STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

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M ∗ ≡ M12 cos 2 α + (M14 + M32) cos α sin α + M34 sin 2 α (4.19) The threshold current is defined as the condition for which the power dissipated by the cavity is exactly compensated by the power deposited by the beam and occurs when ˙ U = 0. Applying this condition to Eq. (4.18) yields qM Ith ∗ sin(ωTr) + 2ωpb 1 k 2 (Rd/Qo)QL and the threshold current for instability is given by 2Vb Ith = − k(Rd/Qo)QLM ∗ sin(ωTr) 87 = 0 (4.20) (4.21) where Vb = pb(c/q) is the beam voltage and M ∗ is defined in Eq. (4.19). Despite the simplicity of Eq. (4.21), it is important for gaining insights into the parametric dependence of the threshold current. The formula depends on parameters that characterize the electron beam, the machine optics and properties of the HOM. The threshold is directly proportional to the beam energy. That is, at higher energies, the beam is more rigid and will be deflected less for a given HOM angular kick. This implies that the front end of the linac, where the injected beam is only a few MeV, can be particularly susceptible to BBU. The threshold also depends on the details of the machine lattice. Specifically it is inversely proportional to the matrix elements that transform an HOM-induced kick on the first pass to a displacement on the second pass. Large values of these matrix elements (M12 or M34 with decoupled optics, for example) will lead to large off-axis displacements in the cavity and, given the correct phase sin(ωTr), the beam will deposit energy proportional to the displacement into the HOM. In fact, a poorly designed optics can contribute to a low threshold current just as surely as insuffi- ciently damped HOMs.
The threshold is inversely proportional to the HOM impedance and is also sensitive to the polarization of the mode. The ratio (Rd/Qo) is a quantity that depends purely on the geometry of the cavity structure. The goal in designing RF cavities for high current applications is to provide strong HOM damping and thereby decrease the QL of HOMs. For a physically viable solution, the threshold current must be positive, which requires that the quantity M ∗ sin(ωTr) be less than zero. The regions for which the formula is valid are discussed more fully in Section 4.2.1. As a brief aside, note that there can sometimes arise confusion when it comes to defining the ratio (R/Q) for dipole HOMs [66]. Codes such as MAFIA track particles a distance, a, off-axis through the cavity and use the accelerating voltage to compute (Rd/Qo) as Rd Qo = V 2 a ωU 88 (4.22) with units of Ω/cm 2 . This can be misleading because it implies that (Rd/Qo) is a function of the off-axis displacement. The convention throughout this dissertation uses the definition Rd Qo = V 2 ⊥ ωU (4.23) where V⊥ is the transverse deflecting voltage and (Rd/Qo) is in units of Ohms. The transverse voltage is proportional to the magnetic field, which for a TM110 mode, is constant for small values of the off-axis displacement. Converting between the two definitions is achieved using the relation where k (= ω/c) is the wavenumber. Va = (ka)V⊥ (4.24)
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STUDIES OF ENERGY RECOVERY LINACS A
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DEDICATION To my wife Danielle and
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2 CEBAF with Energy Recovery . . .
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5.2 HOM Power . . . . . . . . . . .
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ACKNOWLEDGMENTS First and foremost
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6.1 Summary of the measured effects
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2.10 Illustration of quadrupole sca
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5.1 Successive frames in time (prog
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6.8 A plot of 1/Qeff versus average
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ABSTRACT An energy recovering linac
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CHAPTER 1 Introduction An increasin
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FIG. 1.1: Schematic of a generic li
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FIG. 1.2: A CEBAF 5-cell cavity wit
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The solution to Eq. (1.3) is U(t) =
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y reducing the impedance of HOMs, a
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Despite its success, this method of
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design parameters, most notably ach
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1.4.2 Machine Optics The second cat
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analytic model elucidates many impo
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CHAPTER 2 CEBAF with Energy Recover
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FIG. 2.1: Energy versus average cur
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FIG. 2.3: Additional hardware insta
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FIG. 2.4: A picture of the energy r
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dipoles and beam diagnostics such a
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FIG. 2.7: Horizontal (red) and vert
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FIG. 2.8: Illustration of the cryom
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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 and 102: 4.1 The Pillbox Cavity Although the
Page 103 and 104: FIG. 4.2: Electric field (red) and
Page 105: where the full 4×4 transfer matrix
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
Page 153 and 154: FIG. 5.18: A plot of the three valu
Page 155 and 156: the beam’s response in regions wh
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CHAPTER 6 BBU Suppression: Beam Opt
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FIG. 6.1: Schematic of a FODO cell
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plane [85]. Equations (6.7) and (6.
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6.2.3 Discussion The method of poin
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FIG. 6.3: Beam envelopes (horizonta
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FIG. 6.6: Beam position monitor rea
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FIG. 6.8: A plot of 1/Qeff versus a
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⎛ ⎞ ⎜ ⎝ 0 0 0 0 0 −1/K 0
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FIG. 6.11: A plot of 1/Qeff versus
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FIG. 6.12: Threshold current for no
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FIG. 6.14: Threshold current utiliz
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TABLE 6.1: Summary of the measured
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CHAPTER 7 BBU Suppression: Feedback
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FIG. 7.1: A schematic of the feedba
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FIG. 7.4: A coaxial 3-stub tuner us
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All of these considerations are con
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FIG. 7.6: Generic layout for a feed
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in Section 4.2.1, however, in the p
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FIG. 7.7: The threshold current as
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FIG. 7.8: Threshold current versus
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FIG. 7.10: The threshold current as
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CHAPTER 8 Conclusions The work pres
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le were experimentally measured. Du
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APPENDIX A The Pillbox Cavity Start
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FIG. A.1: A pillbox cavity exhibiti
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Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ
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FIG. B.1: Relationship of the S-par
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FIG. C.1: Impedance and frequency o
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FIG. C.5: Impedance and frequency o
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BIBLIOGRAPHY [1] M. Tigner, Nuovo C
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[22] L. Merminga, in Proceedings of
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[50] C. Hernandez-Garcia et al., in
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[79] L. Merminga et al., in Proceed
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VITA Christopher D. Tennant Christo
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