Accelerator Physics NMI and Synchrotron Radiation. G. A. Krafft Old Dominion University Jefferson Lab Lecture 16

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1 Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17

2 Oscillation Frequency nq I n i Z c E Re Z 1 mode has positive imaginary part instability Resistive impedance has positive real part "Resistive wall instability" If Re Z (e.g. space charge impedance at long wavelengths) stability/instability depends on sign of RHS Im Z (inductive, stable if,unstable if ) Im Z (capacitive, space charge is this way, Later case is negative mass instability c stable if,unstable if ) c c c Graduate Accelerator Physics Fall 17

3 Impedance? NMI Growth time er e r r r c r r Er e e E r r b b rb rb B e r r b c r rb r r 1 1 ln / z c b 4 z e n i n t in in V r r I r r 1 ln / 1 ln / SC n c b n c b c n c c i Z 1 ln r / r nq I n q I SC c b E 4 c E r c r b Δz Graduate Accelerator Physics Fall 17

4 Stabilization by Beam Temperature? Canonical variables, p/ t e n i n t in n n innt e c current perturbation is I q d n n n p Graduate Accelerator Physics Fall 17

5 Dispersion Relation c 1 de / E dt 1 i c q Z / d n 3 c E n recover before b / / Nn b d n n n N n Graduate Accelerator Physics Fall 17

6 Landau Damping Use our favorite analytic distribution 1 1 ˆ ˆ c qz I n ˆ 1 i ˆ c E q Z I n 1 i c E ˆ n n ni n ni ˆ V iu n n n ˆ d Graduate Accelerator Physics Fall 17

7 u u Fe b it it e 1 1 u F 1 dnb N d it e u F d it e u F d Graduate Accelerator Physics Fall 17

8 LD from another view Single Oscillator it u u Fe it Fe 1 1 ut Many oscillators distributed in frequency N 1 dn Nd ui i1 U N it Fe 1 1 U d for U it Fe d Graduate Accelerator Physics Fall 17

9 Resonance Effect it Fe U i PV.. d it U Fe ip. V. d For our analytic Lorentzian it it Fe Fe U i i Energy goes in! Where does it go? Graduate Accelerator Physics Fall 17

10 Inhomogeneous Solution F u t asin t sint Solution with zero initial excitation F a F u sint sin t No energy flow F u t cost sint Resonant particles capture energy and oscillation generated out of phase Graduate Accelerator Physics Fall 17

11 Oscillators Similtaneously Excited u i t it u u Fe 1 it Fe 1 1 ut Many oscillators distributed in frequency N 1 dn Nd ui i1 U N it Fe 1 1 U d for U it Fe d Graduate Accelerator Physics Fall 17

12 Synchrotron Radiation Accelerated particles emit electromagnetic radiation. Emission from very high energy particles has unique properties for a radiation source. As such radiation was first observed at one of the earliest electron synchrotrons, radiation from high energy particles (mainly electrons) is known generically as synchrotron radiation by the accelerator and HENP communities. The radiation is highly collimated in the beam direction From relativity ct ' ct z x' x y' y z ' ct z Graduate Accelerator Physics Fall 17

13 Lorentz invariance of wave phase implies k μ = (ω/c,k x,k y,k z ) is a Lorentz 4-vector kc k k x k k y x y k / c k z kx ky k x k y k z sin sin cos / c / c / c / c / c k 1 cos / c z z z / c / c k 1 cos / c z Graduate Accelerator Physics Fall 17

14 sin sin 1 cos Therefore all radiation with θ' < π /, which is roughly ½ of the photon emission for dipole emission from a transverse acceleration in the beam frame, is Lorentz transformed into an angle less than 1/γ. Because of the strong Doppler shift of the photon energy, higher for θ, most of the energy in the photons is within a cone of angular extent 1/γ around the beam direction. Graduate Accelerator Physics Fall 17

15 Larmor s Formula For a particle executing non-relativistic motion, the total power emitted in electromagnetic radiation is (Larmor, verified later) 1 q 1 e Pt a p 6 c 6 m c 3 3 Lienard s relativistic generalization: Note both de and dt are the fourth component of relativistic 4-vectors when one is dealing with photon emission. Therefore, their ratio must be an Lorentz invariant. The invariant that reduces to Larmor s formula in the non-relativistic limit is e du du P 6 c d d Graduate Accelerator Physics Fall 17

16 Pt e 6 c 6 For acceleration along a line, second term is zero and first term for the radiation reaction is small compared to the acceleration as long as gradient less than 1 14 MV/m. Technically impossible. c For transverse bend acceleration r ˆ Pt ec Graduate Accelerator Physics Fall 17

17 Fractional Energy Loss E e For one turn with isomagnetic bending fields E E beam 4 r 3 e 3 3 r e is the classical electron radius: cm Graduate Accelerator Physics Fall 17

18 Radiation Power Distribution Consulting your favorite Classical E&M text (Jackson, Schwinger, Landau and Lifshitz Classical Theory of Fields) dp d 3 e 8 c / c K 5/3 x dx Graduate Accelerator Physics Fall 17

19 Critical Frequency Critical (angular) frequency is 3 c 3 c Energy scaling of critical frequency is understood from 1/γ emission cone and fact that 1 β ~ 1/( γ ) t A c A B t t B 3 3 c c c t B 3 c c c 1/γ Graduate Accelerator Physics Fall 17

20 Photon Number dp 3 e e c c 5/3 d 8 6 P d K x dxd dn 1 dp d d 4 dn d d dn d d c 5 c 5 e 1 n n c 137 Graduate Accelerator Physics Fall 17

21 Insertion Devices (ID) Often periodic magnetic field magnets are placed in beam path of high energy storage rings. The radiation generated by electrons passing through such insertion devices has unique properties. Field of the insertion device magnet ˆ B x y z B z y B z B z,, cos / ID Vector potential for magnet (1 dimensional approximation) B A x y z A z x A z z ID,, ˆ sin / ID Graduate Accelerator Physics Fall 17

22 Electron Orbit Uniformity in x-direction means that canonical momentum in the x-direction is conserved ea z K vx z csin z / m ID v 1 K x z dz z v x ID cos / Field Strength Parameter z z ID K eb ID mc Graduate Accelerator Physics Fall 17

23 Average Velocity Energy conservation gives that γ is a constant of the motion 1 z x z x z 1 z z Average longitudinal velocity in the insertion device is Average rest frame has z 1 1 K K / Graduate Accelerator Physics Fall 17

24 Relativistic Kinematics In average rest frame the insertion device is Lorentz contracted, and so its wavelength is ID / The sinusoidal wiggling motion emits with angular frequency c / Lorentz transformation formulas for the wave vector of the emitted radiation k k1 cos k k k x y z k k x y k k sin cos k sin sin cos Graduate Accelerator Physics Fall 17

25 ID (or FEL) Resonance Condition Angle transforms as Wave vector in lab frame has cos k cos z k 1 cos k k c 1 cos 1 cos ID In the forward direction cos θ = 1 e ID ID 1 K / Graduate Accelerator Physics Fall 17

Fundamental Concepts of Particle Accelerators III : High-Energy Beam Dynamics (2) Koji TAKATA KEK. Accelerator Course, Sokendai. Second Term, JFY2012

.... Fundamental Concepts of Particle Accelerators III : High-Energy Beam Dynamics (2) Koji TAKATA KEK koji.takata@kek.jp http://research.kek.jp/people/takata/home.html Accelerator Course, Sokendai Second