Lecture 3 RF Acceleration in Linacs Part 1 Outline • Transit-time factor • Coupled RF cavities and normal modes • Examples of RF cavity structures • Material from – Wiedemann, Chapters 15 and 6 – Wangler, Chapters 2 and 3 Transit Time Factor • We now consider the energy gained by a charged particle that traverses an accelerating gap, such as a pillbox cavity in TM010 mode. • The energy-gain is complicated by the fact that the RF field is changing while the particle is in the gap. Transit Time Factor • We will consider this problem by considering successively more realistic (and complicated) models for the accelerating gap, where in each case the field varies sinusoidally in time. • We also must consider the possibility that the energy gain depends on particle radius. +q E Acceleration by Time-Varying Fields Consider infinite parallel plates separated by a distance L with sinusoidal voltage applied. Assume uniform E-field in gap (neglect holes) L Ez Ez (t) E0 cos(t ) where at t=0, the particle is at the +q center of the gap (z=0), and the phase x z of the field relative to the crest is . E But t is a function of position t=t(z), with y - z dz + t(z) V 0 v(z) 0 The energy gain in the accelerating gap is L/ 2 L/ 2 W q Ezdz qE0 cos(t(z) )dz L/ 2 L/ 2 Energy Gain in an Accelerating Gap • Assume the velocity change through the gap is small, so that t(z) = z/v,and z 2c z 2z t v c L/ 2 W qE (cost cos sint sin)dz This is an 0 L/ 2 odd-function of z L/ 2 2z L/ 2 2z W qE cos cos dz qE sin sin dz 0 0 L/ 2 L/ 2 L/ 2 2z W qE0 cos sin 2 L/ 2 Energy Gain and Transit Time Factor sin(L / ) W qE Lcos 0 L / W qV0T cos sin(L / ) T L / • Compare to energy gain from W W T cos static DC field: DC • T is the transit-time factor: a factor that takes into account the time- variation of the field during particle transit through the gap. • is the synchronous phase, measured from the crest. Transit-Time Factor 1.2 1 0.8 0.6 0.4 0.2 Transit-Time Factor Transit-Time 0 0 1 2 3 4 5 -0.2 -0.4 L/ For efficient acceleration by RF fields, we need to properly match the gap length L to the distance that the particle travels in one RF wavelength, . Transit Time Factor for Real RF Gaps • The energy gain just calculated for infinite planes is the same as that for an on-axis particle accelerated in a pillbox cavity neglecting the beam holes. • A more realistic accelerating field depends on r, z Ez Ez (r, z,t) E(r, z)cos(t ) • Calculate the energy gain as before: L/ 2 L/ 2 W q Ezdz q E(0, z)cos(t(z) )dz L/ 2 L/ 2 L/ 2 W q E(0, z)(cost cos sint sin)dz L L/ 2 Transit-time Factor • Choose the origin at the electrical center of the gap, defined as L/ 2 E(0, z)sint(z)dz 0 • This gives L/ 2 L/ 2 E(0, z)costdz L/ 2 W qV T cos q E(0, z)dz L / 2 cos 0 L / 2 L / 2 E(0, z)dz L/ 2 • From which we identify the general form of the transit-time factor as L / 2 E(0, z)costdz L / 2 T L / 2 E(0, z)dz L / 2 Transit-time Factor • Assuming that the velocity change is small in the gap, then z 2z t kz v • The transit time factor can be expressed as 1 L/ 2 T(k) T(0,k) E(0, z)cos(kz)dz V0 L/ 2 2 L/ 2 k V0 E(0, z)dz L/ 2 Radial Dependence of Transit-time Factor • We calculated the Transit-time factor for an on-axis particle. We can extend this analysis to the transit-time factor and energy gain for off-axis particles. • This is important because the electric-field in a pillbox cavity decreases with radius (remember TM010 fields): 1 L/ 2 T(r,k) E(r, z)cos(kz)dz V0 L/ 2 T(r,k) T(k)I0 (Kr) • Here I0 is the modified Bessel function of order zero, and 2 K • Giving for the energy gain W qV0T(k)I0 (Kr)cos which is the on-axis result modified by the r-dependent Bessel function. Realistic Geometry of an RF Gap • Assume accelerating field at drift-tube bore radius (r=a) is constant within the gap, and zero outside the gap within the drift tube walls Eg 0 z g / 2 E(r a, z) L/ 2 0 g / 2 z 1 T(r,k) E(r, z)cos(kz)dz V0 L/ 2 • Using the definition of transit-time factor: • we get Eg g sin(kg/ 2) Eg g V0T(k) V0 I0 (Ka) kg/ 2 J0 (2a / ) • Finally, J0 (2a / ) sin(g / ) T(r,k) T(k)I0 (Kr) I0 (Kr) I0 (Ka) g / What about the drift tubes? • The cutoff frequency for a cylindrical waveguide is c 2.405c / R • The drift tube has a cutoff frequency, below which EM waves do not propagate. • The propagation factor k is 2 2 2.405 2.405 k 2 0 z R R c hole • So the electric field decays exponentially with penetration distance in the drift tube: i(kzt) i(i k zt) k z it Ez E0e E0e E0e e • Example: 1 GHz cavity with r=1cm beam holes: Power and Acceleration Figures of Merit U • Quality Factor: Q – “goodness” of an oscillator P 2 • Shunt Impedance: V0 rs – “Ohms law” resistance P 2 • Effective Shunt Impedance: (V0T) 2 r rsT – Impedance including TTF P r E 2 • Shunt Impedance per unit Z s 0 L P / L length: r (E T)2 • Effective Shunt Impedance/unit ZT 2 0 length: L P / L • “R over Q”: r (V T)2 0 – Efficiency of acceleration per unit Q U of stored energy Power Balance • Power delivered to the beam is: IW PB q • Total power delivered by the RF power source is: PT P PB Coupled RF Cavities and Normal Modes Now, let’s make a real linac • We can accelerate particles in a pillbox cavity • Real linacs are made by stringing together a series of pillbox cavities. • These cavity arrays can be constructed from independently powered cavities, or by “coupling” a number of cavities in a single RF structure. Coupling of two cavities • Suppose we couple two RF cavities together: • Each is an electrical oscillator with the same resonant frequency • A beampipe couples the two cavities • Remember the case of mechanical coupling of two oscillators: • Two mechanical modes are possible: – The “zero-mode”: A- B=0, where each oscillates at natural frequency – The “pi-mode”: A- B=, where each oscillates at a higher frequency Coupling of electrical oscillators • Two coupled oscillators, each with same resonant frequency: 2 1 0 • Apply Kirchoff’s laws to each LC circuit: 1 V i1( jL) i1 i2 ( jM ) 0 jC i1 i2 • Gives C C 2 M i (1 0 ) i 0 1 2 2 L 2 M i (1 0 ) i 0 2 2 1 L • Which can be expressed as: 1/ 2 k / 2 i 1 i 0 0 1 1 2 2 2 k M / L k /0 1/0 i2 i2 • You may recognize this as an ~ 1 eigenvalue problem MX q 2 X q q Coupling of electrical oscillators • There are two normal-mode eigenvectors and associated eigenfrequecies • Zero-mode: 1 0 X 0 0 1 1 k • Pi-mode: 1 0 X 1 k 1 • Like the coupled pendula, we have 2 normal modes, one for in- phase oscillation (“Zero-mode”) and another for out of phase oscillation (“Pi-mode”). • It is important to remember that both oscillators have resonant frequencies , different from the natural (uncoupled) frequency. Normal modes for many coupled cavities • N+1 coupled oscillators have N+1 normal-modes of oscillation. • Normal mode spectrum: 0 q 1 k cos(q / N) where q=0,1,…N is the mode number. • Not all are useful for particle acceleration. • Standing wave structures of coupled cavities are all driven so that the beam sees either the zero or mode. Example for a 3-cell Cavity: Zero-mode Excitation t=0 t=/4 t=/2 t=3/4 t= t=5/4 t=3/2 t=7/4 t=2 Example for a 3-cell Cavity: Pi-mode Excitation t=0 t=/4 t=/2 t=3/4 t= t=5/4 t=3/2 t=7/4 t=2 Synchronicity condition in multicell RF structures TM010 Cavities Drift spaces 1 2 3 4 5 l1 l2 l3 l4 l5 • Suppose we want a particle to arrive at the center of each gap at =0. Then we would have to space the cavities so that the RF phase advanced by – 2 if the coupled cavity array was driven in zero-mode, – Or by if the coupled cavity array was driven in pi-mode. Synchronicity Condition 2c 2c l Zero-mode: t t n 2 cn ln n • RF gaps (cells) are spaced by , which increases as the particle velocity increases. 2c 2c l Pi-mode: t t n cn ln n / 2 • RF gaps (cells) are spaced by /2, which increases as the particle velocity increases.