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emittances are related simply by ɛN = βγɛg 37 (2.6) where β = v/c and γ = 1/ 1 − β 2 is the Lorentz factor. In an ideal machine the normalized emittance would remain constant from the source through the entire transport. In reality non-linear magnetic fields and wakefields, among other things, act to degrade the emittance. Emittance measurements were made for three different beam regimes; the beam in the injector, the first pass accelerating beam in each of the arcs and the beam after energy recovery. Measuring the emittance of the energy recovered beam proved to be most challenging as it had to be performed in the presence of the accelerating beam. The following section describes the details of this measurement. 2.4.1 Emittance Measurement Using a Quadrupole Scan One of the most common beam line configurations used to make emittance measurements consists of a quadrupole followed by a drift of length L to a beam profile monitor. The premise of the quadrupole scan emittance measurement is to determine the horizontal (or vertical) beam size with the profile monitor as function of the strength of the upstream quadrupole. Performing the measurement for three different quadrupole strengths is sufficient to calculate the horizontal (or vertical) emittance. Denote the betatron functions just prior to entrance of the quadrupole as β1 and α1 and those at the downstream observation point by β2 and α2. For a non- dispersive region the beam size squared is σ 2 2 = β2ɛg (2.7)
where σ2 is the rms beam size measured by a beam profile monitor and ɛg is the rms geometrical emittance. By knowing how the Twiss parameters propagate, β2 can be related to the beta function upstream, β1, via ⎛ ⎜ ⎝ β2 α2 γ2 ⎞ ⎛ ⎟ ⎠ = ⎜ ⎝ M 2 11 −2M11M12 M 2 12 −M11M21 M12M21 + M11M22 −M12M22 M 2 21 −2M21M22 M 2 22 ⎞ ⎛ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ β1 α1 γ1 ⎞ ⎟ ⎠ 38 (2.8) where the Mij are the elements of the transfer matrix that propagates beam from the quadrupole to the wire scanner. Using the result of Eq. (2.8) in Eq. (2.7) gives σ 2 2 = β2ɛg = ɛg 2 M11β1 − 2M11M12α1 + M 2 12γ1 (2.9) For the specific case of a quadrupole-drift, the Mij elements are found by multiplying the transfer matrices for a quadrupole (in the thin lens approximation) and a drift of length L ⎛ ⎜ ⎝ ⎞ ⎛ 1 L ⎟ ⎜ 1 ⎠ ⎝ 0 0 1 1 1 f ⎞ ⎛ ⎟ ⎜ ⎠ = ⎝ 1 + L f 1 f L 1 ⎞ ⎟ ⎠ (2.10) Plugging the appropriate matrix elements from Eq. (2.10) into Eq. (2.9), the relationship between the beam size and the beta function prior to the quadrupole entrance can be expressed as σ 2 2 = β2ɛg = ɛg 1 + L 2 β1 − 2L 1 + f L α1 + L f 2 γ1 (2.11) Inspection of Eq. (2.11) shows that the beam size squared varies quadratically with the quadrupole strength, k (= 1 ). Measuring the beam size for three values of the f quadrupole strength results in three equations which is sufficient to solve for the three unknowns; (β1ɛg), (α1ɛg) and (γ1ɛg). This process is known as a quadrupole
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STUDIES OF ENERGY RECOVERY LINACS A
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DEDICATION To my wife Danielle and
Page 5 and 6: 2 CEBAF with Energy Recovery . . .
Page 7 and 8: 5.2 HOM Power . . . . . . . . . . .
Page 9 and 10: ACKNOWLEDGMENTS First and foremost
Page 11 and 12: 6.1 Summary of the measured effects
Page 13 and 14: 2.10 Illustration of quadrupole sca
Page 15 and 16: 5.1 Successive frames in time (prog
Page 17 and 18: 6.8 A plot of 1/Qeff versus average
Page 19 and 20: ABSTRACT An energy recovering linac
Page 21 and 22: CHAPTER 1 Introduction An increasin
Page 23 and 24: FIG. 1.1: Schematic of a generic li
Page 25 and 26: FIG. 1.2: A CEBAF 5-cell cavity wit
Page 27 and 28: The solution to Eq. (1.3) is U(t) =
Page 29 and 30: y reducing the impedance of HOMs, a
Page 31 and 32: Despite its success, this method of
Page 33 and 34: design parameters, most notably ach
Page 35 and 36: 1.4.2 Machine Optics The second cat
Page 37 and 38: analytic model elucidates many impo
Page 39 and 40: CHAPTER 2 CEBAF with Energy Recover
Page 41 and 42: FIG. 2.1: Energy versus average cur
Page 43 and 44: FIG. 2.3: Additional hardware insta
Page 45 and 46: FIG. 2.4: A picture of the energy r
Page 47 and 48: dipoles and beam diagnostics such a
Page 49 and 50: FIG. 2.7: Horizontal (red) and vert
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: 2.4 Transverse Emittance One of the
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 and 106: 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
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FIG. 5.18: A plot of the three valu
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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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