STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

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Equations (A.5) and (A.6) are sometimes referred to as the Helmholtz equations. Assume solutions of the following form, which can be used to describe either traveling or standing waves E(x, y, z) = E(x, y)e ±ikz H(x, y, z) = H(x, y)e ±ikz 185 (A.7) (A.8) Note that the Laplacian operator can be separated into a transverse and longitudinal component and rewritten as ∇ 2 = ∇ 2 t + ∂2 ∂z (A.9) Using Eq. (A.9) and the form of the solutions in Eq. (A.7) and Eq. (A.8), the Helmholtz equation reduces to a 2D wave equation where 2 ∇t + γ 2 ⎧ ⎪⎨ ⎪⎩ γ 2 ≡ ɛoµoω 2 − k 2 ⎫ E ⎪⎬ = 0 (A.10) H ⎪⎭ (A.11) It is useful to rewrite the Maxwell equations in terms of components parallel and transverse to the z-axis. The transverse components of E and H can then be expressed in terms of Ez and Hz, thereby simplifying the problem considerably Et = i γ2 k∇t Ez − µ0ω ˆz × ∇t Hz Ht = i γ 2 k∇t Hz + ɛ0ω ˆz × ∇t Ez (A.12) (A.13)
FIG. A.1: A pillbox cavity exhibiting azimuthal symmetry. The resonator has a cross sectional radius R and a length d. Considering only TM modes, for which Hz = 0 everywhere, Eq. (A.12) and Eq. (A.13) reduce to Et = ik γ 2 ∇t Ez Ht = iɛ0ω γ 2 ˆz × ∇t Ez 186 (A.14) (A.15) Thus given the longitudinal electric field component, Ez, the remaining electric and magnetic field components can be derived. Up to this point, a cylindrical waveguide of arbitrary (but constant) cross section has been considered. An accelerating cavity is a resonator which is created by placing end plates at z = 0 and z = d as shown in Fig. A.1. Due to the reflections at the end plates, standing waves are created and the z-dependence can be described by
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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
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2.4 Transverse Emittance One of the
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where σ2 is the rms beam size meas
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eams. The effects of varying the qu
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FIG. 2.12: A typical wire scan near
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quadratic fit and a multiple regres
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ting the data is difficult. Without
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primary source of error is measurin
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identified, although the phase dela
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TABLE 2.3: Comparison of Twiss para
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the results of the fits. The vertic
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FIG. 2.18: Schematic illustrating t
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FIG. 2.19: The GASK signal measured
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FIG. 2.20: The measured normalized
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CHAPTER 3 The Jefferson Laboratory
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FIG. 3.1: Schematic of the 10 kW FE
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FIG. 3.2: Layout of the DC photocat
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accelerating gradient at the front
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eason for making the endloops achro
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FIG. 3.7: Illustration of path leng
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3.5 Longitudinal Dynamics This sect
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FIG. 3.9: The effect of a thin focu
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Under the constraint that each orde
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form of beam breakup not only occur
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4.1 The Pillbox Cavity Although the
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FIG. 4.2: Electric field (red) and
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where the full 4×4 transfer matrix
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The threshold is inversely proporti
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4.3 BBU Simulation Codes: Particle
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6. The second pass beam bunch then
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which excites it. The BBU instabili
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Equation (4.41) is a dispersion rel
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FIG. 4.4: Output from MATBBU showin
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FIG. 4.5: Setup for measuring cavit
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Consequently, depending on the exte
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The projection of the beam displace
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TABLE 4.1: Experimental measurement
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FIG. 4.10: A plot showing the effec
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these cryomodules. Modes from these
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CHAPTER 5 Experimental Measurements
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threshold current - preferably with
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occurred at approximately 2 mA of a
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FIG. 5.5: FFT of a pure 2106.007 MH
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FIG. 5.6: Illustration to show the
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5.4 Measuring the Threshold Current
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for the HOM-beam system and is deri
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FIG. 5.10: Schematic of the experim
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FIG. 5.12: A plot of 1/Qeff versus
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measured HOMs in zone 3, a BTF meas
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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
Page 157 and 158: CHAPTER 6 BBU Suppression: Beam Opt
Page 159 and 160: FIG. 6.1: Schematic of a FODO cell
Page 161 and 162: plane [85]. Equations (6.7) and (6.
Page 163 and 164: 6.2.3 Discussion The method of poin
Page 165 and 166: FIG. 6.3: Beam envelopes (horizonta
Page 167 and 168: FIG. 6.6: Beam position monitor rea
Page 169 and 170: FIG. 6.8: A plot of 1/Qeff versus a
Page 171 and 172: ⎛ ⎞ ⎜ ⎝ 0 0 0 0 0 −1/K 0
Page 173 and 174: FIG. 6.11: A plot of 1/Qeff versus
Page 175 and 176: FIG. 6.12: Threshold current for no
Page 177 and 178: FIG. 6.14: Threshold current utiliz
Page 179 and 180: TABLE 6.1: Summary of the measured
Page 181 and 182: CHAPTER 7 BBU Suppression: Feedback
Page 183 and 184: FIG. 7.1: A schematic of the feedba
Page 185 and 186: FIG. 7.4: A coaxial 3-stub tuner us
Page 187 and 188: All of these considerations are con
Page 189 and 190: FIG. 7.6: Generic layout for a feed
Page 191 and 192: in Section 4.2.1, however, in the p
Page 193 and 194: FIG. 7.7: The threshold current as
Page 195 and 196: FIG. 7.8: Threshold current versus
Page 197 and 198: FIG. 7.10: The threshold current as
Page 199 and 200: CHAPTER 8 Conclusions The work pres
Page 201 and 202: le were experimentally measured. Du
Page 203: APPENDIX A The Pillbox Cavity Start
Page 207 and 208: Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ
Page 209 and 210: FIG. B.1: Relationship of the S-par
Page 211 and 212: FIG. C.1: Impedance and frequency o
Page 213 and 214: FIG. C.5: Impedance and frequency o
Page 215 and 216: BIBLIOGRAPHY [1] M. Tigner, Nuovo C
Page 217 and 218: [22] L. Merminga, in Proceedings of
Page 219 and 220: [50] C. Hernandez-Garcia et al., in
Page 221 and 222: [79] L. Merminga et al., in Proceed
Page 223 and 224: VITA Christopher D. Tennant Christo
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