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Lecture 14 - Radiofrequency Cavities II Professor Emmanuel Tsesmelis Directorate Office, CERN Department of Physics, University of Oxford Accelerator Physics Graduate Course John Adams Institute for Accelerator Science 24 November 2011 Table of Contents II Group Velocity Dispersion Diagramme for Waveguide Iris-loaded Structures Resonant Cavities Rectangular and Cylindrical Cavities Quality Factor of Resonator Shunt Impedance and Energy Gain Transit-Time Factor Kilpatrick Limit Software for Cavity Design Group Velocity Energy (and information) travel with wave group velocity. Interference of two continuous waves of slightly different E E sin k dk x d t E sin k dk x ( d )t 0 0 E sin kx t cosdk x d t 0 frequencies 2 E f x, t f ( x, t )0 1 2 described by: Group Velocity Mean wavenumber & frequency represented by continuous wave f 1 ( x , t ) sin k x t Any given phase in this wave is propagated such that kx – ωt remains constant. Phase velocity of wave is thus f1 x , t t vp f1 ( x , t ) x k Envelope of pattern described by f 2 ( x, t ) cos[dkx dt ] Any point in the envelope propagates such that x dt – t dω remains constant and its velocity, i.e. group velocity, is f 2 ( x, t ) t d vg f 2 ( x , t ) x dk Dispersion Diagramme for Waveguide Description of wave propagation down a waveguide by plotting graph of frequency, ω, against wavenumber, k = 2π/λ Imagine experiment in which signals of different frequencies are injected down a waveguide and the wavelength of the modes transmitted are measured. Measurables Phase velocity for given frequency: ω/k Group velocity: slope of tangent Dispersion Diagramme for Waveguide Observations However small the k, the frequency is always greater than the cut-off frequency. The longer the wavelength or lower the frequency, the slower is the group Dispersion diagramme for waveguide velocity. is the hyperbola At cut-off frequency, no c 2 2 energy flows along the k 2 waveguide. c c Also v p h v g c 2 Iris-loaded Structures Acceleration in a •Need to modify waveguide to reduce waveguide is not possible phase velocity to match that of the particle as the phase velocity of (less than speed of light). the wave exceeds that of light. •Install iris-shaped screens with a constant Particles, which are separation in the waveguide. travelling slower, undergo acceleration from the passing wave for half the period but then experience an equal deceleration. Averaged over long time interval results in no net transfer of energy to the particles. Iris-loaded Structures Recall that the dispersion relation in a waveguide is 2 2 c kz 2 c With the installation of irises, curve flattens off and crosses boundary at vφ=c at With suitable choice of iris separation d the phase velocity can be set to kz=π/2 any value Iris-loaded Structures Waveguides cannot be used for sustained acceleration as all points on dispersion curve lie above Dispersion diagramme for a loaded waveguide diagonal in dispersion diagramme. The k-value for each space harmonic is 2n Phase velocity > c k k n 0 d An iris-loaded By choosing any frequency in dispersion diagramme it will intercept dispersion curve structure slows down at k values spaced by 2nπ/d the phase velocity. First rising slope used for acceleration. Resonant Cavities General solution of wave equation W (r , t ) Aei (t k r ) Be i (t k r ) Describes sum of two waves – one moving in one direction and another in opposite direction If wave is totally reflected at surface then both amplitudes are the same, A=B, and W (r , t ) Aeit (eikr e ikr ) 2 A cos( k r )eit Describes field configuration which has a static amplitude 2Acos(k·r), i.e. a standing wave. Resonant Cavities Resonant Wavelengths Stable standing wave forms in fully-closed cavity if z lq with q 0, 1, 2,... 2 where l = distance between entrance and exit of waveguide after being closed off by two perpendicular sheets. only certain well-defined wavelengths λr are present in the cavity. General resonant condition 1 11q 2 2 r c 4 l 2 Near the resonant wavelength, resonant cavity behaves like electrical oscillator but with much higher Q-value and corresponding lower losses of resonators made of individual coils and capacitors. Exploited to generate high-accelerating voltages Rectangular Resonant Cavities Inserting 2 2 m n 2 2 a b c into the resonance condition yields 2 r with m, n, q int egers 2 2 2 m n q a b l Integers m,n,and q define modes in resonant cavity. Number of modes is unlimited but only a few of them used in practical situations. m,n,and q between 0 and 2 Cylindrical Resonant Cavities Inserting the expression for cut-off frequency into general resonance condition yields 2 2 1 x 1q 1 with q 0,1,2,... r D 4 l 2 where x1=2.0483 is the first zero of the Bessel function. For the case of q=0, termed the TM010 mode, the resonant wavelength reduces to D r x1 Bessel Functions Pill-box Cylindrical Cavity Cylindrical pill-box cavity with holes The simplest RF cavity type for beam and coupler. The accelerating modes of this cavity are TM0lm Indices refer to the polar co-ordinates φ, r and z TM010 TM011 Lines of force for the electrical field. Pill-box Cylindrical Cavity The modes with no φ variation are: 2 E 2 E 0 1 E 1 E 2 E r 2 2 2 E 0 r r r r r z P m E z E0 J 0 0 l r r cos h z 0 m r0 P0l m E r E0 J1 r sin h z P0l h r0 2 P m 2 2 0 lm 0l r h 0 l indicates the radial variation while m controls the number of wavelengths in the z-direction. P0l is the argument of the Bessel function when it crosses zero for the lth time. J0(P0l) = 0 for P0l = 2.405 Pill-box Cylindrical Cavity TM010 Mode 2.405 2.405 010 r r ; 010 r ; 010 E E0 J 0 0 0 010 1 010 ; 010 2 010 Quality Factor of Resonator, Q Ratio of stored energy to energy dissipated per cycle divided by 2 Q W s W s W d P d Ws = stored energy in cavity Wd = energy dissipated per cycle divided by 2 Pd = power dissipated in cavity walls = frequency Quality Factor of Resonator, Q Stored energy over cavity volume is E 2 2 H 0 W 2 s dv W s 2 0 dv where the first integral applies to the time the energy is stored in the E-field and the second integral as it oscillates back into the H-field. Quality Factor of Resonator, Q Losses on cavity walls are introduced by taking into account the finite conductivity of the walls. Since, for a perfect conductor, the linear density of the current j along walls of structure is j = n H we can write R2 H surf P d 2 s ds with s = inner surface of conductor Quality Factor of Resonator, Q Rsurf = surface resistance δ = skin depth f 1 R surf 0 For Cu, Rsurf = 2.61 10-7 Ω Shunt Impedance - Rs Figure of merit for an accelerating cavity Relates accelerating voltage to the power Pd to be provided to balance the dissipation in the walls. Voltage along path followed by beam in electric field Ez is V = path |Ez(x,y,z)| dl from which (peak-to-peak) 2 V R s 2 Pd Shunt Impedance - Rs z 1 2 2 sin D / 2 Rs 5.12 10 8 p 2.61 z 1 D / 2 with z (phase velocity) c h (h thickness, d iris separation ) d p number of irises per wavele ngth (equal to mode number) 2 D 1 p Energy Gain Energy gain of particle as it travels a distance through linac structure depends only on potential difference crossed by particle: U K PRF lRs where PRF supplied RF power l length of linac structure Rs shunt impedance K correction factor ( 0.8) Analogous to Electrical Oscillator Cavity behaves as an electrical oscillator but with very high quality factor (sharp resonance) Electrical response of cavity Q r R s described by parallel circuit containing C, L, and Rs Z On resonance the impedance is r resonant frequency 1 Δ = frequency shift at which amplitude Z L is reduced by -3 dB relative to resonance peak C Transit-Time Factor The RF Gap Accelerating gap Space between drift tubes in linac structure Space between entrance and exit orifices of cavity resonator Field is varying as the particle traverses the gap E z E 0 c o s( t ) Makes cavity less efficient and resultant energy gain Field is uniform along gap axis which is only a fraction of and depends sinusoidally on time the peak voltage Phase refers to particle in middle of gap z=0 at t=0 Transit-Time Factor Transit-Time Factor is ratio of energy actually given to a particle passing the cavity centre at peak field to the energy that would be received if the field were constant with time at its peak value The energy gained over the gap G is: G sin( G 2 c ) E 0 G cos 2 V E 0 cos( t ) dz G G / 2c 2 Transit-Time Factor The Transit Gap Factor is defined as sin( G 2 c ) Transit G ap Factor G / 2c Defining a transit angle T ra n sit A n g le G c 2 G the Transit Gap Factor becomes sin 2 with 0 < < 1 2 The Transit-Time Factor Observations At relativistic energies, cavity dimensions are comparable with /2 Reduction in efficiency due to transit-time factor is acceptable. At low energies, this is not the case Cavities have strange re-entrant configuration to keep G short compared to dimensions of its resonant volume. The Transit-Time Factor Field in resonant cavity Compromise cavity design ‘Nose-cones’ Increasing ratio of volume/surface area Reduces ohmic losses Increases Q factor Minimise gap factor Kilpatrick Limit RF breakdown observed at very high fields. Kilpatrick Limit expresses empirical relation between accelerating frequency and E-field f = 1.64Ek2e − 8.5 / Ek Software for Cavity Design Structures usually solved by Finite Element Analysis