STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA
187 Ez(z) = A sin kz + B cos(kz) (A.16) The boundary conditions require that the tangential components of E must vanish at z = 0 and z = d. These conditions are satisfied when Ez(ρ, φ, z) = Eo cos pπz ψ(ρ, φ) (p = 0, 1, 2, ...) (A.17) d where ψ(ρ, φ) describes the azimuthal dependence of the fields. It follows from Eq. (A.14) and Eq. (A.15) that the transverse components of the electric and magnetic field are given by Et = − pπ sin dγ2 Ht = iɛoω cos γ2 pπz d pπz d ∇tψ(ρ, φ) (A.18) [ˆz × ∇tψ(ρ, φ)] (A.19) The problem can be simplified further by taking advantage of the azimuthal sym- metry in the problem and writing ψ(ρ, φ) = ψ(ρ)e ±imφ where m is an integer. Substituting Eq. (A.20) into Eq. (A.10) yields (A.20) d2ψ 1 dψ + dρ2 ρ dρ + γ 2 − m2 ρ2 ψ = 0 (A.21) which is Bessel’s equation and whose solutions are Bessel functions of order m and denoted as Jm(γρ). One of the solutions diverges for ρ = 0 which is physically unacceptable and so this solution is disregarded. as From Eq. (A.17), Eq. (A.20) and the solution to Eq. (A.21), Ez can be written
Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ)e imφ 188 (A.22) Applying the condition that at Ez(ρ = R) = 0 requires Jm(γR) = 0. Define xmn ≡ γmnR as the n th root of the Bessel function of order m. Finally, the two components of primary interest, Ez and Hφ for TMmn0 modes, are given by Ez(ρ, φ, z) = E0Jm Hφ(ρ, φ, z) = = iE0 xmnρ iɛoωmn0 cos(mφ) R ∂Ez ∂ρ (A.23) (A.24) cos(mφ) (A.25) γ2 mn ɛ0ωmn0R J xmn ′ xmnρ m R where the eigenfrequencies are determined by combining Eq. (A.11) and the defini- tion of xmn to give ωmn0 = 1 xmn √ µoɛo R The two important properties of TM110 modes in particular, are that 1. Ez ∝ ρ 2. Hφ is nonzero for ρ = 0 (A.26) The first property is due to the fact that J1 is linear for small values of ρ which means that Ez grows linearly with off-axis displacement. The second property implies that a beam bunch traveling on-axis can still be deflected by the magnetic field since J ′ 1(0) = 0.
- 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: FIG. A.1: A pillbox cavity exhibiti
- 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
Ez(ρ, φ) = ψ(ρ, φ) = E0Jm(γρ)e imφ<br />
188<br />
(A.22)<br />
Applying the condition that at Ez(ρ = R) = 0 requires Jm(γR) = 0. Define<br />
xmn ≡ γmnR as the n th root of the Bessel function of order m. Finally, the two<br />
components of primary interest, Ez and Hφ for TMmn0 modes, are given by<br />
Ez(ρ, φ, z) = E0Jm<br />
Hφ(ρ, φ, z) =<br />
= iE0<br />
xmnρ<br />
iɛoωmn0<br />
<br />
cos(mφ)<br />
R<br />
∂Ez<br />
∂ρ<br />
<br />
(A.23)<br />
(A.24)<br />
cos(mφ) (A.25)<br />
γ2 mn <br />
ɛ0ωmn0R<br />
J<br />
xmn<br />
′ <br />
xmnρ<br />
m<br />
R<br />
where the eigenfrequencies are determined by combining Eq. (A.11) and the defini-<br />
tion of xmn to give<br />
ωmn0 = 1 xmn<br />
√<br />
µoɛo R<br />
The two important properties of TM110 modes in particular, are that<br />
1. Ez ∝ ρ<br />
2. Hφ is nonzero for ρ = 0<br />
(A.26)<br />
The first property is due to the fact that J1 is linear for small values of ρ which means<br />
that Ez grows linearly with off-axis displacement. The second property implies that<br />
a beam bunch traveling on-axis can still be deflected by the magnetic field since<br />
J ′ 1(0) = 0.
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