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voltages from these cavities and taking the FFT of the signals yielded the frequencies 1786.206 MHz and 1881.481 MHz for cavities 3 and 8, respectively. According to the results of earlier HOM measurements, these two modes have relatively low impedances and simulations predict that the threshold current due to these modes is at least an order of magnitude higher than that of the 2106 MHz mode in cavity 7. From analyzing the data, it was discovered that the Schottky diode signals from cavities 3 and 8 had nearly the same growth rate as the signal from cavity 7. This suggested that the other modes are being driven by the 2106 MHz mode after it goes unstable. At the onset of BBU, the transverse beam displacement is deflected at the frequency of 2106.007 MHz. This frequency is aliased to sideband frequencies which, for a bunch repetition frequency of 37.425 MHz, appear at ±10.207 MHz around the beam harmonics. As the instability grows the sidebands become sufficiently strong to the point that they are able to resonantly excite modes which lie at the sideband frequencies. To within several kilohertz (the error in the measurement of the frequencies), the 1786.206 MHz mode corresponds to the lower sideband frequency of the 48 th beam harmonic (48 × 37.425 MHz - 10.207 MHz) while the 1881.481 MHz mode corresponds to the upper sideband frequency of the 50 th beam harmonic (50 × 37.425 MHz + 10.207 MHz). This effect is illustrated in Fig. 5.6. Note that the 1786 MHz and 1881 MHz modes themselves are not unstable, but rather they are driven unstable by the 2106 MHz mode. This phenomenon of sidebands driving otherwise stable HOMs unstable was verified through simulations. Verification by Simulations Consider a bunch repetition frequency of 1497 MHz so that the sidebands produced when the 2106 MHz mode becomes unstable do not lie at the frequencies of 119 1786.206 MHz and 1881.481 MHz. Also consider a bunch frequency of 37.425 MHz
FIG. 5.6: Illustration to show the effect of sideband frequencies driving otherwise stable modes unstable. for which, as previously mentioned, the sidebands produced by the 2106 MHz mode lie exactly at 1786.206 MHz and 1881.481 MHz. Figure 5.7 shows a plot of the HOM voltage squared of four modes in zone 3. The simulation was performed with an average beam current that exceeds the threshold current. Therefore the voltage corresponding to the 2106 MHz mode grows rapidly. The other modes plotted correspond to 1786 MHz, 1881 MHz and 2114 MHz (which is pseudo-stable because M ∗ sin(ωTr) > 0). While they exhibit growth, the magnitude of their voltages are 14, 12 and 9 orders of magnitude, respectively, less than that of the 2106 MHz mode after 15 ms. Figure 5.8 shows the results of repeating the simulation while changing only the bunch frequency from 1497 MHz to 37.425 MHz. Modes that correspond to the sideband frequencies generated by the 2106 MHz mode are now resonantly driven. After 15 ms, the magnitude of the voltages for the 1786 MHz and 1881 MHz modes are now only 5 and 3 orders of magnitude, respectively, less than that of the 2106 MHz mode while the magnitude of the voltage for the 2114 MHz mode remains nearly 9 orders of magnitude less than that of 2106 MHz. Thus only HOM frequencies that correspond to these sideband frequencies are affected. 120
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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
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 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: FIG. 5.5: FFT of a pure 2106.007 MH
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
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
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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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