STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

More documents

Recommendations

Info

enchmarked against both existing codes and was the primary code used to study the effects of BBU in the FEL Upgrade Driver. 4.4 Alternate Derivation of the BBU Threshold Current As electron bunches travel through the accelerator, they generate electromag- netic fields which interact with their surrounding environment (e.g. the vacuum chamber and RF cavities) and are called wakefields. After passing through an RF cavity the transverse momentum of a trailing test charge is affected by the fields generated by the source particle. Consider a wakefield generated by an exciting charge, qe, traveling with coordinates r = (x ′ , y ′ , z ′ ) which, in turn, applies a force to the test charge, qt, following at a distance cτ behind the source. Restricting the beam motion to the x direction, the wake force is c qt The wake function is defined as dpx dz = Ex(r, z/c + τ; d) − cBy(r, z/c + τ; d) (4.28) W1(τ) ≡ 93 c ∆px(τ, d) (4.29) qeqtd Solving for the change in transverse momentum in Eq. (4.28) and plugging into Eq. (4.29) yields W1(τ) = 1 ∞ Ex(r, z/c + τ; d) − cBy(r, z/c + τ; d) qed −∞ dz (4.30) For several cases, analytic solutions of the wake function can be derived. The net effect on the beam due to wakefields is described by the wake potential and is determined by the convolution of the wake function with the charge distribution
which excites it. The BBU instability is generated by dipole HOMs which couple to the beam through the dipole moment of the current. Thus the net effect on the beam, imparted by the transverse deflecting voltage, is given by the integral [67] V (t) = t −∞ W1(t − t ′ )I(t ′ − Tr)r2(t ′ )dt ′ 94 (4.31) where W1(t) is the delta functional dipole wake function and I(t)r2(t) is the dipole moment of the beam bunch when it passes the cavity at time t. Equation (4.31) represents the starting point for an alternate derivation of the BBU threshold current using a wake function formalism. For a single dipole higher-order mode with angular frequency ω and loaded quality factor QL, the long-range wake function can be expressed analytically as W1(t − t ′ ) = Rd Qo ωk 2 e− ω 2Q L (t−t ′ ) sin(ω(t − t ′ )) (4.32) where k is the wavenumber. The bunch displacement at the cavity on the second pass can be written in terms of the voltage kick due to the accumulated wake excited by all preceding bunches and in terms of the appropriate matrix elements describing a single recirculation from the cavity back to itself as r2(t ′ ) = V (t′ − Tr) (M12 cos 2 α + (M14 + M32) sin α cos α + M34 sin 2 α) (4.33) Vb where α is the mode polarization and Vb = pb(c/q) is the beam momentum at the cavity. The beam bunches are approximated by delta functions I(t ′ − Tr) = Ioto δ(t ′ − Tr − nto) (4.34) n where to is the bunching period. Plugging in Eqs. (4.32), (4.33) and (4.34) into Eq. (4.31) leads to an integral equation. Assuming a normal mode solution of the form
Page 1 and 2:
STUDIES OF ENERGY RECOVERY LINACS A
Page 3 and 4:
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 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 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: 6. The second pass beam bunch then
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
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 and 204:
APPENDIX A The Pillbox Cavity Start
Page 205 and 206:
FIG. A.1: A pillbox cavity exhibiti
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
show all

STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?