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PART IV Synchrotron Radiation 1. Generation of Synchrotron Radiation! 2. Effects of Synchrotron Radiation on Beams
I. Generation of synchrotron radiation I. Maxwell Feynmann II. Syncrotron III. II. Effects of synchrotron radiation on beams I. II. III.
K-J.Kim s text book アメリカ物理学会加速器学校のテキスト シンクロトロン放射を真 面 目に取り扱う 位相の取り扱いに取り組むも 未完のようだ J.Schwinger s paper 1948~9 Schwinger Classical Electrodynamics (Jackson) Kim K.J., Characteristics of Synchrotron Radiation, AIP Conference Proceedings 184, vol. 1 p567 (American Institute of Physics, NewYork, 1989). PhysRev.75.1912.pdf Notes http://www.archive.org/details/classicalelectrodynamics M. Sands slac-r-121.pdf Notes.pdf Notes
I. Maxwell Feynman 1.Maxwell 2. Feynman
Maxwell Maxwell
Lorentz
Green s δ
unit aming vector
t : " t :
t t (t)
Heaviside-Feynmann Coulomb Mixture " unit aiming vector n
II. Syncrotron 1. " 2.
ParametricPlot@r@tpD, 8tp, -tst, tst<, A PlotRange Æ 880, r<, 8-r, r<<d 10 r@tp_d := 8r H1 - Cos@Cmov tp ê rdl, r Sin@Cmov tp ê rd<; H* curvature,observer,energy *L r = 10; robs = 80, 100<; grel = 50; H* physical consts.*l Clv = 299 792 458; 1 e 0 = 4 Pi 10 ^H-7L Clv ; 2 qelec = 1.602176462 10 ^-H19L; H* particle velocity *L 1 Cmov = Clv SqrtB 1 - grel F; 2 tst = HPi ê 4L r ê Clv; 5 0-5 n 2 4 6 8 10-10
R@tp_D := Sqrt@Hrobs - r@tpdl.hrobs - r@tpdld; tobs@tp_d := tp + R@tpD ê Clv; Plot@tobs@tpD, 8tp, -tst, tst<, PlotRange Æ AllD n@tp_d := Hrobs - r@tpdl ê R@tpD; ParametricPlot@8tp, n@tpd@@1dd<, 8tp, -tst, tst <, PlotRange Æ All, AspectRatio Æ 1D -2. 10-8 -1. 10-8 1. 10-8 2. 10-8 3.36 10-7 -0.005 3.35 10-7 -0.010 3.34 10-7 3.33 10-7 -0.015 3.32 10-7 -2. 10-8 -1. 10-8 1. 10-8 2. 10-8 k@tp_d = D@tobs@tpD, tpd; Plot@k@tpD, 8tp, -tst, tst<d 0.30 ParametricPlot@8tobs@tpD, n@tpd@@1dd<, 8tp, -tst, tst <, PlotRange Æ All, AspectRatio Æ 1D -0.005-0.020-0.025-0.030 n x 3.32 10-7 3.33 10-7 3.34 10-7 3.35 10-7 3.36 10-7 0.25 0.20-0.010 0.15-0.015 0.10-0.020 0.05-2. 10-8 -1. 10-8 1. 10-8 2. 10-8 -0.025 n x
ParametricPlot@8tobs@tpD, n@tpd@@1dd<, 8tp, -tst, tst <, PlotRange Æ All, AspectRatio Æ 1D n x -0.005-0.010-0.015 3.32 10-7 3.33 10-7 3.34 10-7 3.35 10-7 3.36 10-7 5 10 7 0 n x 3.32 10-7 3.33 10-7 3.34 10-7 3.35 10-7 3.36 10-7 H* dnht'l dt = 1 kht'l dndt@tp_d = -0.020-0.025-0.030 dnht'l dt' *L 1 k@tpd D@n@tpD, tpd; -5 10 7-5.0 10 20 3.33562 10-7 3.33563 10-7 3.33564 10-7 3.33565 10-7 3.33566 10-7 3.33567 10-7 H* d2 nht'l 1 d dt 2 = 2 nht'l HkHt'LL 2 dt' 2 - k' dnht'l HkHt'LL 3 *L dt' d2ndt2@tp_d = 1 D@k@tpD, tpd D@n@tpD, 8tp, 2<D - Hk@tpDL2 Hk@tpDL 3-1.0 10 21 D@n@tpD, tpd; -1.5 10 21 ParametricPlot@8tobs@tpD, dndt@tpd@@1dd<, 8tp, -tst, tst <, AspectRatio Æ 1D -2.0 10 ParametricPlot@8tobs@tpD, d2ndt2@tpd@@1dd<, 8tp, -tst ê 10, tst ê 10 21 <, PlotRange Æ All, AspectRatio Æ 1D n x
Feynmann H* EHx,tL= q 4pe 0 : n R 2 + R c d J n dt R 2 N+ 1 d 2 n c 2 dt 2 > *L dnr2dt@tp_d = DB n@tpd, tpf ì k@tpd; R@tpD2 Efield@tp_D = qelec n@tpd 4 Pi e 0 R@tpD + R@tpD 2 Clv dnr2dt@tpd + 1 Clv d2ndt2@tpd ; 2 Table@ParametricPlot@8tobs@tpD, Efield@tpD@@iDD<, 8tp, -tst ê 10, tst ê 10 <, PlotRange Æ All, PlotPoints Æ 1000, AspectRatio Æ 1D, 8i, 1, 2<D 3.33562 10-7 3.33563 10-7 3.33564 10-7 3.33565 10-7 3.33566 10-7 3.33567 10-7 -5. 10-6 1.4 10-9 1.2 10-9 -0.00001 1. 10-9 -0.000015 8. 10-10 -0.00002 6. 10-10 -0.000025-0.00003-0.000035 4. 10-10 3.33562 10-7 3.33563 10-7 3.33564 10-7 3.33565 10-7 3.33566 10-7 3.33567 10-7
Fourier Nbin = 2^10; eqtime = Table@ FindRoot@ tobs@tpd ä Htobs@-tst ê 10D H1 - i ê NbinL + tobs@tst ê 10D Hi ê NbinLL, 8tp, -tst ê 10, tst ê 10<, AccuracyGoal Æ 24, WorkingPrecision Æ 34, MaxIterations Æ 50D, 8i, 0, Nbin<D; Edata = Efield@tpD ê. eqtime; ListPlot@Transpose@EdataD@@1DD, PlotRange Æ AllD Fdata = Fourier@Transpose@EdataD@@1DDD; ListPlot@Abs@FdataD, PlotRange Æ 880, Nbin ê 10<, All<D -5. 10-6 -0.00001-0.000015-0.00002-0.000025-0.00003-0.000035 200 400 600 800 1000 0.000025 0.00002 0.000015 0.00001 5. 10-6 0 20 40 60 80 100
z t << "FourierSeries`" r@tp_d := :- K und Clv Sin@w u tpd grel w u, 1-1 + K 2 und 2 Clv tp - K 2 und Clv Sin@2 w u tpd >; 2 grel 2 I8 grel 2 M w u z 0.4 0.3 H* curvature,observer,energy *L K und = 1; l u = 0.04; N u = 10; robs = 80, 100<; grel = 50; H* physical consts.*l Clv = 299 792 458; 1 e 0 = ; 4 Pi 10^H-7L Clv 2 qelec = 1.602176462 10^-H19L; 0.2 0.1 H* undulater frequency, velocity of moving charge, time duration of observation *L 2 Pi Clv w u = ; Cmov = Clv SqrtB 1 - l u tst = l u N u ê Cmov; 1 F; grel 2 x -0.0001-0.00005 0.00005 0.0001 ParametricPlot@r@tpD, 8tp, 0, tst<, AspectRatio Æ 2, PlotRange Æ All, PlotPoints Æ 5000D
R@tp_D := Sqrt@Hrobs - r@tpdl.hrobs - r@tpdld; tobs@tp_d := tp + R@tpD ê Clv; Plot@tobs@tpD, 8tp, 0, tst ê N u <, PlotRange Æ AllD t 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 t 2. 10-11 4. 10-11 6. 10-11 8. 10-11 1. 10-10 1.2 10-10 k@tp_d = D@tobs@tpD, tpd; Plot@k@tpD, 8tp, 0, tst ê N u <, PlotPoints Æ 1000D κ 4. 10-8 3.5 10-8 3. 10-8 2.5 10-8 t 2. 10-11 4. 10-11 6. 10-11 8. 10-11 1. 10-10 1.2 10-10
unit aiming vector n@tp_d := Hrobs - r@tpdl ê R@tpD; Table@ParametricPlot@8tp, n@tpd@@idd<, 8tp, 0, tst ê N u <, AspectRatio Æ 1, PlotRange Æ AllD, 8i, 1, 2<D 1. 10-6 n x Table@ParametricPlot@8tobs@tpD, n@tpd@@idd<, 8tp, 0, tst ê N u <, AspectRatio Æ 1, PlotRange Æ All, PlotPoints Æ 1000D, 8i, 1, 2<D 1. 10-6 n x 5. 10-7 5. 10-7 -5. 10-7 2. 10-11 4. 10-11 6. 10-11 8. 10-11 1. 10-10 1.2 10-10 t -5. 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 t -1. 10-6 -1. 10-6 1 n z n1 z 1 1 1 1 1 1 t 2. 10-11 4. 10-11 6. 10-11 8. 10-11 1. 10-10 1.2 10-10 t 3.33564 10-7 3.33564 10-7 3.33564 10-7
Feynmamm 1.5 10-19 E Hx, tl = 6. 10-12 q 4 pe 0 : n R 2 + R c d dt n + 1 d2 n > R 2 c 2 dt 2 Ex Ex Ex 0.0010 1. 10-19 4. 10-12 5. 10-20 2. 10-12 0.0005 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 -5. 10-20 -2. 10-12 3.33564 10-7 3.33564 10-7 3.33564 10-7 -1. 10-19 -4. 10-12 -1.5 10-19 -6. 10-12 -0.0005 1.442 10-13 1.4415 10-13 Ez Ez 1.4 10-9 1.3 10-9 1.2 10-9 -0.0010 Ez 1. 10-9 1.441 10-13 1.1 10-9 1. 10-9 5. 10-10 1.4405 10-13 9. 10-10 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7 3.33564 10-7
Nbin = 2^15; eqtime = Table@FindRoot@tobs@tpD ä Htobs@0D H1 - i ê NbinL + tobs@tstd Hi ê NbinLL, 8tp, 0, tst<, AccuracyGoal Æ 14, WorkingPrecision Æ 24, MaxIterations Æ 50D, 8i, 0, Nbin<D; Edata = Efield@tpD ê. eqtime; ListPlot@Transpose@EdataD@@1DD, PlotRange Æ All, Joined Æ TrueD 0.0010 Fdata = Fourier@Transpose@EdataD@@1DDD; ListPlot@Log@Abs@FdataDD, PlotRange Æ 880, Nbin ê 50<, All<, Joined Æ TrueD 0.0005-5 -0.0005 5000 10 000 15 000 20 000 25 000 30 000-10 -0.0010-15 -20-25 100 200 300 400 500 600
Nbin = 2^15; eqtime = Table@FindRoot@tobs@tpD ä Htobs@0D H1 - i ê NbinL + tobs@tstd Hi ê NbinLL, 8tp, 0, tst<, AccuracyGoal Æ 14, WorkingPrecision Æ 24, MaxIterations Æ 50D, 8i, 0, Nbin<D; Edata = Efield@tpD ê. eqtime; ListPlot@Transpose@EdataD@@1DD, PlotRange Æ All, Joined Æ False, PlotStyle Æ PointSize@0.0003DD 0.0010 Fdata = Fourier@Transpose@EdataD@@1DDD; ListPlot@Log@Abs@FdataDD, PlotRange Æ 880, Nbin ê 100<, All<, Joined Æ FalseD 0.0005-5 5000 10 000 15 000 20 000 25 000 30 000-10 -0.0005-15 -0.0010-20 -25 50 100 150 200 250 300
III. 1. " 2.
Analytical method of radiation calculation General formula(another expression) Far-field approximation
Acceleration gives electric field far from the source where in other form
Power and spectrum Power Spectrum where
Photon number Amplitude of electric field E^2 gives finding probability of photons.
Polarization vector
Electric field of each polarization in time domain
Electric field calculated in frequency domain
Analytic calculation of radiation from bending Emitter time t and observer time t.
Power spectrum of bending radiation where e^2 N N (Ne)^2 Ne^2 "
σ-polarization %-polarization Total
IV. Effects of Synchrotron Radiation on Beams" 1. " 2. " 3.
Transfer matrix
Transfer matrix Transfer matrix Transfer matrix
Determinant
"
u i ( ) t i ( )
t t t
FEL i, j
emittance