#### Transcript Constant Gradient Structure

Linear Accelerator (LINAC) Juwen Wang 王聚文 SLAC National Accelerator Laboratory July 30, 2014 全球华人物理和天文学会 第九届加速器学校 Xiuning, Anhui Outline 1. Introduction: Brief History and RF Accelerator System. 2. Basic Ideas: Modes, Dispersion Curves and Structures Types. 3. RF Parameters for Accelerating Mode: Shunt Impedance, Q, Filling Time, Phase & Group Velocity, Transient Time Factor, Attenuation Factor, Coupling Coefficient. 4. Basic Beam Dynamics: Acceleration, Bunching and Beam Loading. 5. Wakefield: Longitudinal and Transverse Wakefield. 6. How to Make a Linac: Machining, Chemical Cleaning, Diffusion Bonding & Brazing, Tuning & Microwave Measurement, Vacuum Baking, Fiducialization, High Power Processing. * .Some topics are mainly for room temperature RF structures. 2 1. Introduction • High Voltage Linac and RF Linac. • Brief History of RF Linac. • Building Blocks of RF Linac 3 High Voltage Accelerator and Radio Frequency RF Accelerator Van de Graff Accelerator RF Accelerator 4 Brief History of RF Linac The first formal proposal and experimental test for a RF linac was by Rolf Wideröe in 1928. The linear accelerator for scientific application did not appear until after the development of microwave technology in World War II, stimulated by Radar program. 1955 Luis Alvarez at UC Berkeley, Drift-Tube Linac (DTL). 1947 W. Hansen at Stanford, Disk-loaded waveguide linac. 1970 Radio Frequency Quadruple (RFQ) 5 Building Blocks of RF Linac RF Control System Vacuum System Water Cooling System RF Power System 6 2. Basic Ideas • Electromagnetic Wave and Waveguide • Wave Propagation Equations • Waveguide Modes • Dispersion Properties • Periodic Structure and its Dispersion Properties 7 Basic Data and Formulae E total Energy E0 rest energy W kinetic energy m mass, m0 rest mass γ relative mass factor c speed of light v velocity β normalized velocity Speed of light c 2.998x108 m/s Elementary charge e 1.6x10-19 C Electron mass me 0.51 MeV/c2 Proton mass mp 938.3 MeV/c2 p momentum FLorentz Lorentz force q electrical charge 𝐸 electric field 𝐵 magnetic field 8 Electron and Proton Velocities vs. Kinetic Energies 9 Wave Propagation Equations Wave equation for propagation characteristics: 2 Ek E 0 2 k / c 2 / ω is the angular frequency 2πf and k is wave number (radian per unit length) of plane wave in free space. In a cylindrically symmetric waveguide, the transverse magnetic field does not have φ dependence for most simple accelerating modes. All field components can be derived from Ez in cylindrical coordinates and it satisfies the wave equation: 2 Ez r 2 2 1 E z 2 E z 0 r r c β=2π/g is propagation constant and g is called guide wavelength. For perfect metal boundary condition at waveguide wall, Ez = 0, this boundary condition decides the ωc, which is called cut-off frequency. 2 2 c kc c c We will use that ω and β to characterize the wave properties – Dispersion Property 2 2 10 Solution of Wave equation The solution for TM01 mode (lowest mode) is as followings: E z E0 J 0 kc r e j (t z ) Er jE0 1 (c / ) 2 J1 (kc r )e j (t z ) H jE0 J1 ( kc r )e j (t z ) On the axis: J 0 (0) 1 E z (0) Max. On the wall: J 0 (kc b) 0 E z (b) 0 The first zero of J0: Hf EZ r kc b 2.405 c kc c 2.405c / b For example, in a cylinder with wall diameter 2r = 2b= 9 cm, fc = 2.55 GHz, A electromagnetic wave with f =2.856 GHz (S-Band) 𝜷 = 𝟐𝝅 𝒇 𝒄 𝟐 𝒇𝒄 − 𝒄 𝟐 >𝟎 Therefore, this wave can propagate as TM01 mode in this cylindrical waveguide. 11 Simple TM01 Mode 50 0 0 0.2 J 0 kc r J1 kc r 0.5 1 /c r 1.5 2 0.1 Bf (T) For a case of ω ~ 1.2ωc EZ (MV/m) 100 0 2.5 12 Dispersion Curve for TM01 Mode in a Cylinder The phase velocity Vp is the speed of RF field phase along the accelerator, it is given by Vp Vp>c Group velocity is defined as energy propagation velocity. For wave composed of two components with different frequency ω1 and ω2 wave number β1 and β2 , the wave packet travels with the velocity: Vg 1 2 d 1 2 d Hyperbola ω-β diagram for guided wave in a uniform (unloaded) waveguide. For uniform waveguide, it is easy to find: Vp c vp=c VpVg c2 2 kc 2 c 2 In order to use RF wave to accelerate particle beam, it is necessary to make simple cylinder “loaded”. The variety of accelerator structures have been created. 13 Dispersion Curves for Periodic Structures Brillouin (ω-β) diagram showing propagation characteristics for uniform and periodically loaded structures with load period d. Floquet Theorem: When a structure of infinite length is displaced along its axis by one period 2π/d, it can not be distinguished from original self. For a mode with eigen frequency ω: jd E( r , z d ) e E( r , z) r x x y y where βd is called phase advance per period. Make Fourier expansion for most common E z an J 0 (k rn r )e j (t nt ) accelerating TM01 mode: Each term is called space harmonics. 2 2n 2n 2 2 k k n 0 14 rn n The propagation constant is d VPo d Discussion on Some Facts for Better Understanding • It is interesting to notice that for the fundamental harmonic n = 0 travels with Vp = c, then kr0 = 0 , β0 = k and J0(kr0 r)=1, the acceleration is independent of the radial position for all synchronized particles. • Each mode with specific eigenfrequency has unique group velocity for all space harmonics. The total field pattern or distribution is decided by the coefficients of those components – decided by the cell profiles like iris size, disk thickness. (later, we will see how these space harmonics add together). •We need design the structure have higher effective fundamental harmonics (later, we will talk about high transient factor structures) • Every higher order space harmonics does not have contribution to acceleration, but takes RF power. (later, we will know that they contribute to wakefield). 15 Electrical Field Patterns for Periodic Structures with Different Modes Mode is defined as the “phase shift” or “phase advance” per structure period: Phase shift / cavity = 2π/(cavity number per wavelength) 16 SW & TW Structures pulsed RF Power source d pulsed RF Power source Constant Impedance Structure (CI) Standing Wave Structure d Traveling Wave Structure RF load Constant Gradient Structure (CG) 17 Evolution from Single to Bi-period Structures π π π 18 3. RF Parameters for Accelerating Mode • Shunt impedance, • Q, • Filling time, • Phase & Group velocity, • Transient Time Factor, • Attenuation factor, • Coupling coefficient. 19 Particle Acceleration in a Cavity Time various accelerating field Acceleration integrated in a cavity Ez Ez (r, z, t ) Ez (r, z) cos(t f ) L/2 W q L/2 E dz q E ((0, z) cos(t ( z) f )dz z L / 2 z L / 2 Choose the field is maximum while particle in the center of cavity (z=0) L/2 E ( 0 , z ) cos( t ) dz L / 2 L/ 2 z qV T W q E z (0, z )dz 0 L / 2 L / 2 E z (0, z )dz L / 2 L/2 T is defined is as the Transient Time Factor E (0, z) cos(t )dz z T L / 2 L / 2 Ez (0, z)dz L / 2 -- True integration of acceleration -- Integration of amplitude of acceleration field 20 Main RF Parameters – Shunt Impedance Shunt impedance per unit length r and R for structure length L which measure the accelerating quality of a structure and is defined as 2 Ea r Unit of MΩ/m or Ω/m dP / dz where Ea is the synchronous accelerating field amplitude and dP/dz is the RF power dissipated on the accelerator walls per unit length; for a certain structure with length of L, the shunt impedance is 2 Ea L2 V2 R (dP / dz) L Pd Ez (r, z) r 0 Ez (0, z) 2 Unit of MΩ or Ω (some code called z0) 2 L / 2 L / 2 E ( 0 , z ) cos( t ) dz E ( 0 , z ) dz z z 2 Ea T 2 L / 2 zT 2 r L / 2 2 2 dP / dz dP / dz L dP / dz L L/2 E (0, z)Costdz z T where T is defined as L / 2 L / 2 E (0, z)dz z L / 2 often it is calculated and listed in some codes. 21 Main RF Parameters – Factor of Merit and Group Velocity Factor of merit Q, which measures the quality of an RF structure as a resonator. • For standing wave structure W is the RF energy stored in a cavity. Q W Pd w • For traveling wave structure, w is the Q dP / dz RF energy stored per unit length and dP/dz is power dissipated per unit length. Group velocity Vg, which is the speed of RF energy flow along a TW accelerator: Power flow = Group velocity times Stored energy per unit length: P Vg w Vg P P d w QdP / dz d 22 Main RF Parameters – Attenuation Factor Attenuation factor ԏ, which is the measure of power reduction due to RF Ohm loss along a Traveling Wave accelerator. dE ( z ) E dz P out e 2 Pin α(z) is the attenuation coefficient in nepers per unit length. dP 2 ( z ) P dz ԏ is the attenuation factor in nepers of the total structure. • For a constant-impedance section with a length L, the attenuation is uniform: dP / dz w / Q const 2P 2Vg w 2Vg Q L L 2Vg Q • For a constant-gradient section (E=const): the attenuation constant α is a function of z: Pin (1 e 2 ) dP / dz 2 ( z ) P const L • For any non-uniform structures: ( ( z) dP 2P const ) dz It is not a constant 2Vg ( z )Q as a function. L ( z )dz 0 23 Main RF Parameters – Filling Time Filling time tF • For traveling wave structure, the field builds up “in Space”. The filling time is the time needed to fill the whole section of either constant impedance or constant gradient, which is given by L L L dz Q dp / dz Q 2Q dz 2 ( z )dz V P g 0 0 0 tF Before to talk about the filling time for standing wave cavity, let briefly discuss about coupling parameters for standing wave cavity in a coupling system including cavity and external circuit. System Q or Loaded QL=(stored Energy)/Dissipated energy in both cavity and external circuit): Coupling coefficient β of the cavity to the input microwave network is Q 0 Qe P P 1 1 1 0 e QL W0 W0 Q0 Qe Q0 is the unloaded Q value, Qe is external Q value, QL is loaded Q value, QL W0 P0 Pe Q0 = ωW0 /P0 , Qe = ωW0 /Pe = Q0/β , QL = ωW0 /(P0+Pe)=Q0/(1+β) . • The field in SW structures builds up “in Time”. The filling time is defined as the time needed to build up the field to (1 1 ) = 0.632 times the steady-state field: e tF 2QL 2Q0 (1 ) 24 Main RF Parameters – r/Q Ration and Frequency r/Q Ratio is a important factor, which is only depending on geometry of the structure to evaluate its accelerating ability. r E2 dP / dz E 2 Q dP / dZ w w E2/w is independent with material, machining quality of the structure. Working Frequency is a first and important parameter to choose in accelerator design. Almost all basic RF parameters have frequency dependence, they are scaled as the following: 1 size f r f Q 1/ f r f Q 25 4. Basic Beam Dynamics • Acceleration • Bunching • Beam Loading 26 Acceleration for Constant Impedance TW structures • For a constant-impedance section with a length L, the attenuation is uniform: From the α definition: dP 2（z ) P 2 0 P dz P( z) P0e20 z Integration result: E 2 r ( dP z ) 2 0 rP E 2 0 rP0 e 0 dz dE 0 E dz z V ( z ) E ( z )dz 2 0 (1 e 0 z ) P0 rL Per unit length 0 2V Q const g 0 L Total length L 2Vg Q E( z) E0e0 z P( z) P0e20 z E(0) E0 V (0) 0 E( L) E0e2 (1 e ) V ( L) 2 P0 rL 27 Acceleration for Constant Gradient TW structures • For a constant-gradient section (E=const): the attenuation constant α is a function of z: P0 (1 e 2 ) dP / dz 2 ( z ) P const L z P( z ) P0 1 1 e 2 Liner reduction along the structure L 1 e Combine above ( z ) 1 2L z two equations: 2 1 1 e 2 z 1 1 e 2 L L Vg ( z ) 1 e 2 2Q ( z ) Q E 0 r 2 L dP rP0 (1 e 2 ) dz L V E0 L 1 e 2 P0 rL 28 Summary of Acceleration in TW structures 1. The energy gain V of a charged particle is given by CI: V 2 (1 e 0 z ) P0 rL CG: V 1 e 2 P0 rL 2. The RF energy supplied in the time period tF can be derived from above: CI: 2 2 V P0t F 1 e r L Q CG: 2 P0t F 2 1 e 2 V r L Q 3. The energy W stored in the entire section at the end of the filling time is l CI: W Q dp dz P0 Q (1 e 2 ) CG: 0 dz l P Q W dz P0 (1 e 2 ) v 0 g 29 Acceleration for SW structures P Ps Ps e 2L Ps e 4L Ps ..... 1 e 4L The slightly higher energy gain for SW is paid by field building up time. The energy gain of a charged particle is given: V (1 e t / tF ) 2 c 1 c Pin rL (1 e t / t ) PdisrL F Where βc is coupling coefficient between waveguide and structure. 30 An Example – Field Plot by SUPERFISH Code Location of Maximum Field Meshes and electrical field lines in one and half cell for a SLAC 2π/3 mode, 2856 MHz structure. 31 An Example – Calculation of RF Parameters by SUPERFISH Code Note: Often, the computer calculation codes give parameters for SW case – field is a snapshot of TW case at certain moment to meet boundary conditions. Therefore, interpretation for TW is different. For example, shunt impedance needs to have factor of 2, because the backward wave does not have contribution to acceleration. 32 Longitudinal Dynamics Speed of particle: 𝛽𝑒 = 𝑉𝑒 /𝑐 e 2 1 / Phase speed of acceleration RF: 𝛽𝑝 = 𝑉𝑝 /𝑐 m Ve Normalized momentum of particle: p e 2 1 and 2 p 2 1 Energy of particle: ( u m0 c 2 1 e2 m0 c m0 c 2 1 1 2 c 1 1 1 1 )dz ( ) dz k ( p e )dz Vp Ve Vp Ve The longitudinal motion is described by the following two equations: du eE z Sin dz The reference phase is θ=0 without acceleration d 2 1 dz p 2 1 Combining above two formula: 2 1 du eEo sin d 2 p 1 We have the solution: kmo c 2 p 2 1 cos p const eE0 p 33 Longitudinal Phase Space for βp<1 Stable particles stay within the structure circulating Phase velocity less than c (βp<1) 2 1 with phase extreme m while e p const cos m 2m0c 2 cos cos m eE0 p km0c eE0 2 1 p 2 p p 2 1 1 p2 p p When 1>cosθ>-1 the particles oscillate in p and θ plan with elliptical orbits. If an assembly of particles with a relative large phase extent and small momentum extent enters such a structure, then after traversing ¼ of a phase oscillation it will have a small phase extent and large momentum extent, we call this action as bunching. 34 Longitudinal Phase Space for βp=1 Phase velocity equals c (βp=1) When βp=1, dθ/dz(βe<1) is always negative, and the orbits become openended as shown in the figure. The orbit equation becomes 2m0 c 2 cos cos m eE0 2m0 c 2 p 1 p eE0 2 1 e 1 e where θm has been renamed θ∞ to emphasize that it corresponds to p ∞ . 35 Longitudinal Dynamics - continued The threshold accelerating gradient for capture is cosθ-cosθ∞= 2, or For example, the field of 15.3MV/m m0c 2 at 2856 MHz can capture dark E (threshold ) p 2 1 p 0 0 e 0 current (starting with p0~0). Let us discuss an interesting case: a particle entering the structure with a phase θ0=0, has an asymptotic phase θ∞= -π/2, thus the an assembly of particles will get maximum acceleration and maximum phase compression. For small phase extents ±Δθ0 around θ0=0, ( 0 ) 2 8 Let us consider a practical example at SLAC (λ=10.5 cm). Over a wide range of electrons enter an accelerator section with optimized accelerating gradient and have the above idea bunching to better than 5O bunch. Asymptotic bunching process in Vp=c constant-gradient accelerator section with value of accelerating gradient E optimized for entrance condition. 36 Beam Loading - I The effect of the beam on the accelerating field is called BEAM LOADING. The superposition of the accelerating field established by external generator and the beam-induced field needs to be studied carefully in order to obtain the net Phase and Amplitude of acceleration. Steady-state Phasors in a complex plane for beam loaded structure: Vg generator-induced voltage Vb beam-induced voltage Vc net cavity voltage In order to obtain a basic physics picture, we will assume the synchronized bunches in a bunch train stay in the peak of RF field for both TW and SW analysis. 37 Beam Loading - II The RF power loss per unit length is given by: dP dP dP ( ) wall ( )beam dz dz dz E 2 2rP dP 2P dz E2 r dP / dz dE d dP E 2 d E2 E rP r r r EI dz dz dz 2r dz r where I is average peak current, E is the amplitude of synchronized field. dE 1 d E 1 rI 2 dz dz 2 38 Beam Loading for Constant Impedance Structure For constant impedance structure: dE 0 E 0 rI dz E( z) E(0)e0 z Ir(1 e0 z ) E (0) 2 0 rP0 The total energy gain through a length L is L V E ( z )dz 0 2 (1 e 0 z ) P0 rL IrL (1 1 e ) where P0 is input RF power in MW, r is shunt impedance per unit length in MΩ/m, L is structure length in m, I is average beam current in Ampere, V is total energy gain in MV. The first term is unloaded energy gain, and loaded energy decreases linearly with the beam current. 39 Beam Loading for Constant Gradient Structure For constant gradient structures: The attenuation coefficient is After integration: ( z) E E0 dE rI dz (1 e 2 ) / 2 L 1 (1 e 2 )( z / L) rI z ln1 (1 e 2 ) 2 L The complete solution including transient can be expressed as: tF t 2tF 2 t rI Le L Q V (t ) E 0 L t ( 1 e ) 2 2 Q (1 e 2 ) 1 e 2e 2 1 2 1 e Steady case after two filling time. t 2t F rIL V (t ) E0 L 2 Transient beam loading in a TW CG structure. 40 Beam Loading for Standing Wave Structure For a standing wave structure with a Coupling coefficient βc, The energy gain V(t) is t / t F 2 c V (t ) (1 e ) PinrL (1 et / tF ) Pdis rL Without beam loading 1 c t t b irL t / t (1 e tF ) With beam loading V (t ) (1 e F ) Pdis rL 1 If the beam is injected at time tb and the coupling coefficient meets the following conditio Pb c 1 Pdis We will have:Pin Pdis Pb There is no reflection from the structure to power source with beam. From above formula, the beam injection time is Vb 2 c t b t F ln(1 ) t F ln V0 c 1 Transient beam loading in a standing wave structure. 41 Example of Beam Loading Compensation for TW Structure Beam injection is started before filling complete structure 42 5. Wakefields • • • • What is Wakefield Longitudinal Wakefield Transverse Wakefield Examples of Wakefield Mitigation and Measurement 43 Wakefields The wakefield is the scattered electromagnetic radiation created by relativistic moving charged particles in RF cavities, vacuum bellows, and other beam line components. These fields effect on the particles themselves and subsequent charged particles. Electric field lines of a bunch traversing through a three-cell disc-loaded structure. No disturbance ahead of moving charge ----- CAUSALTY. Wakefields behind the moving charge vary in a complex way – in space and time. The fields can be decomposed into MODES. 44 Longitudinal Wakefields - I We define the longitudinal delta-function potential Wz(s) as the potential (in Volt/Coulomb) experienced by the test particle following along the same path with distance s behind the unit driving charge. 1 zs Wz ( s ) E z ( z , )dz Q0 c L Each mode of wakefiekds has its particular FIELD PATTERN and oscillates with its own eigenfrequency. For simplified analysis, the modes are orthogonal, i.e. the energy contained in a particular mode does not has energy exchange with other modes. Notations for a point charge traversing through a discontinuity. 45 Longitudinal Wakefields - II The longitudinal wakefields are dominated by the m=0 modes, TM01, TM02,…. 0( s 0) s Wz ( s) kn cos( n ) 1( s 0) c n 2( s 0) kn Vn 2 n Rn ( ) 4U n 4 Qn The loss factor kn: where Un is the stored energy for nth mode. Vn is the maximum voltage gain from nth mode for a unit test particle with speed of light. The total amount of energy deposited in all the modes by the driving charge: U Q 2 kn n • Longitudinal wakefields are approximately independent of the transverse positions of both the driving and testing charges. • Impact of Short range longitudinal wakefields --- Energy spread within a bunch. • Impact of Long range longitudinal wakefield --- Beam loading effect. 46 Longitudinal Wakefields - III Computed longitudinal δ- function wake potential per cell for S-Band SLAC structure: Solid line: Total wake Dashed line: 450 modes Dot-dashed line: Accelerating mode 47 Transverse wakefields - I The transverse wake potential is defined as the transverse momentum kick experienced by a unit test charge following at a distance s behind on the same path with a speed of light. L 1 W dz E ( v B) Q0 t zs c The transverse wakefields are dominated by the dipole modes (m=1), For example, HEM11, HEM21,… Approximately: 2k1n 1n s r' W ( ) x Sin( ) s0 a c n 1n a / c E Field B Field In phase quadrature Schematic of field Pattern for the lowest frequency mode -- HEM11 Mode. where r´ is the transverse offset of driving charge and the charge is on x axis. a is the tube radius of the structure. k1n for m=1 nth dipole mode has similar definition like m=0 case. The unit of transverse potential V/Coulomb/mm. The transverse wakefields depend on the driving charge as the first power of its offset r’, the direction of the transverse wake potential vector is decided by the position. 48 Transverse wakefields - II Single Bunch Emittance Growth (HeadTail Instability) due to the short range transverse wakefields Computed transverse δ-function wake potential per cell for S-Band SLAC structure. Solid line: Total wake Dashed line: 495 modes Dot-dashed line: lowest frequency dipole mode (λ=7 cm) Multi-bunch Beam Breakup due to the long range transverse wakefields. 49 Long Range Dipole Mode Suppression - Idea of Detuning of Dipole Modes Cells for a Detuned Structure have profiles with Gaussian dimensional distribution. k dn df1 In frequency domain, dipole mode distribution for a Detuned Structure In the time domain, the excited wakefields by the cells with Gaussian distribution dipole frequencies has Gaussian amplitude profile. 50 Experimental Proof of Transverse Wakefield Suppression 10 Calculation ASSET Data Wake (V/pC/mm/m) Comparison of the measurement with error bars (red) and calculated wakefields (black) for a pair of dipole modes Interleaved 60 cm Damped Detuned X-Band Structures. 2 10 10 10 10 1 0 -1 -2 0 5 10 15 20 Time (ns) 25 30 35 51 6. How to Make a Linac • • • • • • • Machining Chemical Cleaning or Electrical Polishing. Diffusion Bonding and Brazing Tuning and Microwave Characterization Vacuum Baking Fiducialization and Assembly High Power Processing and Testing 52 Computer Numeric Control (CNC) lathe and Some Machined Linac Parts 53 Chemical Cleaning – Critical Step for All High Power RF and High Vacuum Parts • Decreasing • Chemical Etching (Contaminations, Mechanical Defects, Surface Roughness …) • Anti-Oxidation Treatment 54 Diffusion Bonding of Linac Body Pressure: 60 PSI (60 LB for this structure disks) Holding for 1 hour at 1020º C 55 Brazing Linac in a Hydrogen Furnace 56 Non-Resonant Perturbation Measurement Reflected wave amplitude is E 2 ( z) Er ( z ) K Ei ( z ) P( z ) where K is a constant, which depends on the bead, E(z) is the forward power flowing across the structure at z, Ei is the incident wave amplitude. The reflection coefficient is defined as: E ( z) ( z) r Ei ( z ) For constant gradient structure: Er (0) E 2 ( z) (0) K S11 Ei (0) P(0) 57 Tuning and Structure Characterization • Set the bead pulling frequency with correction for string perturbation, operation temperature correction, dry nitrogen environment. • Record amplitude and phase of S parameters while bead puling through the structure. • Program to calculate the detuning amount of cell n th based on the backward reflection difference from (n-1) th disc and (n+1) th disc. • Tuning each cell to the target value. 58 Microwave Tuning and Characterization for a CLIC Prototype Structure 59 Example of Phases and Amplitudes along the Axis of CLIC Prototype Structure Phases and amplitudes plotted in a complex plane (2π/3 mode structure, 2x120º=240º per cell for reflection) Electrical field amplitudes along the structure. 60 Final Parameters Measurement Very Small Reflection from Input and Output Ends and Required Transmission Input End S11 Transmission S12 Output End S22 61 Some Very Useful Relations for Any TW Structures Analysis L L dz Tf V 0 g f dz 0 If the operation temperature (or resonant frequency for every cell) changes with Δω. f d dz dz T f 0 d V 0 g L L The total phase drift due to frequency change is: f 2 f T f Example: If particles with speed c passing a Vp =c and T=100 ns structure. Case I: Operation frequency Δ f=1 MHz, phase slippage ΔΦ=36° Case II: Temperature change to cause heat expansion and frequency change. f 1.66105 / C O f Let’s assume the structure frequency is 11424 MHz, the resonant frequency change is 1.66105 11424103 0.19MHz / C O Use above formula, the total phase slippage is 7O per Cº . Vacuum Baking of Two Linac Structures 650° C for10 days 63 Fiducialization and Alignment In a CMM machine for an X-Band Deflector 64 Complete Instrumentation for RF High Gradient Experiments Outgassing RGA Vacuum Gauge X-Ray PMTs Scintillators Ion Chamber Dark Current Profile Monitor Spectrometers Faraday Cup RF Breakdown Forward/Reflected RF Instruments Surface Damages SEM, XPS, EDX 65 Wish You All Great Success in your Career! 66