Accelerator Physics NMI and Synchrotron Radiation G. A. Krafft Old Dominion University Jefferson Lab Lecture 16 Graduate Accelerator Physics Fall 17
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 6 4 4 Graduate Accelerator Physics Fall 17
Fractional Energy Loss E e 6 3 4 For one turn with isomagnetic bending fields E E beam 4 r 3 e 3 3 r e is the classical electron radius:.8 1-13 cm Graduate Accelerator Physics Fall 17
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
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
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 8 15 3 c 5 c 5 e 1 n n 3 3 4 c 137 Graduate Accelerator Physics Fall 17
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
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
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 1 1 1 K / Graduate Accelerator Physics Fall 17
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
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