Index. Accelerator model 8 Adiabatic damping 32, 141 Air-bag model 338 Alternating explicit time scheme 112 Azimuthal modes, see Modes

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1 Index Accelerator model 8 Adiabatic damping 32, 141 Air-bag model 338 Alternating explicit time scheme 112 Azimuthal modes, see Modes Beam breakup in linacs dipole mode 136 higher modes 160 quadrupole mode 153 with solenoid focusing 149 Beam signal spectrum betatron oscillation 174 bunch spectrum 81, 117 finite bunch length 166 fm sidebands 168 multiple bunches 207 point bunch 165 synchrotron oscillation 168 Beam transfer function (BTF) bunched beams 265 unbunched beams 267 various spectral distributions, see Landau damping zero beam intensity with external focusing 231 without external focusing 256 Bessel function, properties 297 p-functions 9 BNS damping dipole beam breakup in linacs general bunch distribution 144 multiple bunches 144 two-particle model 142 quadrupole beam breakup 156, 159 strong head-tail 188 Boussard criterion 249, 262 Broad-band resonator model, see Impedance Bunch lengthening mode coupling instability 329 potential-well, see Potential-well distortion scaling law 330 Chebyshev modes, see Sacherer formalism Chromaticity 143 Classical radius of particles 4 Condon method of mode expansion application to two parallel plates lt 32 formalism 99 Conductivity definition 42 values 43 Decoherence effect 223 Diffraction model 92 Dispersion relation bunched beams 241 unbunched beams 253,255,256,362 bunched beams 235 unbunched beams 244, 246, 364

2 368 = INDEX Effective impedance 320 Gaussian model 321 parabolic model Gaussian model 346 uniform model 346 Emittance 14, 36 Energy depression 27 Energy extraction efficiency in linacs 130 Energy spread, wake induced in linacs resistive-wall 133 resonator impedance 133 rf phase offset 135 space charge 131 two parallel plates 134 Envelope equation Gaussian distribution 37 KV distribution 30 second moments 36 Faraday s law 21 Fast wave 245 Fokker-Planck equation 278 Fourier transform, pairs 45 Fresnel integrals 94 Fundamental theorem of beam loading 50, d\ Gamma function, properties Gaussian model Harmonic number 162 Head-tail instability air-bag model 350 head-tail phase 199, 339 head-tail instability 358 strong, see Strong head-tail instability two-particle model, see Macroparticle Hermite modes, see Sacherer formalism Impedance careless limit 88 cavity structure, rough estimates 84 definition, m = 0 68, m # diffraction model 96 effective, see Effective impedance free space 4, 69 frequency-domain calculation, see Condon method infinite cavity array analytical extension 109 optical resonator model 109 measurement, using beam in accelerator by beam transfer function, see Beam transfer function by betatron frequency shift 186 by head-tail instability growth rate 201, 203, 348 by parasitic loss 120, 283 by strong head-tail threshold 186 by kick to beam 184, 266 optical resonator model 109 properties 76 resistive-wall 70 resonator 72, 75 broad-band, m = 0 87 broad-band, m = 1 90 generalized 79 Q=;78 sharply peaked, 75 sharply peaked, 75 small hole on beam pipe 91 space charge 71 stripline monitor 83 time-domain calculation 115 Isochronous accelerator 139 Jacobi polynomials, properties 310 Keil-Schnell criterion 259 KV distribution 28 Laguerre polynomials, properties 316 Landau damping bunched beams, one-particle model 233 unbunched beams various spectral distributions 228, 257 Laslett tune shift, see Tune shift Legendre, see Sacherer formalism

3 INDEX = 369 Liouville theorem 277 Loss factor definition 98 mode expansion method 100, 102 two parallel plates 105 Macroparticle one-particle parasitic loss in linacs 129 rigid beam instability 172 Robinson instability 162 two-particle beam breakup in linacs 136 energy spread in linacs 129 head-tail instability 198 strong head-tail instability 179 two-slice quadrupole beam breakup in linacs, see Beam breakup in linacs quadrupole mode strong head-tail instability 192 Maxwell equations Coulomb gauge 98 differential form 40 integral form 111 Metal, definition 42 Method of the steepest descent 151 Microwave instabilities bunched beams 262 unbunched beams 259 bunched beams 249 unbunched beams 247 Mode coupling instabilities Gaussian model 327 parabolic model 328 water-bag model 323, air-bag model 353 Mode expansion method, see Condon method Modes azimuthal multiple bunches, see Multibunch instabilities radial, and Hermite, Legendre, Chebyshev, see Sacherer formalism Momentum compaction factor 9 Multibunch instabilities general 359 one-particle model 209 Robinson, see Robinson instability general 360 one-particle model 203, 205 resistive-wall, see Resistive wall Negative mass effect 10 instability 255 One-particle, see Macroparticle Optical resonator model, see Impedance Panofsky-Wenzel theorem 59, 70 Parabolic model 309 Parasitic loss definition 80, 117, 125 resistive wall 56, 118 resonator impedance 119 two parallel plates 121 Phase space 11, 293, 334 Poisson sum formula 125 Potential-well distortion capacitive impedance, protons and electrons 289 Gaussian beam 282 inductive impedance for electrons 287 for protons 285 resistive impedance 290 synchrotron frequency shift 165, 290, 302 multiple bunches 210 potential-well distortion 358 Principal value 77

4 370 = INDEX Q-value de-q of higher order cavity modes rf cavity 110 Radial modes, see Sacherer formalism Reactive feedback 188 Resistive wall conductivity, see Conductivity effects parasitic heating, see Parasitic loss stability condition, unbunched beams 261 synchrotron frequency shift 133 effects feedback 208 multiple bunches 208 single bunch 177 stability condition 247 unbunched beams 245 wake fields m = 0 44, 47 m#o54 wake function, see Wake functions Resonator model, see Impedance rf bucket 280 Rigid beam instability, see Macroparticle Robinson instability 1 = 1 mode, one-particle model > 1 modes, parabolic model > 1 modes, Gaussian model > 1 modes, water-bag model 302 multiple bunches 172, = 0 mode, one-particle model > 0 modes, air-bag model 349 Sacherer formalism, m = 0 Chebyshev modes for water-bag model 302, 308 Hermite modes for Gaussian model 316 integral equation 306 interaction matrix 307 Legendre modes for parabolic model 311 solution by factorizability 304, 327, m = 1 Chebyshev modes for air-bag model 339 Hermite modes for Gaussian model 345 integral equation 343 interaction matrix 344 Legendre modes for uniform model 345 solution by factorizability 342 Scaling law for mode coupling instabilities, see Bunch lengthening 356 Simplified stability criteria bunched beams 241, 262 unbunched beams 261 bunched beams 239, 249 unbunched beams 248 Skin depth 42 Slippage factor 9 Slow wave 245 Space charge effects forces 21, 25 mode frequency shift, parabolic model 315 negative mass instability, see Negative mass tune shift, see Tune shift wake function, see Wake function forces 13, 19 instability in transport line 14 tune shift, see Tune shift wake function, see Wake function Stability boundary diagrams bunched beams 237 unbunched beams 260, bunched and unbunched beams 237

5 INDEX 371 Strong head-tail instability, see Macroparticle Synchro-betatron coupling effects 173, 342 strong head-tail 189 Synchronous particle 8 Transition below, above, crossing 9 Tunes, definitions betatron 9 synchrotron 10 Tune shift incoherent betatron 12 space charge direct 14 Laslett 18 24, 290 with boundary walls 17 Turbulence instability, see Mode coupling instability Two-particle model, see Macroparticle Two stream instability 256 Uniform model 344 Vlasov equation derivation 276 linearized multiple bunches 359 unbunched beams 361, 363 Vlasov-Maxwell equation 273 Wake field free space 4 perfectly conducting smooth pipe 6 resistive-wall, see Resistive wall space charge, see Space charge Wake field accelerator 65 transformer ratio 66 Wake functions cavity structure, rough estimate 84 definition 58, 70 diffraction model 96 mode expansion 98, 102 resistive-wall 59 resonator model 73 space charge 60 stripline monitor 83 time-domain calculation 110 Wake potentials 58, 63 Water-bag model 296 azimuthal modes, see Modes radial modes, see Sacherer formalism unperturbed modes 299