摘要

電子束生命期和亮度一直是同步輻射光源用戶的考量。對中低能量的第三代同步輻射光源而言其電子束生命期主要受限於Touschek散射，而拉長電子束的縱向長度是可以在不影響亮度下，減低電子散射的機率，進而提升電子束的生命期。實際的作法是加入被動式諧波共振腔，利用電子束在諧波共振腔感應的高頻諧波電壓降低電子束團的回復力，拉長電子束長度。然而，非對稱性的電子束團填充模式會產生電子束團同步相位的改變，造成電子束團序列裡諧波電壓的不均勻，因而顯著地影響束團拉長的效果和生命期的有效延長。在之前的文獻，諧波共振腔因非對稱性電子填充模式造成的性能低落只能利用數值方式來模擬。造成性能低落的機制被推測是因為諧波共振腔的非線性效應，並且認為只能使用數值方式分析。
本論文撰寫一個程式可以重現前人的模擬結果，根據此程式，我們在有穩定解的假設下發展了解析解，用來分析電子束團拉長效果降低的機制。解析解的結果和前人文獻的模擬數據相當吻合，無論是常溫共振腔或低溫超導共振腔。此解析解解釋了非對稱性電子束團填充模式在有諧波共振腔的系統下會放大電子束團位移量的原因。其原因是來自於電子束團產生之電壓的變動和所需之電子束團同步相位的位移兩者之間互相競爭，以達到能量平衡狀態，意即每個束團從高頻產生器獲得的能量等於所需回補的能損。當被動式諧波共振腔操作在拉長電子束團的模式，電子束團如需要往獲得較大高頻能量的相位移動，其移動的同時會感應更多的電壓，意即損失較多的能量在共振腔，如此電子束團又得往獲得較大高頻能量的相位移動，直到兩者達成平衡。電子束團在共振腔感應的電壓變化的傾向和需求相反，因此需要更多的位移量來達到能量平衡。反之當被動式諧波共振腔操作在縮短電子束團的模式，電子束團在共振腔感應的電壓變化的傾向和需求一致，需要的位移量就小的多。

Abstract

For a low-to-medium energy 3rd generation synchrotron light source, the beam lifetime is dominated by Touschek scattering which can be improved by lengthening the bunch length. Physically, the longitudinal restoring force can be reduced by introducing a harmonic voltage generated by passive harmonic cavities. The gap transient induced by an asymmetric filling pattern of bunch train can cause a considerable shift of synchronous phase that result in a huge variation of the harmonic voltage along the bunch train. Therefore, the bunch lengthening as well as the lifetime improvement is greatly degraded owing to deviation of the harmonic voltage from its optimal target in a degree strongly dependent on the size of filling gap. In this thesis, a tracking code is developed which reproduces the simulation results of previous studies. An analytic model, under assumption of stationary circulation of bunch trains around the storage ring, is conceived benefiting from the tracking code as a test bench for probing the degradation mechanism of bunch lengthening. The analytic model explain that the amplification of synchronous phase shift by implementing the passive harmonic cavity with an asymmetric filling pattern of bunch train reflects a competition between the variation of beam induced harmonic voltage owing to transient beam loading and the required shift of synchronous beam phase to gain opposite amount of rf generator voltage, in addition for maintaining the energy balance of individual stationary bunches along the bunch train.

Contents

Abstract(Chinese) i
Abstract ii
Contents iv
List of Figures v
List of Tables vii
Chapter 1 Introduction 1
Chapter 2 Beam Dynamics with Passive Harmonic Cavity 5
2.1 Touschek Lifetime 5
2.2 Synchrotron Motion 6
2.3 Bunch Lengthening with Harmonic Cavity 12
2.4 Performance of Passive Harmonic Cavity 14
2.5 Beam Loading and Robinson Instability[9] 15
2.5.1 Equivalent Circuit of RF System 16
2.5.2 Static Beam Loading and Compensation 19
2.5.3 DC Robinson Instability 20
2.5.4 AC Robinson Instability 22
Chapter 3 Algorithm 26
3.1 Development of Tracking Code 26
3.2 Verification of the Tracking Code 28
3.2.1 Synchrotron Motion 29
3.2.2 DC Robinson& AC Robinson Instabilities 30
3.2.3 Gap Transient Effect 34
Chapter 4 Analytic Model and Discussion 38
4.1 Derivation of the Analytic Model 38
4.2 Verification of the Analytic Model 42
4.3 Phenomena Explanation with Analytic Model 46
4.4 Applicable Region of the Analytic Model 47
Chapter 5 Conclusions 49
Bibliography 51
Appendix 53
A. Tracking code 53
B. Code for analytic model 60

List of Figures

Figure 2.1 A diagram presenting Touschek effect 6
Figure 2.2 General mechanism of synchrotron motion in a storage ring 9
Figure 2.3 Phase diagram of synchrotron motion 11
Figure 2.4 The potential well and seperatrix in phase diagram 11
Figure 2.5 (a) Voltage seen by the beam versus the phase relative to synchronous particle. (b) The potential well without and with harmonic cavity. (c) The bunch density without and with harmonic cavity. 14
Figure 2.6 The amplitude and tuning angle of the cavity versus frequency. The cavity parameters are as follows: Q=36000, Rs=3MΩ,Wcav = 500MHz. 17
Figure 2.7 Circuit model representing an RF generator current source driving an RF cavity with a beam loading current . 18
Figure 2.8 Phasor diagram for an accelerating cavity and arbitrary tuning angle 20
Figure 2.9 The effect restoring force depends on the gradient of (a) The bunch sees a nonzero slope of . (b) The bunch lies at the crest of , where the slope is zero. Thus no restoring force to pull the bunch back to equilibrium position. 21
Figure 2.10 Qualitative treatment of the AC Robinson instability[16] 24
Figure 3.1 The reference plane of energy deviation and phase deviation and concept of turn by turn tracking. 27
Figure 3.2 (a) Phase deviation of the first bunch with radiation damping turn for TLS, Ib=300mA falls in DC Robinson stable region. (b) Phase deviation of the first bunch with radiation damping turn for TLS, Ib=700mA falls in DC Robinson unstable region. 31
Figure 3.3 (a) Synchrotron frequency versus beam current of NewSUBARU (b)Spectrum of NewSUBARU with varying currents. One observe the coherent synchronous frequency shift with beam current, while the incoherent is invariable. 32
Figure 3.4 Results of AC Robinson instability of TLS at beam current equals to 650mA. (a)Cavity frequency tuned above, i.e. positive tuning angle. (b)Cavity frequency tuned below, i.e. negative tuning angle 34
Figure 3.5 The synchronous phase deviation versus different gap percentages for TLS. The solid lines are plot from Peterson’s model. 35
Figure 3.6 A comparison with J.M. Byrd’s results. (a) Synchronous phase shift (b) Harmonic voltage Dashed lines are got from tracking code. 36
Figure 4.1 The illustration of beam voltages along the bunch train. 40
Figure 4.2 The flow chart for solving the analytic equations. 41
Figure 4.3 Simulation result of ALS with 17%gap and harmonic tuning angle is 85.8112°. (a) Error plotting versus guessing synchronous phase shift. (b) Harmonic voltage along the bunch train. 42
Figure 4.4 Simulation result of ELETTRA with 20%gap and harmonic tuning angle is 89.9985°. (a) Error plotting versus guessing synchronous phase shift. (b) Harmonic voltage along the bunch train. 43
Figure 4.5Convergence test of tracking code for superconducting cavity in ELETTRA. (a) The harmonic voltage with different initial conditions. (b) The harmonic phase with variant initial conditions. 45
Figure 4.6 Convergence test of analytic solution for superconducting cavity in ELETTRA. (a) The harmonic voltage with small perturbations. (b) The harmonic phase with small perturbations. 45
Figure 4.7 The amplification of gap effect. ALS parameters are used for both diagrams except the tuning angle is positive (85.8˚) in the left diagram and negative (-85.8˚) in the right diagram. 46
Figure 4.8 The diagram for amplification of gap shift with opposite cavity detuning 47
Figure 4.9 The phase deviation wiggles when the harmonic cavities are parked. 48



List of Tables

Table 2.1 Performance of harmonic cavity for lifetime improvement 15
Table 3.1 The machine parameters of different storage ring used in tracking code 29
Table 3.2 A comparison of tracking code and the real case for synchrotron frequency and rms bunch length. 30

Bibliography

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