ビーム不安定性ー 子雲、イオン、CSR.pdfVIP

CSR BINP-PSR CERN-ISR [2] 11 e-p (R/Q) mA A () Q (mω0 + ωβ) (ωc) 0.1 80 () e-p [4] 90 (z1) (z2) [6] 1 B A PF PF [5, 8] KEKB ILC 2 CSR x = (x, y, z) 8 500 mA B KEKB d2x ds2 + Kx(s)x = ? N r0 γL ∫∞ ?∞ Wx(z ? z )ρ1,x(z )dz d2y ds2 + Ky (s)y = ? N r0 γL ∫∞ ?∞ Wy (z ? z )ρ1,y (z )dz d2z ds2 + μs2 L2 z = ? N r0 γL ∫∞ ?∞ W0(z ? z )ρ0(z (1) )dz px, py (p0) px(y) = x (y ) z pz = δp/p0, z = s ? ct c (z 0) (z 0) z () K μs = 2πνs J-PARC SNS LHC superKEKB SuperB L Ls K → (μx/L)2 W 2 e-p W (z) = 0 z 0 (2) CSR 2 xy ? W 1 (1 (s) L x?y (z ) (z) z ? z z (z ) z ? x?y (ρ1,x, ρ1,y, ρ0) ∫ ρ1,x(z) = xΨ(x, px, y, py, z, pz)dxdpxdydpydpz 2 ∫ ρ1,y(z) = yΨ(x, px, y, py, z, pz)dxdpxdydpydpz ∫ (3) ρ0(z) = Ψ(x, px, y, py, z, pz)dxdpxdydpydpz 3 3.1 Ψ Vlasov (x ? y ) ?Ψ = ?[Ψ, H] = ∑ ? ?Ψ dxi + ?Ψ dpi ?s i=xyz ?xi ds ?pi ds (4) 1 y(x) [] H H = px2 + py2 ? αmpz2 2 Kxx2 + + Ky y2 ? zμ2s 2 αm L2 + N r0 γ ∫∞ ?∞ 2 [W0(z ? z )ρ0(z ) (5) +xWx(z ? z )ρx(z ) + yWy(z ? z )ρy(z )] dz [] exp ? Jy(a cos mφs) ? Jz 2εy εz (8) Jy(?y) = γy(y ? ?y)2 + 2αy(y ? ?y)py + βyp2y (9) m = 1, 2, 3 φs () y ρ0 φs = tan αmpz/μsz Vlasov (1) dx dy dz ds = px ds = py ds = αmpz dp ds = N r0 γL ∫∞ Wi(z ?∞ ?z )ρi(z )dz Vlasov x = [x, H] H Vlasov (6) 4 (7) () () y py → My?1 y py φs → φs + μs (10) (11) (4,7) (4) OHO ( ) m = 0, M ? 1 m m n 1 ym(t) = a[n] exp ( nm 2πi M ? ) iωβ t 2 n = 0, 1, 2 (12) M (n) 3 y 1 0 -1 -1 z0 1 0 ?p/p 1 -1 y 1 0 -1 -1 z0 1 -1 0 ?p/p 1 y 1 0 -1 -1 z0 1 -1 0?p/p 1 1: m=0 t = ?mT0/M t=0 m y(t) = ∑∞ ( )( ym ? mT0 M δ t ? ) mT0 M (13) = m=0 ∑ a[n] exp [ 2πim (n M + ] νβ ) δ ( t ? mT0 M ) m ∫ y(t)e?iωtdt = ∑∞ { exp im [(n + νβ )ω0 + ω] } T0 M m=0 (14) ∑∞ ∑∞ exp(imωT0) = ω0 δ(ω ? pω0) m=?∞ p=?∞ (15) n pM ω0 ω = (n + νβ + pM )ω0 (16) (16) 4.1 RF OFF [( )] 2πn Ψ(y, py, s) = δ y ? a cos ? ωt L [ ( )] 2πn δ py ? b sin ? ωt L (17) 4 2: z = s ? ct Vlasov ) ?Ψ + ?Ψ dy ? ?Ψ dpy = 0 ?s ?y ds ?py ds b = ak, ω ? nω0 = ωβ py = ?ωβ2 y ωβ (z ? s (18) ω 5 2 2 2 3 y 1 0 -1 -1 3: 1 0 ?p/p 0 z 1 -1 M →∞ n ω = (n + νβ)ω0 (19) 5 y(