STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
and is used to accelerate charged particles due to the fact that the electric field is strongest on-axis, ρ = 0. The TM110 modes are given by Ez(ρ, φ, z) = E0J1 Hφ(ρ, φ, z) = iE0 x11ρ R ɛ0ω110R x11 J ′ 1 R 83 cos(φ) (4.5) x11ρ cos(φ) (4.6) The electric field vanishes on-axis, but for small off-axis displacements grows linearly. These modes are typically referred to as dipole HOMs due to the behavior of the magnetic field. A plot of the electric and magnetic fields in ρ−φ space is shown in Fig. 4.2. With these field configurations in mind, the mechanism which facilitates BBU can be understood. The magnetic field deflects a particle on the first pass (even if it travels on-axis) and is transformed into a displacement, the magnitude of which depends on the machine optics, through the cavity on the second pass. As the par- ticle travels off-axis through the cavity, energy can be exchanged with the electric field depending on the phase of the beam relative to the field. Under certain condi- tions the beam can couple energy to the HOM which in turn more strongly deflects trailing particles traveling through the cavity. Hence a feedback loop is generated between the recirculated beam and the cavity dipole HOM fields which can become unstable if the average beam current exceeds the threshold current. 4.2 Derivation of the BBU Threshold Current In this section, the threshold current is derived by equating the energy dissi- pated by the cavity to the energy deposited by the beam into the HOM [65]. In Section 4.4 an alternate derivation is outlined which uses the concept of the wake
FIG. 4.2: Electric field (red) and magnetic field (blue) in the ρ−φ plane for a TM110 mode in a pillbox cavity. function to describe the interaction between the HOM and beam. Consider a two-pass energy recovering linac with a single RF cavity which contains a single dipole HOM. While it is true that dipole HOMs occur in orthogonal pairs, for reasons that will be discussed in Section 4.7, one polarization can be safely neglected for the derivation. The change in the stored energy of a dipole HOM due to the passage of a bunch of charge q is given by 84 r ∆U = −qVa cos φ (4.7) a where Va is the accelerating HOM voltage at the beam pipe radius a induced by all previous bunches, r is the off-axis displacement of the bunch and φ is the phase of the bunch relative to the maximum HOM electric field. In previous analytic treatments of BBU, the off-axis displacement had always been assumed to be collinear with an HOM polarized at either 0 ◦ or 90 ◦ . Now
- Page 51 and 52: FIG. 2.8: Illustration of the cryom
- Page 53 and 54: linac and θNL is the RF phase. The
- Page 55 and 56: 2.4 Transverse Emittance One of the
- Page 57 and 58: where σ2 is the rms beam size meas
- Page 59 and 60: eams. The effects of varying the qu
- Page 61 and 62: FIG. 2.12: A typical wire scan near
- Page 63 and 64: quadratic fit and a multiple regres
- Page 65 and 66: ting the data is difficult. Without
- Page 67 and 68: primary source of error is measurin
- Page 69 and 70: identified, although the phase dela
- Page 71 and 72: TABLE 2.3: Comparison of Twiss para
- Page 73 and 74: the results of the fits. The vertic
- Page 75 and 76: FIG. 2.18: Schematic illustrating t
- Page 77 and 78: FIG. 2.19: The GASK signal measured
- Page 79 and 80: FIG. 2.20: The measured normalized
- Page 81 and 82: CHAPTER 3 The Jefferson Laboratory
- Page 83 and 84: FIG. 3.1: Schematic of the 10 kW FE
- Page 85 and 86: FIG. 3.2: Layout of the DC photocat
- Page 87 and 88: accelerating gradient at the front
- Page 89 and 90: eason for making the endloops achro
- Page 91 and 92: FIG. 3.7: Illustration of path leng
- Page 93 and 94: 3.5 Longitudinal Dynamics This sect
- Page 95 and 96: FIG. 3.9: The effect of a thin focu
- Page 97 and 98: Under the constraint that each orde
- Page 99 and 100: form of beam breakup not only occur
- Page 101: 4.1 The Pillbox Cavity Although the
- Page 105 and 106: where the full 4×4 transfer matrix
- Page 107 and 108: The threshold is inversely proporti
- Page 109 and 110: 4.3 BBU Simulation Codes: Particle
- Page 111 and 112: 6. The second pass beam bunch then
- Page 113 and 114: which excites it. The BBU instabili
- Page 115 and 116: Equation (4.41) is a dispersion rel
- Page 117 and 118: FIG. 4.4: Output from MATBBU showin
- Page 119 and 120: FIG. 4.5: Setup for measuring cavit
- Page 121 and 122: Consequently, depending on the exte
- Page 123 and 124: The projection of the beam displace
- Page 125 and 126: TABLE 4.1: Experimental measurement
- Page 127 and 128: FIG. 4.10: A plot showing the effec
- Page 129 and 130: these cryomodules. Modes from these
- Page 131 and 132: CHAPTER 5 Experimental Measurements
- Page 133 and 134: threshold current - preferably with
- Page 135 and 136: occurred at approximately 2 mA of a
- Page 137 and 138: FIG. 5.5: FFT of a pure 2106.007 MH
- Page 139 and 140: FIG. 5.6: Illustration to show the
- Page 141 and 142: 5.4 Measuring the Threshold Current
- Page 143 and 144: for the HOM-beam system and is deri
- Page 145 and 146: FIG. 5.10: Schematic of the experim
- Page 147 and 148: FIG. 5.12: A plot of 1/Qeff versus
- Page 149 and 150: measured HOMs in zone 3, a BTF meas
- Page 151 and 152: FIG. 5.16: HOM voltage measured fro
and is used to accelerate charged particles due to the fact that the electric field is<br />
strongest on-axis, ρ = 0.<br />
The TM110 modes are given by<br />
Ez(ρ, φ, z) = E0J1<br />
Hφ(ρ, φ, z) = iE0<br />
x11ρ<br />
R<br />
ɛ0ω110R<br />
x11<br />
<br />
J ′ 1<br />
R<br />
83<br />
<br />
cos(φ) (4.5)<br />
<br />
x11ρ<br />
<br />
cos(φ) (4.6)<br />
The electric field vanishes on-axis, but for small off-axis displacements grows linearly.<br />
These modes are typically referred to as dipole HOMs due to the behavior of the<br />
magnetic field. A plot of the electric and magnetic fields in ρ−φ space is shown in<br />
Fig. 4.2.<br />
With these field configurations in mind, the mechanism which facilitates BBU<br />
can be understood. The magnetic field deflects a particle on the first pass (even if<br />
it travels on-axis) and is transformed into a displacement, the magnitude of which<br />
depends on the machine optics, through the cavity on the second pass. As the par-<br />
ticle travels off-axis through the cavity, energy can be exchanged with the electric<br />
field depending on the phase of the beam relative to the field. Under certain condi-<br />
tions the beam can couple energy to the HOM which in turn more strongly deflects<br />
trailing particles traveling through the cavity. Hence a feedback loop is generated<br />
between the recirculated beam and the cavity dipole HOM fields which can become<br />
unstable if the average beam current exceeds the threshold current.<br />
4.2 Derivation of the BBU Threshold Current<br />
In this section, the threshold current is derived by equating the energy dissi-<br />
pated by the cavity to the energy deposited by the beam into the HOM [65]. In<br />
Section 4.4 an alternate derivation is outlined which uses the concept of the wake
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