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Wave velocity: the dispersion relation The relation ω(k) can be calculated from the standard theory of waveguides: ω2 = k2c2 + ω 2 c Plotting g this curve (hyperbola), ( yp ), we see that: k=2π/λ p 0 ω 1) There is a “cut-off frequency”, below which a wave will not propagate. It depends on dimensions (λc=2.61a for the cylindrical waveguide). vph>c ( c y g ) vph=c tg α = ω/kz = vph v ph=ω/k = (c 2 +ω c 2 /k 2 ) 1/2 v g=dω/dk kz 2) At each excitation frequency is associated a phase velocity, the velocity at which a certain phase travels in the waveguide. vp=∞ at k=0, ω=ωc and then h decreases d towards d vp=c ffor kk,ω￫∞. 3) Energy (and information) travel at group velocity dω/dk, which is between 0 and c. This velocity has to respect the relativity principle! 4) A particle traveling inside our cylinder has to travel at v = v ph to see a constant accelerating Efield → should travel at v > c !!! 5) To use the waveguide to accelerate particles, we need a “trick” to slow down the wave. 8
Slowing down waves: the disc- loaded waveguide Discs inside the cylindrical waveguide, spaced by a distance l , will ind induce mmultiple ltipl reflections fl ti ns bbetween t n th the dis discs. s 9
Page 1 and 2: Introduction to RF Linear Accelerat
Page 3 and 4: (v/c)^2 ( 1 Proton and Electron Vel
Page 5 and 6: Example: Superconducting Proton Lin
Page 7: RF input λ p TM01 field configurat
Page 11 and 12: eam Traveling wave linac structures
Page 13 and 14: mode 0 mode π/2 mode 2π/3 mode π
Page 15 and 16: Comparing traveling and standing wa
Page 17 and 18: Disc-loaded structures operating in
Page 19 and 20: Examples p of DTL Top; CERN Linac2
Page 21 and 22: eam Multigap linac structures: the
Page 23 and 24: Proton linac architecture - cell le
Page 25 and 26: Multi-gap Superconducting linac str
Page 27 and 28: Quarter Wave Resonators Simple 2-ga
Page 29 and 30: Focusing solenoids Examples: p an e
Page 31 and 32: Electron linac architecture EXAMPLE
Page 33 and 34: Heavy y Ion Linac Architecture EXAM
Page 35 and 36: Longitudinal g dynamics y → Ions
Page 37 and 38: Transverse dynamics - Space charge
Page 39 and 40: Transverse equilibrium in ion and e
Page 41 and 42: x_rms beaam size [m] High-intensity
Page 43 and 44: 55. DDouble bl periodic i di accele
Page 45 and 46: Mechanical errors differences in f
Page 47 and 48: The Side Coupled p Linac To operate
Page 49 and 50: The Cell-Coupled Drift Tube Linac W
Page 51 and 52: The Radio Frequency Quadrupole (RFQ
Page 53 and 54: RFQ Qpproperties p - 2 3. The dista
Page 55 and 56: How to create a quadrupole RF mode
Page 57 and 58: Electron sources: give energy to th
Page 59 and 60:
Accelerating structure: the choice
Page 61 and 62:
Mains Transforms mains power into D
Page 63:
Bibliography g p y 1. Reference Boo
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