϓςΟΫεΛ࠷దԽͨ݁͠Ռ͕ਤ 87

3 km ͱͳΔɻͦͷ΄͔ʹՃ଎ۭಎʹ͸ RF ʹΑΔऩଋྗ͕ಇ͘ɻ͜Ε͸ɺप‫ظ‬తͳఆࡏ೾͕ RF ۭಎͷத

Ͱ͋ΔɻՃ଎͢ΔϏʔϜͷΦϓςΟΫεͷΈΛࣔͯ͠ʹ্ཱ͕͍ͪͬͯΔ࣌ʹɺMaxwell ํఔࣜʹैͬͯి͍ͯΔɻ·ͣɺ300 m ෇ۙ·Ͱͷ্ྲྀଆ͸ɺՃ଎ϏʔࢠΛԣํ޲ʹΩοΫ͢ΔΑ͏ͳి࣓৔͕ൃੜ͢Δ΋ͷϜʹ߹Θͤͯઃ‫͠ܭ‬ɺਤ 86 ͷΦϓςΟΫεΛͦͷ·Ͱ͋Δɻ࠷΋Ճ଎Ͱ͖ΔҐ૬ʢΦϯΫϨετՃ଎ʣͷ·ͭͳ͍͛ͯ͘ɻԼྲྀଆ͸ಉ༷ʹ‫ݮ‬଎ϏʔϜʹ߹Θͤసૹߦྻ͸ɺ

ΔͨΊɺ1.5 GeV ͔Β 3 GeV ͷՃ଎ϏʔϜͷϕʔλRF focus :(12.18)͍Δ͕ɺ‫ݮ‬଎ϏʔϜ΋શ͘ରশͳΦϓςΟΫεʹͳͬؔ਺͕େ͖͘ͳͬͯ͠·͏ɻͦͷϕʔλؔ਺Λগ͠Ͱ㸲㸫 

ਤ 87: ઢ‫ܗ‬Ճ଎‫ث‬ͷ 3 GeV ʹՃ଎͢ΔϏʔϜͷϕʔλؔ਺ɻ‫ݮ‬଎͢ΔϏʔϜ͸͜ΕͷରশͳΦϓςΟΫεʹͳΔɻ΋খ͘͢͞ΔͨΊɺ֤ triplet ͷ K ஋Λඍௐ੔͢Δɻ͜ͷख๏Λ variable triplet ͱ‫ͼݺ‬ɺਤ 87 ͷ 500 m
෇ۙͰ͏ͶΔΑ͏ͳؔ਺ʹͳΔɻ্ྲྀͷΦϓςΟΫε
Ͱਤ 86 ͷप‫ظ‬తͳΦϓςΟΫε͔Β่Ε͍ͯΔͷ͸ͦͷͨΊͰ͋Δɻ͜ͷΦϓςΟΫεͱपճϧʔϓͷҐ૬ͷਐΈ͔ΒɺHOM BBU ͷిྲྀᮢ஋Λ‫ٻ‬ΊΔ͜ͱ͕Ͱ͖Δɻ൚༻ੑͷ͋Δ์ࣹޫ‫ݯ‬ͷͨΊʹฏ‫ ྲྀిۉ‬100 mA Λ໨ࢦ͓ͯ͠ΓɺHOM BBU ͷ໰୊͸ਂࠁͰ͋Δɻ௒
఻ಋۭಎ͸ ILC Ͱ։ൃ͞Ε͍ͯΔ 1.3 GHz ͷ 9 ηϧۭಎͷ‫ڀݚ‬։ൃ͕ਐΜͰ͍Δ͕ɺిྲྀᮢ஋͕਺ 10mA ʹ੍‫͞ݶ‬Εͯ͠·͏‫ڪ‬Ε͕͋Δɻͦ͜ͰɺERL
Ͱ͸ɺILC ۭಎΛϕʔεʹ‫ܗ‬ঢ়Λมۭ͑ͨಎΛ։ൃ͠
͍ͯΔɻ‫ࡏݱ‬ɺcERL Ͱ͸ͦͷࢼ‫ݧ‬Ճ଎ۭಎΛΠϯετʔϧͯ͠ϏʔϜӡసΛߦ͓ͬͯΓɺ҆ఆʹՔಇͯ͠
͍Δɻͨͩ͠ɺcERL Ͱ͸਺୆ͷۭಎ͔͠Πϯετʔਤ 88: ओՃ଎ۭಎ෇ۙͷσβΠϯΦϓςΟΫεɻϧͰ͖ͣɺHOM BBU ͸ൃੜ͠ͳ͍ͨΊɺϏʔϜෆ҆ఆੑ͸‫͜ى‬Βͳ͍ͱࢥΘΕΔɻۭಎઃ‫ܭ‬΍ӡసঢ়‫گ‬Δɻ͔͠͠ɺपճϏʔϜ΋௨ա͢Δ͜ͱΛߟྀ͠ɺऩ

ʹ͍ͭͯ͸ɺࡕҪࢯͷࢿྉΛࢀߟʹ͞Ε͍ͨɻଋྗʹ্‫ݶ‬ΛܾΊ͍ͯΔɻҰํͰɺΞʔΫͷΦϓςΟ
Ϋε͸෼ࢄؔ਺ͷ੍໿͕େ͖͍ͨΊɺೖࣹϏʔϜͱ

cERL ͷઢ‫ܗ‬Ճ଎‫ث‬ͷΦϓςΟΫε
σβΠϯ͸ಠཱʹઃ‫ܭ‬Λߦ͏ɻͦͷ‫ޙ‬ɺΞʔΫग़‫͔ޱ‬ΒμϯϓcERL ɹͷՃ଎ۭಎपลͷϨΠΞ΢τΛਤ 89 ʹࣔ࠷దԽ͢Δɻͭ·ΓɺQMAD01-04 ͷ 4 ୆͸ 20 MeV͢ɻ͜͜Ͱ͸ɺ3 MeV Ͱೖࣹ෦͔Β߹ྲྀͨ͠ϏʔϜϏʔϜ΋௨ա͢Δ͕ɺ3 MeV ͷμϯϓʹ޲͔͏ϏʔϜ

12.3

·Ͱͷ‫ݮ‬଎ϏʔϜͷΦϓςΟΫεΛɺQMAD01-04 ͱ

QMAM05-08 ͷ 8 ୆Ͱϕʔλؔ਺͕খ͘͞ͳΔΑ͏ʹ͕ओՃ଎ۭಎͰ 20 MeV ·ͰՃ଎͞ΕΔɻ·ͨɺपʹ࠷దԽ͢Δɻ࠷‫ʹޙ‬ɺ20 MeV ʹՃ଎ͨ͠ϏʔϜ͸ճͯ͠໭͖ͬͯͨϏʔϜ͸ೖࣹϏʔϜͱ߹ྲྀͨ͠‫ ޙ‬3

QMAD ΑΓ‫ޙ‬ͷ QMAC01-04 ͱ QMAM01-04 ͷ 8

MeV ·Ͱ‫ݮ‬଎ͯ͠μϯϓʹ༠ಋ͞ΕΔɻೖࣹ߹ྲྀγέΠϯ΍μϯϓγέΠϯͰೖࣹɾपճϏʔϜͷ‫ي‬ಓΛ୆ͰΞʔΫ෦ͱͷϚονϯάΛߦ͏ɻओՃ଎ۭಎग़‫ޱ‬߹ྲྀɾ෼‫͖Ͱذ‬ΔΑ͏ʹɺӡಈྔൺ͸ 1:7 ʹઃఆͯͨ͠Ίɺϕʔλؔ਺͕େ͖͘ͳΓ͕ͪͰ͋ΔɻҎ্͸ɺ͍Δɻઃ‫ܭ‬ΦϓςΟΫεΛਤ 88 ʹࣔ͢ɻ࠷ॳʹɺೖࣹΦϓςΟΫεΛઃ‫͢ܭ‬ΔखॱͰ͋Δ͕ɺ࣮ࡍͷϏʔϜ߹ྲྀ෦௚‫ޙ‬ͷ QMAG01-08 ͷ࢛‫࣓ిۃ‬ੴ͸ۭؒిՙௐ੔΋ࣅͨखॱͰߦ͍ͬͯΔɻॳ‫ʹظ‬ϏʔϜΛपճ͞ޮՌͷେ͖͍ 3 MeV ͷೖࣹϏʔϜʹରͯ͠࠷దԽ͢

ͤͨ࣌ʹɺμϯϓϥΠϯͷϏʔϜͰ QMAD01-04 ͷ͔Β਺ m ͷؒ͸‫͍ڧ‬ऩଋྗΛ༩͑Δ͜ͱ͕Ͱ͖ͳ͍

㸲㸫 

ਤ 89: cERL ͷՃ଎ۭಎ෇ۙͷϨΠΞ΢τɻऩଋྗΛܾΊΔɻͦͷ‫ޙ‬͸ɺQMAD01-04 Λม͑ͣʹɺपճϧʔϓͷϚονϯάΛ࣮ࢪ͢Δɻͦͷ‫ޙ‬ͷϦλʔϯΞʔΫΛ໭ͬͨϏʔϜ͸ QMAM05-08 ͷ 4 ୆ͷΈΛ࢖ͬͯௐ੔͢Δɻ

㸲㸫 

13

पճϧʔϓͷઃ‫ܭ‬ɾௐ੔

13.1

γϯΫϩτϩϯ์ࣹʹΑΔ์ࣹྭ‫ى‬[59, 61]ిࢠͷ‫ي‬ಓ͕‫͛ۂ‬ΒΕΔͱ͖ɺ์ࣹޫΛ์ग़ͯ͠ΤωϧΪʔΛࣦ͏ɻͦͷͱ͖Τϛολϯε͕‫ݮ‬গ͢Δ‫ݱ‬৅ʢ์ࣹ‫ݮ‬ਰʣͱ૿Ճ͢Δ‫ݱ‬৅ʢ์ࣹྭ‫ى‬ʣ͕‫ݱ‬Εɺ஝ੵϦϯάͷΤϛολϯε͸͜ΕΒͷฏߧঢ়ଶͰܾ·ਤ 90: CSR wake ͷ໛ࣜਤɻΔɻҰํͰɺERL ͷిࢠϏʔϜ͸̍౓पճ͢Δ͚ͩͳͷͰɺΤϛολϯεͷมԽ͸஝ੵϦϯάʹൺ΂ͯඇৗʹখ͍͞ɻैͬͯɺిࢠॐͰ௿ΤϛολϯεϏʔϜΛେ͖͘૿Ճͤ͞Δ͜ͱͳ͘पճͤ͞Δ͜ͱ͕ՄೳͰ
͋Δɻ͔͠͠ɺ1 प͔͠͠ͳ͍ͱ͸͍͑ɺͦͷؒʹΤϛολϯεͷྼԽ͸ੜ͡Δɻ‫֨ن‬ԽΤϛολϯε 0.1

mm·mrad Λ໨ࢦ͢ʹ͸ɺͦͷΘ͔ͣͳΤϛολϯε
૿Ճ͕໰୊ͱͳΓ͏Δɻ์ࣹྭ‫ʹى‬ΑΔ̍पͷ‫֨ن‬ԽΤϛολϯε૿Ճ͸
3

6

E
,
5GeV
(13.1)
ͱͳΔɻ͜͜ͰɺE ͸ిࢠͷΤωϧΪʔɺρ ͸ภ޲ిΔεn ≈ 0.54μm · rad

16
Nb

28m
ρ

࣓ੴͷ‫཰ۂ‬൒‫ܘ‬ɺNb ͸ภ޲ి࣓ੴͷ਺Ͱ͋Δɻిࢠ
ͷΤωϧΪʔ E ͷ 6 ৐ʹൺྫ͢ΔͨΊɺΤωϧΪʔ͕ߴ͍΄Ͳ‫཰ۂ‬൒‫ܘ‬Λେ͖͘͢Δඞཁ͕͋Δɻ3 GeV

ਤ 91: Gaussian ෼෍ͷઢిՙີ౓ͱࣜ (13.3) ͷ I0

ͷ ERL Ͱ͸ ρ = 10 m ͱઃఆ͍ͯ͠Δɻपճ෦ͷͷؔ܎ɻภ޲ి࣓ੴ͸͓Αͦ 120 ୆Ͱ͋ΔͷͰɺΔεn Λ‫ੵݟ‬΋Δͱ 0.0002 μm·rad ͱͳΓແࢹͰ͖Δ΄Ͳখ͍͞ɻ

cERL ͸͞ΒʹిࢠΤωϧΪʔ͕খ͍ͨ͞Ίɺ์ࣹྭ‫ى‬͸໰୊ʹͳΒͳ͍ɻෑ஍ʹ‫ݶ‬Γ͕͋ͬͨͷͰɺলεϖʔεͷͨΊʹ ρ = 1 m ͱখ͍ͯ͘͞͠Δɻ์ࣹྭ‫ʹى‬ΑͬͯΤωϧΪʔͷ޿͕Γ΋େ͖͘ͳΓɺిࢠΤωϧΪʔͷ 5/2 ৐ʹൺྫ͢Δɻ

ΔσE≈ 3.2 · 10−5
E

28m
ρ

E
5GeV

ͯɺόϯν௕͕௕͘ͳΔͱɺΤωϧΪʔ޿͕Γ͕େ͖͘ͳͬͯ͠·͏ͨΊɺ௨ৗͷӡసͰ͸όϯν௕Λ͓Α
ͦ 2 ps ͱ͍ͯ͠Δɻ͜Ε͸ɺ஝ੵϦϯάͷ৔߹ͱൺ΂͓ͯΑͦ 1 ܻ୹͍ɻͦ͜Ͱ୹όϯνͳΒͰ͸ͷϏʔϜμΠφϛΫε͕ੜ͡Δɻ
લষͰ৮Εͨ์ࣹޫ͸ҰͭͷిࢠʹΑΔి࣓৔ΛͦͷిࢠͷΈ͕‫͡ײ‬Δ͜ͱʹΑͬͯൃੜ͢ΔͨΊɺͦ

5/2.

(13.2)

3 GeV ERL ͷ৔߹ɺ1 पͰ 3×10−5 ఔ౓Ͱ͋ΔɻҰํͰɺERL ͷόϯν௕͸ 2 ps ఔ౓Ͱ͋Γɺ1.3 GHzͷ‫౓ڧ‬͸ిࢠ਺ʹൺྫ͢Δɻ͔͠͠ɺόϯν௕ΑΓ΋௕͍೾௕ͷ์ࣹޫͰ͸ɺόϯν಺ͷଞͷిࢠʹΑΔ
ి࣓৔Λ‫͡ײ‬Δɻ͜ΕΛίώʔϨϯτ์ࣹʢCoherent

ͷ RF ͷΧʔϒʹΑͬͯ 2×10−4 ͷΤωϧΪʔ޿͕

Synchrotron Radiation, CSRʣͱ‫Ϳݺ‬ɻͦͷ์ࣹޫͷ‫౓ڧ‬͸όϯν಺ͷిࢠ਺ͷ 2 ৐ʹൺྫ͠ɺ௨ৗͷిΓ͕ൃੜ͢Δ͕ɺ͜ͷӨ‫ʹڹ‬ൺ΂ͯे෼খ͍͞ͱ‫ݴ‬ࢠόϯν͸ 108 ∼ 1010 ‫ݸ‬ͷిࢠΛ‫ؚ‬ΉͨΊɺͦͷ‫ڧ‬͑Δɻ౓͸ܻҧ͍ʹେ͖͘ͳΔɻ͜ͷ CSR ͕ిࢠόϯνʹ
༩͑ΔӨ‫ڹ‬Λ CSR wake ͱ‫Ϳݺ‬ɻ໛ࣜਤΛਤ 90 ʹࣔ

13.2

ίώʔϨϯτγϯΫϩτϩϯ์ࣹʹ
ΑΔߤ੻৔ (CSR wake) [48, 62]−4

ERL Ͱ͸ 10 Φʔμʔͷখ͍͞ΤωϧΪʔ޿͕ΓΛ໨ඪͱ͍ͯ͠Δɻ1.3 GHz ͷ RF ͷΧʔϒʹΑͬ͢ɻ೾௕ͷ௕͍ CSR ͸์ࣹ֯౓͕େ͖͍ɻ͢Δͱɺόϯν‫͔ํޙ‬Βͷి࣓೾͕‫ۂ‬ઢΛඳ͘ిࢠόϯνͷલํʹ௥͍͖ͭɺ૬‫༻࡞ޓ‬Λ͢Δ͜ͱʹΑͬͯੜ͡Δɻ
ҰํͰɺ஝ੵϦϯάͷόϯν௕͸νϟϯόʔͷ cutoﬀ㸲㸫 प೾਺ͷ೾௕ΑΓ௕͍ͨΊɺCSR ͸ൃੜ͠ͳ͍ͱ‫ݴ‬ΘΕ͍ͯΔɻ

CSR wake ʹΑΔΤωϧΪʔมԽ͸ɺHOM BBU
ͷͱ͖ͱಉ༷ʹ wake function, W0 (z) Ͱॻ͘͜ͱ͕
Ͱ͖Δɻ

dE= N re m0 c2
cdt

 z
−∞

W0 (z − z  )λ(z  )dz  .

(13.3)

ਤ 92: ΤωϧΪʔͷͣΕʹΑΔਫฏํ޲ͷҐஔมԽͱΤϛολϯεͷ૿Ճɻ

͜͜ͰɺN ɺm0 ɺre ɺλ(z) ͸ͦΕͧΕిࢠͷ਺ɺ࣭ྔɺ‫ݹ‬య൒‫͓ܘ‬Αͼઢిՙີ౓Ͱ͋ΔɻW0 (z) ͸ z 1/3ʹ൓ൺྫ͢Δؔ਺Ͱ͋Γɺ࣍ͷΑ͏ʹॻ͖Լ͢͜ͱ͕Ͱ͖Δɻ

W0 (z) = −1 ∂2.
(3ρ2 )1/3 z 1/3 ∂z(13.4)

͜͜Ͱɺిࢠόϯνͷ rms αΠζ͕ σz ͷ Gaussianਤ 93: Τϛολϯε૿Ճͷ૬ࡴɻ෼෍Ͱ͋Γɺઢిՙີ౓ λ(z) ͕√λ(z) = (1/ 2πσz ) exp(−z 2 /2σz2 ),(13.5)

Ͱ͋Δͱ͖Λߟ͑Δɻ͢Δͱɺࣜ (13.3) ͸࣍ͷΑ͏ʹͳΔɻ

CSR wake ͕ःณ͞ΕΔޮՌ΍୹͍ภ޲ి࣓ੴͰ͸transient eﬀect ͕͋ΔͨΊɺࣜ (13.8) ΍ (13.9) ͸ෳࡶͳ΋ͷͱͳΔɻ·ͨɺCSR wake ʹΑͬͯਫฏํ޲
ͷΩοΫ΋ൃੜ͢Δ͕ɺΤωϧΪʔมԽͷӨ‫ʹڹ‬ൺ΂zdE
2N re m0 c2
,
I0
= −√
cdt
σz
2π(3ρ2 σz4 )1/3
 xdx∂ −x2 /2I0 (x) =
e
.

1/3
∂x
−∞ (x − x )(13.6)

ͯແࢹͰ͖Δ΄Ͳখ͍͞ɻ

(13.7)

13.3

I0 Λਤ 91 ʹࣔ͢ɻόϯνઌ಄ͷҰ෦ͰΤωϧΪʔ

CSR wake ʹΑΔΤϛολϯε૿Ճͱͦͷ཈੍ [63–65]

্͕͕ΔҰํɺத৺͔Β‫͚͔ͯʹํޙ‬ΤωϧΪʔ͸‫ݮ‬Τϛολϯεͱ͸ x, x ͷฏ໘্ʹ޿͕ΔϏʔϜͷ

গ͢ΔɻόϯνશମʹΘͨͬͯੵ෼͠ɺిࢠͷΤωϧ໘ੵʹ૬౰͢ΔɻCSR wake ʹΑͬͯΤωϧΪʔ͕ΪʔมԽ ΔE ͱΤωϧΪʔ޿͕Γ ΔσE ͷมԽΛ‫ٻ‬Θ͔ͣʹ δp ͚ͩͣΕͨ৔߹ʹ͍ͭͯߟ͑Δɻ‫ي‬ಓ͕ΊΔɻ௕͞ LB ͷภ޲ి࣓ੴΛ௨աͨ࣌͠ʹ͸ɺͦΕΤωϧΪʔʹΑͬͯมԽ͠ͳ͍௚ઢ෦Ͱ͸ɺਫฏɾਨ

ͧΕҎԼͷΑ͏ʹॻ͖ද͢͜ͱ͕Ͱ͖Δɻ

௚ํ޲ͷϏʔϜʹ΄ͱΜͲແࢹͰ͖Δɻ͔͠͠ɺภ޲

ΔE = −

2

4/3 1/6

N r e m0 c L B 2
4/3

ρ2/3 σzΔσE ≈ 0.22

32

[Γ(2/3)],π

N r e m0 c 2 L B
4/3

ρ2/3 σz(13.8)

ి࣓ੴ಺ͰΤωϧΪʔͷมԽ͕͋ͬͨ৔߹ɺਤ 92 ʹࣔ͢Α͏ʹਫฏํ޲ͷҐஔ͕มΘͬͯ͠·͏ɻΤωϧΪʔͷͣΕ͸ɺ௕खํ޲ͷҐஔʹΑͬͯҟͳΔͨΊɺ

.

(13.9)

ਫฏํ޲ͷҐஔͷ޿͕Γ͕ൃੜͯ͠͠·͏ɻ͜Ε͕Τϛολϯε૿ՃͷϝΧχζϜͰ͋ΔɻΤωϧΪʔͷมόϯνશମͷ ΔE ͸ෛͰ͋ΓɺΤωϧΪʔ͕Լ͕ͬ

Խ͸ආ͚ΒΕͳ͍͕ɺΤϛολϯεͷ૿Ճ͸ΦϓςΟ

͍ͯΔ͜ͱ͕Θ͔Δɻ֤ిࢠͷ ΔE ͸ N ʹൺྫ͢ΫεͰখ͘͞཈͑Δ͜ͱ͕Ͱ͖Δɻ

ΔͨΊɺόϯνશମͱͯ͠͸ N 2 ʹൺྫͨ͠ΤωϧޮՌతͳํ๏ͱͯ͠ɺϕʔλؔ਺͕प‫ظ‬తͳΦϓΪʔ͕์ग़͞Ε͍ͯΔ͜ͱʹͳΔɻΔE ΍ ΔσE ͸ςΟΫεͰҐ૬ͷਐΈΛ 180 ౓ʹ͢Δํ๏͕͋Δɻ͜

ిࢠͷΤωϧΪʔʹґଘ͠ͳ͍ͷͰɺ3 GeV ERL ʹΕ͸ಉ͡แབྷઢͷϏʔϜϓϩϑΝΠϧ͕‫܁‬Γฦ͢Ұํൺ΂ͯΤωϧΪʔͷ௿͍ cERL ͷ΄͏͕ CSR wakeͰɺ‫ݸ‬ʑͷిࢠͷҐ૬ۭؒͷҐஔ͸‫఺ͱ఺ݪ‬ରশͷ৔ͷӨ‫͕ڹ‬େ͖͍ɻϧʔϓ 1 प෼ͷޮՌ͸ LB = 2πρॴʹҠಈ͍ͯ͠Δ͜ͱΛࣔ͢ɻ͢Δͱɺਤ 93 ʹࣔ͢ͱ͢Ε͹Α͘ɺࣜ (13.8) ΍ (13.9) ʹ୅ೖ͢Δͱ ρ1/3Α͏ʹɺδp ʹΑͬͯҐ૬ۭؒΛҠಈͨ͠ిࢠ͸ɺ࣍ʹൺྫ͢ΔɻcERL Ͱ͸ ρ ͕ 1 m ͱখ͘͞ͳΔ͕ɺͷप‫ظ‬ͷΦϓςΟΫεͰ‫ݩ‬ͷҐஔʹ໭Δɻͨͩ͠ɺδpͦͷޮՌ͸গͳ͍ɻͦͷ΄͔ʹɺνϟϯόʔʹΑͬͯ㸲㸫 ͚Ε͹ͳΒͣɺ͜ͷ৚݅ΛΞΠιΫϩφεͱ‫Ϳݺ‬ɻ͜ΕΒͷ৚݅ΛΈͨͨ͢ΊʹɺΞʔΫ෦͸ 3 ͭͷภ޲ి࣓ੴͰߏ੒͞ΕΔ Triple Bend Achromat, TBA Λ࠾༻ͨ͠ɻਤ 95 ʹࣔͨ͠΋ͷ͸ภ޲ి࣓ੴ͕ 4 ͕ͭͩɺΦϓςΟΫεͷ‫͔఺؍‬Βதԝͷ 2 ͭΛͻͱͭͷి࣓ੴͱಉ౳ͱΈͳΔ͜ͱ͕Ͱ͖Δɻ
ภ޲ి࣓ੴͰ͸ δp ʹΑͬͯ‫ي‬ಓ͕ҟͳΔ͜ͱΛ
ਤ 92 ʹ͕ࣔͨ͠ɺ௕खํ޲ z ͷҐஔ΋ͣΕΔɻసૹߦྻ R56 ͷ੒෼Λ࢖ͬͯਤ 94: Τϛολϯε૿Ճͷ࠷খԽɻΔz = R56

δp+ R566p2

δpp+ ···,(13.10)ͱද͢͜ͱ͕Ͱ͖ΔɻR56 ͷූ߸ͷఆٛ͸ਓʹΑͬͯҧ͏ͷͰɺ஫ҙ͍ͨ͠ɻR56 ͸෼ࢄؔ਺ η Λ༻͍ͯɺ

η
R56 =
ds,
(13.11)
ρਤ 95: ΞʔΫ෦ͷϨΠΞ΢τɻͱද͢͜ͱ͕Ͱ͖Δɻ௚ઢ෦Ͱ͸ ρ = ∞ Ͱ͋ΔͨΊɺ͕มԽ͠ͳ͍ɺͭ·Γόϯν௕͕มԽ͠ͳ͍ɺͱ͍͏

ੵ෼͸ภ޲ి࣓ੴͷΈͱͳΔɻΞΠιΫϩφεͱ͸ɺ৚݅෇͖Ͱ͋ΔɻGeV Ϋϥεͷ ERL Ͱ͸पճϧʔϓ͜ͷ R56 ͕θϩͰ͋Γɺδp ʹΑͬͯ z ํ޲ͷҐஔ͕͕͍͔ͭ͘ͷ‫܁‬Γฦ͠ͷΦϓςΟΫεΛ‫ؚ‬ΉͨΊɺ͜ෆมͰ͋Δ͜ͱΛࢦ͢ɻภ޲ి࣓ੴͷ‫֯͛ۂ‬͸ϨΠΞͷΑ͏ͳํ๏ͰΤϛολϯε૿ՃΛ཈͑Δ͜ͱ͕Մೳ

΢τͨ࣌͠఺ͰܾΊΒΕ͓ͯΓɺΞΫϩϚʔτ΍ R56

Ͱ͋Δɻ͔͠͠ɺ3 GeV ERL Ͱ͸ 77 pC/bunch Ͱͷௐ੔͸ TBA ͷதͷ 3 ͭͷ࢛‫࣓ిۃ‬ੴʢtripletʣΛ

2 ps ͷόϯν௕Λ૝ఆ͓ͯ͠ΓɺτϥοΩϯάͷ݁ՌʹΑΔͱ CSR wake ʹΑΔӨ‫ڹ‬͸େ͖͘ͳ͍ɻҰ

༻͍Δɻਤ 95 ͷภ޲ి࣓ੴ͸͢΂ͯ‫ ֯͛ۂ‬θB ɺ‫཰ۂ‬

ํͰɺcERL Ͱ͸पճϧʔϓ͕খ͍ͨ͞Ίɺप‫ظ‬తͳࢹ͢ΔɻରশͳΦϓςΟΫεΛߟ͑ͯɺTBA த৺Ͱ൒‫ ܘ‬ρB ͱͯ͠ɺ؆୯ͷͨΊʹΤοδͳͲͷޮՌΛແΦϓςΟΫεΛ࡞Δ͜ͱ͕Ͱ͖ͣɺ͜ͷख๏Λ࢖͏͜η  = 0 Ͱ͋Δͱ͢Δɻ͜ͷͱ͖ͷ TBA த৺ͷ෼ࢄؔͱ͕Ͱ͖ͳ͍ʢϦλʔϯΞʔΫग़‫Ͱޱ‬Ωϟϯηϧ͢Δ਺Λ ηc ͱ͢ΔͱɺR56 ͱҎԼͷΑ͏ͳؔ܎ʹͳΔɻ

͜ͱ͸Ͱ͖Δʣɻηc =͔͠͠ɺΤϛολϯεΛ཈੍͢Δํ๏͸ଞʹ΋͋

ρB(R56 − 2θB + 2 sin θB ).
sin θB(13.12)ΔɻͦΕ͸ δp ʹΑͬͯҐ૬্ۭؒΛҠಈ͢Δํ޲

߆ଋ৚͕݅ 2 ͭʹରͯ͠ɺม਺͕ 3 ͭ͋ΔͨΊɺͻͱ

φCSR ͱҐ૬෼෍ͷ௕खํ޲ φphase Λ͋ΘͤΔํ๏
Ͱɺਤ 94 ʹ໛ࣜਤΛࡌͤΔɻશͯͷภ޲ి࣓ੴͰҐͭͷࣗ༝౓͕࢒͍ͬͯΔɻTriplet ͷ‫ۃ‬ੑͷ૊Έ߹Θͤ͸ਫฏํ޲ʹ Focus-Defocus-Focus ͕෯޿͍ R56

૬Λ͋ΘͤΔ͜ͱ͕೉͍͕͠ɺόϯνѹॖͳͲͰ࠷΋ͷௐ੔ʹ޲͘ҰํɺDefocus-Focus-Defocus Ͱ͸࢛‫ۃ‬όϯν͕୹͘ CSR wake ͕େ͖͘ͳΔͱ͜ΖͰ߹Θి࣓ੴͷ K ஋Λখ͘͞཈͑Δ͜ͱ͕Ͱ͖ɺϕʔλؔ

ͤΔͱޮՌతͰ͋Δɻ͜ͷํ๏ʹΑͬͯɺcERL ͷΤ਺ͷϚονϯά΋༰қͰ͋Δɻ

ϛολϯε͕ 1/10 ʹͳΔ͜ͱ΋͋Δɻ

ૠೖޫ‫ݯ‬ͷεϖʔεͰɺͰ͖Δ‫ݶ‬Γϕʔλؔ਺Λখ͘͢͞ΔͨΊʹϚονϯά͕ඞཁͰ͋ΔɻυϦϑτε

13.4

TBA ͷΞʔΫ෦ͱ 3 GeV ERL ͷपճϧʔϓ [1, 53, 59]

3 Gev ERL Ͱ͸ૠೖޫ‫ݯ‬ͷͨΊͷ 6 m ͷεϖʔεΛ 22 ϲॴɺ30 m ͸ 6 ϲॴ༻ҙ͢Δ༧ఆͰ͋Δɻૠೖޫ‫ݯ‬ͷ͋Δ৔ॴ͸෼ࢄؔ਺ ͕θϩͰ͋Δඞཁ͕͋Γɺ͜ͷΑ͏ͳ৚݅ΛΞΫϩϚʔτͱ‫Ϳݺ‬ɻ·ͨɺΤωϧΪʔճऩͷͨΊʹόϯν௕Λม͑ͣʹ༌ૹ͠ͳϖʔεͷϕʔλؔ਺͸ 2 ࣍ؔ਺ͱͳΓɺ‫ۃ‬খ஋ͷϕʔλؔ਺Λ β0 ͱ͢Δͱɺs ͚ͩ཭Εͨ৔ॴͰ͸ɺ

β(s) = β0 +

s2
,
β0

(13.13)

ͱͳΔɻΑͬͯɺ6 m υϦϑτεϖʔεͷத৺Ͱ β0= 3 m ͱͳΔΑ͏ʹઃ‫͍ͯ͠ܭ‬Δɻਤ 96 ʹ 3 GeV
ERL ͷΞʔΫͷҰ෦Λࡌ͍ͤͯΔɻ㸲㸫 

ਤ 96: 3-GeV ERL ͷΞʔΫ෦ͷϥςΟεͱϕʔλؔ਺ͷҰ෦ɻ

13.5

cERL ΞʔΫ෦ͱΦϓςΟΫεͷௐ੔ [66]cERL Ͱ͸ɺภ޲ి࣓ੴͷ‫ ͕֯͛ۂ‬45 ౓Ͱ͋Γɺͻͱͭͷ TBA Ͱ 180 ౓΋‫͍ͯ͛ۂ‬ΔɻϨΠΞ΢τΛ
ਤ 97 ʹࣔ͢ɻෑ஍ʹ‫ݶ‬Γ͕͋ΔͨΊɺ‫཰ۂ‬൒‫ܘ‬͸ ρ=1 m ͱখ͍͞ʢෑ஍ʹ༨༟͕͋ΔΑ͏ʹ‫͑ݟ‬Δ͕ɺਤ 97: cERL ͷΞʔΫ෦෇ۙͷϨΠΞ΢τɻओՃ଎ۭಎͷ௥Ճ΍ 2 ϧʔϓͷ‫ݐ‬ઃΛ૝ఆ͍ͯ͠ΔʣɻͦͷͨΊϕʔλؔ਺ͷϚονϯά͕೉͘͠ͳΔ͜ͱ͔Βɺtriplet ͸ DFD ͱͨ͠ɻ෼ࢄؔ਺͸ɺΞʔΫ಺ͷ࢛‫࣓ిۃ‬ੴͰ͔͠ௐ੔Ͱ͖ͳ͍ɻͦ͜Ͱɺ࣮ࡍͷΦϖϨʔγϣϯͰ͸ɺΦϓςΟΫεͷϚονϯάͷલʹ෼ࢄؔ਺Λௐ੔͍ͯ͠Δɻ෼ࢄؔ਺ͷௐ੔͸ɺओՃ଎ۭ
ಎͷৼ෯Λม͑ͯΤωϧΪʔΛ 1 % ఔ౓ม͑ɺͦͷͱ͖ͷ‫ي‬ಓมԽΛ‫͕ͳݟ‬Βߦ͍ͬͯΔɻ‫ي‬ಓ͸ BPM
ͰϦΞϧλΠϜʹߦ͏͜ͱ͕ՄೳͰ͋Δɻਤ 98 ʹଌఆ݁ՌͱσβΠϯ஋Λൺֱ͍ͯ͠Δ͕ɺΑ͘Ұகͯ͠
͍Δ͜ͱ͕Θ͔ΔɻcERL ͷ৔߹ɺΞʔΫ෦தԝͷ෼ࢄؔ਺ ηc ͕ 0.24 m Ͱ͋Ε͹ΞΠιΫϩφεͰ͋Δɻਤ 98: cERL ෼ࢄؔ਺ͷଌఆ஋ͱσβΠϯ஋ɻਨ௚ํ޲ʹ෼ࢄؔ਺͕࢒͍ͬͯΔͨΊɺΞʔΫ෦ʹεΩϡʔ࢛‫࣓ిۃ‬ੴΛΠϯετʔϧ͢ΔͳͲͷରࡦ͕ඞ͘อͪͳ͕Β༌ૹ͢Δ͔͕ॏཁͱͳΔɻΞʔΫͷΦϓཁͰ͋ΔɻςΟΫε͸ରশੑΛอͭͨΊʹɺத৺Ͱ αx = αy = 0෼ࢄؔ਺ͷௐ੔ͷ‫ʹޙ‬ɺΞʔΫલ‫ޙ‬ͷ࢛‫࣓ిڧ‬ੴͰ

ͱͳΔΑ͏ʹઃ‫͍ͯ͠ܭ‬ΔɻϏʔϜαΠζ͕େ͖͘ͳ

ΦϓςΟΫεͷϚονϯάΛߦ͏ɻϕʔλؔ਺ͷσβΓ͗͢ΔͱɺCSR wake ʹΑΔΤϛολϯεͷ૿Ճ͕Πϯ஋Λਤ 99 ʹࡌͤΔɻΞʔΫ෦͸ΞΫϩϚʔτɺ‫ݦ‬ஶʹͳΔͨΊɺΞʔΫத৺Ͱ βx = βy = 5 m ໨҆ΞΠιΫϩφεͱ͍͏৚݅Λຬͨͨ͢Ίʹɺ݁Ռͱ

ͱ͍ͯ͠Δɻ࣮ࡍ͸ Q scan ๏ͱ‫ݺ‬͹ΕΔख๏Λ‫ʹج‬ͯ͠ਨ௚ํ޲ʹେ͖͘ൃࢄ͢ΔΦϓςΟΫεʹͳͬͯϚονϯάΛߦ͍ͬͯΔɻ͍ΔɻϚονϯάͰ͸͍͔ʹਨ௚ํ޲ͷϏʔϜΛখ͞㸲㸫 

ਤ 99: cERL ΞʔΫ෦෇ۙͷϕʔλؔ਺ɻ

㸲㸫 

14

ϏʔϜͷ‫ݮ‬଎ͱμϯϓ14.1

μϯϓϥΠϯͷઃ‫[ ܭ‬61]μϯϓΤωϧΪʔ·Ͱ‫ݮ‬଎ͨ͠ϏʔϜ͸ภ޲ి࣓ੴ
ͰμϯϓϥΠϯ΁ͱಋ͔ΕΔɻ͜ͷμϯϓϥΠϯͷΤ
ωϧΪʔΞΫηϓλϯε͕ ERL શମͷΤωϧΪʔΞΫηϓλϯεΛܾΊΔͱ͍ͬͯ΋͍͍ɻGeV Ϋϥεͷ ERL Ͱ͸ 10 MeV ͷμϯϓΤωϧΪʔʹର͠ɺ͓Αͦ 20 % ͷΤωϧΪʔΞΫηϓλϯεΛ૝ఆ͍ͯ͠
ΔɻͦͷͨΊɺόϯν௕͕ҰఆͷӡసͰ͸ɺपճΤωϧΪʔʹؔ܎ͳ͘ ΔE ∼ 2 MeV ·Ͱ͕‫ڐ‬༰ൣғͱͳ
Δɻʢόϯν௕͕มΘΔ৔߹ʹ͍ͭͯ͸ɺ‫ޙ‬ड़͢ΔόϯνѹॖΛࢀߟʹ͢Δ͜ͱɻʣ͜Ε·Ͱʹɺ์ࣹྭ‫ى‬΍ CSR wake ͷӨ‫ڹ‬Λ঺հ͕ͨ͠ɺ͜ΕΒʹΑΔΤ
ωϧΪʔ޿͕Γ͸େ͖ͯ͘΋਺ 100 keV ఔ౓Ͱ͋Γɺμϯϓͷ༌ૹʹ͸໰୊ͳ͍ɻ

ਤ 100: μϯϓϥΠϯͷΦϓςΟΫεɻcERL ͷμϯϓϥΠϯͱͦͷΦϓςΟΫεΛਤ 100
ʹࡌͤΔɻऔΓग़͠γέΠϯͷภ޲ి࣓ੴͱ 4 ͭͷओՃ଎ۭಎͰΤωϧΪʔΛճऩ͢Δʹ͸ɺपճ࣌ؒ

࢛‫࣓ిۃ‬ੴ͔ΒͳΓɺਫฏํ޲ͷ෼ࢄؔ਺Λখ͘͞཈Λ 1.3 GHz ͷ൒੔਺ഒʹ߹ΘͤΔඞཁ͕͋ΔɻcERL

͑Δ͜ͱ͕ࠔ೉Ͱ͋Δɻͦ͜Ͱɺϕʔλؔ਺͸ɺਨ௚Ͱ͸प௕Λิਖ਼͢ΔͨΊʹɺ2 ͭͷํ๏Λ༻͍͍ͯํ޲Ͱ΍΍େ͖Ίʹͳͬͯ͠·ͬͯ΋ɺਫฏํ޲Ͱখ

Δɻͻͱͭ͸ 4 ୆ͷภ޲ి࣓ੴͰߏ੒͞ΕΔप௕ิ

͘͞ͳΔΑ͏ʹ཈͑ͯ༌ૹ͍ͯ͠ΔɻμϯϓϥΠϯͷ

ਖ਼γέΠϯͰ௚ઢ෦ʹόϯϓΛཱͯΔํ๏ɺ΋͏ͻͱΤωϧΪʔ޿͕Γ͸ೖࣹ෦ग़‫ͱޱ‬ಉ౳ɺ10−3 ͷΦʔͭ͸֤ΞʔΫͷ௖఺ʹ 2 ୆Ұ૊ͷεςΞϦϯάΛઃμʔͰ͋ΔɻϏʔϜμϯϓͷ೤ෛՙ௿‫ݮ‬ͷͨΊɺ࢛‫ۃ‬ஔͯ͠ภ޲ి࣓ੴͷ‫ي‬ಓΛม͑Δํ๏Ͱ͋Δɻ໛ࣜਤి࣓ੴͰϏʔϜαΠζΛখ͘͞ߜΓ͗͢ͳ͍Α͏ʹ͠Λਤ 102 ʹࣔ͢ɻਤ 102 (b) ͷ΄͏͕ௐ੔ൣғ͕޿ͯ͘ɺμϯϓϥΠϯͷภ޲ి࣓ੴͰप‫ظ‬తʹϏʔϜ‫ي‬ಓϏʔϜΦϓςΟΫε΁ͷӨ‫͕ڹ‬গͳ͍ɻ͔͠͠ɺ͜ͷΛৼΔରࡦʢϥελϦϯάʣΛߦ͍ͬͯΔɻ

ํ๏͕༗ޮͰ͋Δͷ͸ɺͻͱͭͷภ޲ి࣓ੴͷ‫֯͛ۂ‬͕ 45 ౓ͱେ͖͍৔߹ͷΈͰ͋Δ͜ͱʹ஫ҙ͍ͨ͠ɻ

14.2

ERL Ͱ͸ΤωϧΪʔճऩޮ཰Λ্͛Δ͜ͱ͕ඇৗ
ʹॏཁͰ͋Δɻ‫ي‬ಓɾΦϓςΟΫεௐ੔த͸ฏ‫ྲྀిۉ‬प௕ิਖ਼ͱϏʔϜͷ‫ي‬ಓௐ੔ [66]͜͜ͰɺcERL ͷ‫ݮ‬଎ɾμϯϓϏʔϜͷௐ੔ʹ͍ͭ

͕਺ 100 pA ͱ௿͍ͨΊɺओՃ଎ۭಎͰΤωϧΪʔճ

ͯ঺հ͢Δɻ‫ݮ‬଎ϏʔϜΛՃ଎ۭಎʹ௨͢ͱ͖͸ɺ͢ऩޮ཰Λଌఆ͢Δ͜ͱ͕Ͱ͖ͳ͍ɻͦ͜ͰɺμϯϓγͰʹՃ଎ϏʔϜ͕௨ա͍ͯ͠ΔͷͰɺεΫϦʔϯϞχέΠϯೖ‫ޱ‬ͷภ޲ి࣓ੴ BMAD01 ͷ‫Ͱ֯͛ۂ‬μϯϓλʔͳͲͷഁյతͳํ๏͕࢖༻Ͱ͖ͳ͍ͨΊɺϏʔϜΤωϧΪʔͷਪఆΛߦ͍ͬͯΔɻਤ 103 ʹ BMAD01ʹӨ‫ڹ‬Λ༩͑ͳ͍ BPM Ͱଌఆ͍ͯ͠ΔɻϏʔϜௐ੔ͱμϯϓϥΠϯೖ‫ޱ‬ͷϨΠΞ΢τΛࡌͤΔɻϏʔϜͷͷͱ͖͸ɺύϧε௕ 1 μs ͷۣ‫ܗ‬೾Ͱӡస͍ͯ͠Δɻ

‫࠷͕֯͛ۂ‬େͱͳΔ఺Λ࠷΋ΤωϧΪʔճऩޮ཰͕͍

पճϧʔϓͷ௕͕͓͞Αͦ 100 m Ͱ͋ΔͨΊɺՃ଎ɾ

͍఺Ͱ͋ΔͱΈͳͯ͠ɺप௕Λௐ੔͍ͯ͠Δɻਤ 102‫ݮ‬଎ϏʔϜͷ 2 ͭύϧεʹ͓Αͦ 0.3 μs ͷλΠϛϯ
ͭͷϏʔϜΛ۠ผ͢Δ͜ͱ͕Ͱ͖Δɻ‫ݮ‬଎ϏʔϜͷΦ(b) ͷํ๏Ͱௐ੔ͨ͠ྫΛਤ 104 ʹࡌ͍ͤͯΔɻภ޲ి࣓ੴͷྭ࣓ిྲྀ͔Βɺ‫ݮ‬଎ϏʔϜͷΤωϧΪʔ͸ೖࣹΤωϧΪʔͱಉఔ౓Ͱ͋ΔͱࢥΘΕΔɻओՃ଎ۭಎ

ϓςΟΫεௐ੔͸ɺೖࣹ߹ྲྀ෦खલͷ QMAM05-08͕‫ݮ‬଎ϏʔϜΛΩοΫ͢ΔޮՌ΋‫·ؚ‬ΕΔͨΊɺμϯͰߦ͏ʢਤ 89 ͱ ηΫγϣϯ 12.3 Λࢀߟʣɻͦͷ࢛ϓΤωϧΪʔΛਖ਼֬ʹଌఆ͢Δ͜ͱ͸ࠔ೉Ͱ͋ΔɻάͷͣΕ͕ੜ͡Δɻ͢Δͱɺਤ 101 ʹࣔ͢Α͏ʹɺ2‫࣓ిۃ‬ੴʹ‫ي‬ಓΛௐ੔͢ΔεςΞϦϯά‫ػ‬ೳ΋උ͑ͯ
͍Δɻ

㸲㸫 

ਤ 101: Ճ଎ɾ‫ݮ‬଎ϏʔϜͷ BPM ͷԠ౴ɻਤ 102: (a) ௚ઢ෦ʹઃஔ͍ͯ͠Δप௕ิਖ਼γέΠϯɺ(b) cERL ͷΞʔΫ෦ʹઃஔ͍ͯ͠Δप௕ิਖ਼εςΞϦϯάɻਤ 103: औΓग़͠γέΠϯ෇ۙͷϨΠΞ΢τɻਤ 104: μϯϓΤωϧΪʔͷଌఆɻ

㸲㸫 

15

όϯνѹॖͷ‫ݪ‬ཧͱτϥοΩϯ
ά [53, 62, 67]

ERL ͷಛ௃ͷҰͭ͸୹͍όϯν௕Ͱ͋Γɺ100 fs
ҎԼͷόϯν௕Λ໨ࢦ͍ͯ͠Δɻ͔͠͠ɺిࢠॐ͔Β୹͍όϯνΛ༌ૹ͢Δͱɺ௒఻ಋՃ଎ۭಎͰߴௐ೾
ʹΑΔൃ೤͕૿͑ͯ͠·͏ɻ·ͨɺ௿ΤωϧΪʔͰ͸

CSR wake ͷӨ‫͕ڹ‬େ͖͍ͨΊɺओՃ଎ۭಎͰՃ଎
ͨ͠‫ʹޙ‬όϯν௕Λ୹͘͢Δํ๏Λ࠾༻ͨ͠ɻͦͷ໛ࣜਤΛਤ 105 ʹࣔ͢ɻΞʔΫͰ͸࢛‫࣓ిۃ‬ੴʹΑΔ

ਤ 105: ΦϑΫϨετՃ଎ʹΑΔόϯνѹॖͷ‫ݪ‬ཧऩଋྗ͕͋·Γ‫͍ͳ͘ڧ‬৔߹ɺߴ͍ΤωϧΪʔ͕ԕճʢӈʣՃ଎௚‫ޙ‬ɺʢࠨʣόϯνѹॖ‫ޙ‬ɻΓΛͯ͠஗ΕΔɻ͜ͷͱ͖ɺ༧ΊલํͷిࢠΛΤωϧΪʔͷߴ͍ঢ়ଶʹ͢Δͱɺόϯν௕Λ୹͘͢Δ͜ͱ͕
Ͱ͖Δɻ͜Ε͸Ճ଎Ґ૬Λ φRF ͚ͩͣΒ͢͜ͱͰߦ

σʔλʹΑΔͱɺ࿡‫࣓ిۃ‬ੴʹΑͬͯςʔϧ͕ͳ͘ͳΓे෼ʹόϯνѹॖ͕Ͱ͖͍ͯΔ͜ͱ͕Θ͔Δɻ
όϯνѹॖͨ͠ϏʔϜ΋ΤωϧΪʔճऩΛ͠ͳ͚Ε͏͜ͱ͕Ͱ͖ΔɻओՃ଎ۭಎͷి৔ E Λ z ͷؔ਺Ͱ͹ͳΒͳ͍ɻͦͷͨΊʹ͸ओۭಎͷखલͰόϯν௕ΛE(z) = E0 cos(kRF z + φRF ),(15.1)

৳͹ͯ͠΋ͱʹ໭͠ɺμϯϓγέΠϯͰΤωϧΪʔ޿͕Γ͕ΤωϧΪʔΞΫηϓλϯεҎԼʹͳ͍ͬͯͳ͍

ͱද͢ɻ͜͜ͰɺkRF ͸Ճ଎೾ͷ೾਺Ͱɺ1.3 GHz ͷͱ͍͚ͳ͍ɻͦΕʹ͸ 2 ௨Γͷํ๏͕͋Δɻόϯνѹप೾਺ fRF ͱ͸ɺkRF = 2πfRF /c ͱ͍͏ؔ܎ʹ͋ॖΛͨ࣌͠ͱಉ͡Α͏ʹϦλʔϯͷΞʔΫ΋ R56 ΛΔɻࣜ (15.1) Λ z ʹ͍ͭͯඍ෼͠ɺࣜ (13.10) Λ༻͍Δͱɺ࠷΋όϯν௕͕୹͘ͳΔ৚݅͸

kRF sin φRF =

1
,
R56

ಉ͡ූ߸ʹઃఆͯ͠ɺߴ͍ΤωϧΪʔͷిࢠΛ͞Βʹ஗Βͤͯόϯν௕Λ৳͹͢ํ๏Ͱ͋Δʢਤ 107 (a)ʣɻ

(15.2)

͜Ε͸Φʔόʔόϯνϯάͱ΋‫ݺ‬͹ΕΔɻ͜ͷ৔߹ɺ‫ํޙ‬ͷిࢠ͔ΒΑΓଟ͘ΤωϧΪʔΛճऩͯ͠ɺμϯ

ͱͳΔɻR56 ͕େ͖͍ͱ φRF ͸খͯ͘͞ࡁΈɺόϯϓγέΠϯೖ‫ޱ‬ͷΤωϧΪʔ޿͕ΓΛখ͘͢͞Δඞཁf΋খ͍͞ɻҰํνѹॖ‫ޙ‬ͷΤωϧΪʔ޿͕Γ σδp/pfͰɺόϯνѹॖ‫ޙ‬ͷόϯν௕ σz ͸͕Δɻ͢Δͱɺ௨ৗͷΞΠιΫϩφεͷӡసͱ͸֯౓iσzf ∼ R56 σδp/p,(15.3)

ͱͳΓɺόϯνѹॖલͷΤωϧΪʔ޿͕Γ͕ॏཁͰ͋
Δ͜ͱ͕Θ͔ΔɻR56 ͕খ͍͞ํ͕୹͘͢Δ͜ͱ͕Ͱ͖ΔΑ͏ʹ‫͑ݟ‬Δ͕ɺφRF ͕େ͖͗͢ΔͱՃ଎Ͱ͖
ͳ͘ͳ͘ͳΔͱ͍͏໰୊͕ੜ͡ΔɻcERL Ͱ͸όϯνѹॖͷ࣌͸ R56 = 0.1 ∼ 0.2 m ͰόϯνѹॖΛ͢Δ༧
ఆͰ͋ΔͨΊɺ100 fs ·Ͱόϯνѹॖ͢Δʹ͸ δp/p
͕ 10−4 ͷΦʔμʔͰͳ͍ͱ͍͚ͳ͍ɻ

2 φRF ʹ૬౰͢Δ෼͚ͩप௕Λม͑Δඞཁ͕͋Γɺ‫ݱ‬ࡏͷ cERL ͷप௕ิਖ਼ͷγεςϜͰ͸ରԠ͢Δ͜ͱ
͕ࠔ೉Ͱ͋Δɻ·ͨɺҟͳΔҐ૬ͰΤωϧΪʔճऩΛ
ߦ͏ͱɺRF ͷύϫʔιʔε͕ෆ҆ఆʹͳΔՄೳੑ͕͋Δɻͦ͜ͰɺϦλʔϯΞʔΫͰ͸ߴ͍ΤωϧΪʔ͕ਐΉΑ͏ʹΦϓςΟΫεΛઃ‫͠ܭ‬ɺόϯνѹॖલͷঢ়ଶʹ໭͢ํ๏Λ࠾୒ͨ͠ʢਤ 107 (b)ʣɻϦλʔϯΞʔΫͰ͸ΑΓ‫͍ڧ‬ऩଋྗ͕ඞཁʹͳΔ͕ɺਤ 106 ͷτϥοΩϯάͷ݁ՌʹΑΔͱɺΤωϧΪʔճऩʹ੒ޭ͍ͯ͠Δɻ

1.3 GHz ͷ RF ΧʔϒͷӨ‫ Ͱڹ‬δp/p ͕ z ʹରͯ͠ઢ‫͍ͨͳͰܗ‬Ίɺόϯνѹॖ‫ޙ‬ͷॎํ޲ͷҐ૬ۭؒͷ෼෍͸‫ͳٷ‬ΓͷΑ͏ʹͳΓɺςʔϧ͕ൃੜ͢Δɻ
ਤ 108 ʹͦͷτϥοΩϯάͷਤΛࡌͤΔɻ͜ͷߴ࣍ͷӨ‫ڹ‬Λิਖ਼͢ΔͨΊɺΞʔΫ಺ʹ࿡‫࣓ిۃ‬ੴΛΠϯετʔϧ͢Δ༧ఆͰ͋Δɻ࿡‫࣓ిۃ‬ੴ͸ਫฏҐஔʹΑͬͯҟͳΔऩଋྗΛ༩͑Δ͜ͱ͕Ͱ͖ΔͨΊɺ෼ࢄؔ਺
͕େ͖͍৔ॴʹઃஔ͢Δͱࣜ (13.10) ͷ R566 Λௐ੔͢Δ͜ͱ͕Ͱ͖Δɻਤ 108 ʹࣔͨ͠τϥοΩϯάͷ

㸲㸫 

ਤ 106: όϯνѹॖͷͱ͖ͷϏʔϜϓϩϑΝΠϧͷมԽɻਤ 107: ‫ݮ‬଎ϏʔϜͷҐ૬ɻ

ਤ 108: όϯνѹॖʹ͓͚Δ 6 ‫࣓ిۃ‬ੴͷޮՌɻ

㸲㸫 

16

͓ΘΓʹ

ࢀߟจ‫ݙ‬·ͣ͸͡Ίʹɺओۭಎͷઢ‫ܗ‬Ճ଎‫ث‬ΦϓςΟΫεઃ[1] ӋౡྑҰ, தଜయ༤, ࡔதষ‫ޛ‬, খྛ޾ଇฤू, ίϯύΫτ ERL ͷઃ‫ڀݚܭ‬, KEK Report 2007-7,‫( ܭ‬ηΫγϣϯ 12) ͔Βόϯνѹॖͷ‫ݪ‬ཧ (ηΫγϣϯ 15) ·ͰͷηΫγϣϯ͸ౡాඒൕࢯʹ࡞੒͍ͯͨ͠

JAEA-Research 2008-032, 2008.

͍ͩͨɻౡాࢯ͕த৺ͱͳͬͯਐΊ͖ͯͨ cERL ͷ[2] OHO’2013, “̭ઢࣗ༝ిࢠϨʔβʔ ∼ ̨̡̖̘पճ෦ίϛογϣχϯάͷ࠷৽ͷ݁Ռ΋঺հ͞Ε͓̖ͯ”, 2013.

Γɺྑ͍‫ߘݪ‬Λఏ‫ͱ͍͍ͨͩͨͯ͜͠ڙ‬Λ‫͢ँײ‬Δɻ

ERL Λओ୊ͱͨ͠ OHO’2008 ͷߨٛͷͱ͖͸ɺ࣮

[3] J. Galayda et al., ”The Linac Coherent Light
Source-II Project”, in Proc. of IPAC’14, Dres-ূ‫͋Ͱػ‬Δ cERL ͸·ͩઃ‫ܭ‬ஈ֊ͷ΋ͷͰ͕͋ͬͨɺ

2013 ೥य़͔Β·ͣ cERL ೖࣹ‫ث‬୯ମͰͷϏʔϜίden, Germany, 2014, pp. 935-937.

ϛογϣχϯά͕։࢝͞Εɺ2014 ೥͔Β͸ cERL શ[4] S. Sakanaka et al., “Recent Progress and Oper-ମͰͷӡస͕։࢝͞ΕΔ·ͰʹͳͬͨɻcERL ͷϏʔational Status of the Compact ERL at KEK”,
in Proc. of IPAC’15, Richmond, USA (2015),
TUBC1.

ϜίϛογϣχϯάΛ௨ͯ͠ଟ͘ͷ͜ͱΛֶͿ͜ͱ͕Ͱ͖ɺ͜ΕΛগ͠Ͱ΋ࠓճͷ OHO’2015 ͷߨٛͰ͓఻͑Ͱ͖ͨΒͱࢥ͍ͬͯΔɻcERL ͷ‫ݐ‬ઃɺϏʔϜίϛογϣχϯά͸ඇৗʹଟ͘ͷํͷ‫ྗڠ‬ͷԼͰਐΊΒ[5] ‫ ٶ‬ౡ ɹ ࢘,
“ERL ։ ൃ ͷ ‫ ݱ‬ঢ় ͱ ల ๬”,ઌ୺์ࣹޫ‫ؔ͢ʹݯ‬Δ‫ڀݚ‬ձ
(2014),

͖ͯͨɻࠓճͷߨٛͷ಺༰΋ଟ͘ͷํʹࢧԉͷԼͰߦΘΕͨ͜ͱΛ‫͍ͯ͠ͱૅج‬Δɻ͜͜ʹɺERL ʹؔΘͬ

http://www.jssrr.jp/event/2014/sentansiryou/miyajima.pdf

ͨํօ༷ʹ‫ँײ‬Λਃ্͍͛ͨ͠ɻ
ຊςΩετͰ͸ɺERL ʹ͓͚ΔϏʔϜμΠφϛΫ

[6] S. Y. Lee, Accelerator Physics, World Scientiﬁc,εʹ͍ͭͯɺ࠷্ྲྀͷిࢠॐ͔Β࠷ԼྲྀͷϏʔϜμϯ1999.

ϓ·ͰͷҰ࿈ͷྲྀΕΛ঺հ͖ͯͨ͠ɻ͔͠͠ͳ͕ΒɺຊςΩετͰͷ಺༰͸චऀ͕஌͍ͬͯΔ಺༰ʹ‫ݶ‬ΒΕ[7] H. Wiedemann, Particle Accelerator Physics,͓ͯΓɺภΓ͕͋Δ͜ͱ͸൱Ίͳ͍ɻ͜͜ʹ‫ͨ͠ࡌܝ‬second edition, Springer, 1999.

ࢀߟจ‫ݙ‬Ҏ֎ʹ΋ྑ͍จ‫͕ݙ‬ଟ਺͋ΔͷͰɺͦΕΒΛ

[8] M. Reiser, Theory and design of charged particle beams, Wiley & Sons, 1996.

ࢀরͯ͠ิ͍͚ͬͯͨͩΕ͹޾͍Ͱ͋Δɻ
ຊςΩετΛॻ͘ʹ͋ͨͬͯɺ͜Ε·Ͱ౰ͨΓલͷΑ͏ʹߟ͑ͯɺࣗ෼Ͱ‫͍ͯͭʹͱ͍͜ͳ͍ͯ͠ࢉܭ‬[9] J. D. Lawson, The physics of charged-particle
beams, Oxford Press, 1988.

΋վΊͯษ‫͢ڧ‬Δ‫ػ‬ձ͕ಘΒΕͨɻ͔͠͠ͳ͕Βɺචऀͷ஌Βͳ͍͜ͱ͸·ͩ਺ଟ͋͘Γɺ·ͨචऀͷཧղ͕‫͍ͯͬޡ‬ΔՄೳੑ͕͋Δ͜ͱ͸൱ఆͰ͖ͳ͍ɻ‫ޡ‬Γʹ‫͍͕ͨͭؾ‬৔߹ʹ͸ɺ‫͚͍ͩͨͯ͑ڭ‬Ε͹޾͍Ͱ͋[10] ொా৻ೋ, OHO’2000, ۭؒిՙޮՌ, 2000.
[11] ‫܀‬໦խ෉, OHO’2006, ཻࢠ‫ݯ‬ͷઃ‫ݱͱܭ‬ঢ়, 2006.

ΔɻຊςΩετͷ಺༰ʹ͍ͭͯͷίϝϯτɺ࣭͝໰Λ‫܀‬໦խ෉, OHO’2002, ిࢠ‫ݯ‬, 2002.

‫ܴ͢׻‬Δɻe-mail:tsukasa@post.kek.jp·ͨɺຊςΩετͷߏ૝ஈ֊Ͱ͸ɺೖࣹ‫ۭث‬ಎͷ‫ي‬[12] I. V. Bazarov, G. A. Kraﬀt and L. Merminga, USPAS course on Reciculated and Energy Recovered Linacs, http://casa.jlab.org/
publications/lecture 2005.shtml, 2005.

ಓ‫ʹࠩޡ‬ΑΔΤϛολϯε૿Ճͷղੳ΍ɺcERL ʹ͓͚Δ࣮‫ݧ‬౳͍ͭͯ΋঺հ͢Δ༧ఆͰ͕͋ͬͨɺ࣌ؒͷ౎߹ͰɺຊςΩετͰ͸ׂѪ͢Δ͜ͱʹͨ͠ɻࠓ‫ޙ‬ɺຊςΩετͷվగ൛Λ OHO ηϛφʔͷ web ϖʔδ[13] Xiuguang Jin et al., J. Appl. Phys. 116, 064501(2014).

্ʹެ։͍ͨ͠ͱࢥ͓ͬͯΓɺͦΕʹ‫͖Ͱࡌܝ‬Ε͹ͱߟ͍͑ͯΔɻ

[14] Ivan V. Bazarov, Bruce M. Dunham, Yulin Li,
Xianghong Liu, Dimitre G. Ouzounov, Charles
K. Sinclair, Fay Hannon, and Tsukasa Miyajima, J. Appl. Phys. 103, 054901 (2008).

㸲㸫 

[15] Ivan V. Bazarov, Dimitre G. Ouzounov, Bruce
M. Dunham, Sergey A. Belomestnykh, Yulin Li,
Xianghong Liu, Robert E. Meller, John Sikora,
Charles K. Sinclair, Frank W. Wise, Tsukasa
Miyajima, Phys. Rev. ST Accel. Beams 11,040702 (2008).

[29] C. Ronsivalle et al., ”Simulations of the
Emittance Compensation in Photoinjectors
and Comparison with SPARC Measurements”,
Proc. of EPAC2008, (2008).
[30] M. Ferrario, ”HOMDYN USER GUIDE”,
https://projects.astec.ac.uk /Plone /astec/Software /HOMDYN/HOMDYN%20User%20Manual.pdf.

[16] S. Matsuba, et. al., Jpn. J. Appl. Phys. 51,
046402 (2012).
[17] I. V. Bazarov and C. K. Sinclair, Phys. Rev. ST
Accel. Beams 8, 034202 (2005).
[18] N. Nishimori et al., Phys. Rev. ST Accel. Beams
17, 053401 (2014).
[19] POISSON SUPERFISH, Los Alamos National
Laboratory Report No. LA-UR-96-1834 (revision 14 March 2000).

[31] K. Floetmann, ASTRA,
http://www.desy.de/∼mpyﬂoASTRA dokumentation.
[32] L. M. Young, PARMELA, Los Alamos National
Laboratory Report LA-UR-96-1835.
[33] Pulsar Physics,
http://www.pulsar.nl/gpt/index.html.

[20] V. N. Litvinenko, R. Hajima and D. Kayran,Nucl. Instr. Meth. A 557, 165 (2006).

[34] F. Ciocci et al., Nucl. Instr. Meth. A 393, 434(1997).

[21] R. Hajima, Proc. of 1st Annual Meeting of Par-[35] A. Bacci et al, Proceedings of PAC03, p. 3512

ticle Accelerator Society of Japan and the 29th
Linear Accelerator Meeting in Japan (August
4-6, 2004, Funabashi, Japan), 432 (2004).
[22] R. Hajima, Jpn. J. Appl. Phys. 42, L974 (2003).
[23] R. Nagai et al., “Demonstration of High-ﬂux
Photon Generation from an ERL-based Laser
Compton Photon Source”, TUPJE002, These
Proceedings, IPAC’15, Richmond, USA (2015).
[24] A. W. Chao, Physics of Collective Beam Instabilities in High Energy Accelerator, Wiley(1993).
[25] ৿ాত෉, OHO’2004, ॳ౳ϏʔϜྗֶ͔Βͷ Optics Correction ೖ໳, 2004.

[36] R. Hajima and R. Nagai, Nucl. Instr. Meth. A
557, 103 (2006).
[37] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM
Philadelphia (1992).
[38] A. W. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering, World Scientiﬁc, 2006.
[39] D. Sagan, Proceedings of EPAC06, Edinburgh,
Scotland, 2006, pp. 2829-2831.
[40] I. V. Bazarov and T. MIyajima, ”Calculationof Coherent Synchrotron Radiation in General
Particle Tracer” Proc. of EPAC2008, (2008).

[26] L. seraﬁni and J. B. Rosenweig, Phys. Rev. E
55, 7565 (1997).
[27] S. G. Anderson and J. B. Rosenweig, Phys. Rev.
STAB 3, 094201 (2000).
[28] B.E. Carlsten, Nucl. Instr. Meth. A 285, 313(1989).

[41] G. Bassi, it et al., Nucl. Instr. Meth. A 557, 189(2006).
[42] D. Sagan, Proceedings of EPAC06, Edinburgh,
Scotland, 2006, pp. 2829-2831.
[43] M. Borland, Phys. Rev. ST-AB 4, 070701(2001).

㸲㸫 

[44] L. Giannessi, P. Musumeci, and M. Quattromini, NIM A 436, 443 (1999)[58] ਆ୩ɹ޾लɺʮՃ଎‫ث‬ͷ‫ݪ‬ཧʯɺߴΤωϧΪʔՃ଎‫ث‬ηϛφʔ OHO’84, 1984.

[45] M. Dohlus, A. Kabel and T. Limberg, Nucl. In-[59] ”Energy Recovery Linac Conceptual Design Re-

str. Meth. A 445, 338 (2000)

port”, KEK Report 2012-4 (2012).

[46] C. Mayes, private communication.

[60] J. Rosenzweig and L. Seraﬁni, ”Transverse par-[47] P. Emma and R. Brinkmann, Proceedings of
PAC97, Vancouver, B.C., Canada, 1997, pp.
1679-1681.
[48] Ya. S. Derbenev. et al., TESLA FEL-Report
1995-05.
[49] T. Miyajima et al., “Low Emittance Electron
Beam Transportation in Compact ERL Injector”, in Proc. of IPAC’14, Dresden, Germany,

ticle motion in radio-frequency linear accelerators”, Phys. Rev. E 49, p.1599-1602 (1994).
[61] Project Deﬁnition Design Report, ”Cornell Energy Recovery Linac”, 2012.
[62] ౡాɹඒൕɺʮίώʔϨϯτγϯΫϩτϩϯ์ࣹޫͱϏʔϜμΠφϛΫεʯɺߴΤωϧΪʔՃ଎‫ث‬ηϛφʔ OHO’08, 2008.

[63] D. Douglas, ”Suppression and enhancement of2014, pp. 3104-3106.

CSR-driven emittance degradation in the IRFEL driver”, JLAB-TN-98-012 (1998).

[50] T. Takahashi et al., “Development of a 1.3GHz Buncher Cavity for the Compact-ERL”,
in Proc. of IPAC’14, Dresden, Germany, 2014,
pp. 3866-3868.

[64] R. Hajima, ”Emittance compensation in a return arc of an energy-recovery linac”, Nucl. Instrum. and Meth. A 528, p335-9, (2004).

[51] ຊా༸հଞ,“cERL ೖࣹ‫ث‬௒఻ಋۭಎͷߴ࣍ϞʔυʹΑΔϏʔϜҐஔଌఆ”, THP089, in these[65] M. Shimada et al., ”Lattice and beam optics design for suppression of CSR-induced emittance
growth at the KEK-ERL test facility”, Nucl.
Intrum. Meth. A 575, p.315-320 (2007).

proceedings.
[52] S. Matsuba et al., “Deﬂecting Cavity for Bunch
length Diagnostics at compact ERL Injector”,
in Proc. of IPAC’10, Kyoto, Japan, 2014, pp.
951-953.

[66] ౡాɹඒൕଞɺʮίϯύΫτ ERL ίϛογϣχ
ϯάͷਐ௙ঢ়‫گ‬ʯɺՃ଎‫ ث‬11, pp.1-10 (2014).

[67] N. Nakamura et al., ”Simulation study on bunch[53] ԣ୩ɹ֜ɺʮERL ೖ໳ʯɺߴΤωϧΪʔՃ଎‫ث‬ηϛφʔ OHO’03, 2003.
[54] A. Rahman et al., ”Benchmarking the multipass
beam-breakup simulation code BI”, OAG-TN2008-029.
[55] G. H. Hoﬀstaetter and I. Bazarov, ”Beambreakup instability theory for energy recovery
linacs”, Phys. Rev. STAB, 7, 054401 (2004).
[56] E. Pozdeyev, ”Regenerative multipass beam
breakup in two dimensions”, Phys. Rev. STAB,
8, 054401 (2005).
[57] I. V. Bazarov et al, ”Linac optics for energy
recovery linac”, Proceedings of PAC, pp.334749, Chicago, (2001).

㸲㸫 

compression and decompression for the compact ERL”, Proceedings of IPAC’15, pp.1591-3,
Richmond, (2015).