High acceleration gradient. Critical applications: Linear colliders e.g. ILC X-ray FELs e.g. DESY XFEL

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1 High acceleration gradient Critical applications: Linear colliders e.g. ILC X-ray FELs e.g. DESY XFEL

2 Critical points The physical limitation of a SC resonator is given by the requirement that the RF magnetic field at the inner surface has to stay below the critical field of the superconductor (about 190 mt for niobium). 59 MV/m have been reached in a single cell using particular shapes!! The main reasons for the performance degradation are excessive heating caused by impurities on the inner surface or by field emission of electrons. The cavity becomes partially normal-conducting, associated with strongly enhanced power dissipation. Because of the exponential increase of surface resistance with temperature this may result in a run-away effect and eventually a quench of the entire cavity. Field emission of electrons from sharp tips is the most severe limitation in high-gradient superconducting cavities. Small particles on the cavity surface act as field emitters

3 Field enhancement The surface of an accelerating structure will have a number of imperfections at the surface caused by grain boundaries, scratches, bumps etc. As the surface is an equipotential the electric fields at these small imperfections can be greatly enhanced. In some cases the field can be increase by a factor of several hundred

4 Breakdown Breakdown occurs when a plasma discharge is generated in the cavity. This is almost always associated with some of the cavity walls being heated until it vaporizes and the gas is then ionized by field emission. The exact mechanisms are still not well understood. This is the major limitation to gradient in most pulsed RF cavities and can permanently damage the structure.

5 Kilpatrick Limits A rough empirical formula for the peak surface electric field is It is not clear why the field strength decreases with frequency. As a SCRF cavity would quench long before breakdown, we only see breakdown in normal conducting structures.

6 Multipacting Accelerated primary electron strikes the surface material and extract secondary electrons Field inversed and accelerated secondary electrons hurt the opposite surface and eject other electrons

7 Secondary emission yield For multipactor to occur each electron striking the surface must generate more than one secondary on average. The ratio of primary to secondary electrons is given by the secondary emission yield (SEY). SEY is strongly dependant on impact energy and only gives an SEY>1 for a finite range of low energies

8 Multipacting in cavity Electron trajectory Two point multipacting can also occur in pillbox and elliptical cavities. The electron trajectories are bent by the RF magnetic field to form semi-circles. In pillbox cavities trajectories are resonant if the cyclotron conditions are met B o m e In elliptical cavities the resonant condition varies with radius as smaller arcs are required causing any resonant trajectories to break up, preventing multipacting

9 SCRF vs NCRF SCRF More efficient (even when including cryogenic losses) Higher CW gradient Long pulse or CW only (higher Luminosity) Complex system needing cryostats and cryogenics Only frequencies below 4 GHz. NCRF Less efficient. Higher pulsed gradient Simpler systems, water cooled More reliable Lower capital costs Smaller apertures mean higher wakefields

10 Limit of a single resonator For a single resonator Each cavity has its own power source, all at the same frequency and amplitude We also adjusted all the phases so that the ultrarelativistic particle sees always the field at its maximum

11 Travelling wave 0 is the phase in the first cell In the center of every cell (z=nd) the field is equal to the field of a travelling wave with constant phase velocity v f =/k

12 Transverse Travelling Wave t = Transverse Standing Wave

13 Transverse Travelling Wave t = T / Transverse Standing Wave

14 Transverse Travelling Wave t = T / Transverse Standing Wave

15 Transverse Travelling Wave t = 3T / Transverse Standing Wave

16 Transverse Travelling Wave t = T / Transverse Standing Wave

17 Transverse Travelling Wave t = 5T / Transverse Standing Wave

18 Transverse Travelling Wave t = 3T / Transverse Standing Wave

19 Transverse Travelling Wave t = 7T / Transverse Standing Wave

20 Transverse Travelling Wave t = T 0.5 Transverse Standing Wave

21 Advantages of acceleration with E.M. wave Using a travelling wave has several benefit: Transit time factor = 1 Easier power system. Remember that small differences in the driving frequency resolve in strong differences in the phase

22 But Unfortunately the Lawson-Woodward theorem states that the energy gain of a charged particle, interacting with e.m. field in the vacuum without any static or magnetic field or boundaries, is zero if the average is performed over a long distance (ideally infinity) Also if we consider an e.m. propagating with an angle (the Electric field is orthogonal to the wave propagation) with respect to the charge motion the phase velocity>c 0 0 /cos

23 Group velocity There are no truly monochromatic waves in nature. A real wave exists in the form of a wave group, which consists of a superposition of waves of different frequencies and wave numbers. If the spread in the phase velocities of the individual waves is small, the envelope of the wave pattern will tend to maintain its shape as it moves with a velocity that is called the group velocity.

24 Phase velocity vs group velocity The exponential factor describes a traveling wave with a mean frequency and mean wave number, and the first factor represents a slowly varying modulation of the wave amplitude. The mean phase velocity is

25 Starting equations

26 Equation for E z In the case of a plane wave with only z component different from zero

27 Equation for k

28 Spatial solution

29 Frequency cutoff If the wavenumber k z is complex, then the amplitude of the wave travelling through the waveguide falls off exponentially, i.e. loss-free wave propagation is not possible. Loss-free propagation occurs only when k z is real.

30 Dispersion relation In the regime of loss-free wave propagation in the waveguide, the wavelength z is always larger than the wavelength in free space. This means that the phase velocity of the wave within the waveguide is greater the speed of light

31 Example cylindrical waveguide In a lossless uniform waveguide with azimuthal symmetry, the axial electric field for the lowest transverse magnetic mode is

32 Brillouin Plot Plot: each frequency corresponds to a certain phase velocity, the phase is always larger than the speed of light, At = c the propagation constant k z goes to zero and the phase velocity v ph = /k z becomes infinite, It is impossible to accelerate particles in a circular waveguide because synchronism between the particles and the RF is impossible (particles would have to travel faster than light to be synchronous with the RF),

33 TM and TE modes in a waveguide

34 Periodic loaded structure 2b L 2a One might expect that converting the uniform guide to a periodic structure might perturb the field distribution by introducing a z- periodic modulation of the amplitude of the wave. The complicated boundary conditions cannot be satisfied by a single mode, as was the case with an empty cavity, but by a whole spectrum of so-called 'space harmonics', which is in fact a Fourier series applied to a periodic case.

35 Space Harmonics

36 New dispersion

37 More details For a given mode, there is a limited pass band of possible frequencies At both ends of the pass band, the group velocity is zero For a given frequency, one has an infinite series of space harmonics All space harmonics have the same group velocity, but different phase velocities When the group and phase velocity are in the same direction, we speak about forward waves; if the directions are opposite, we speak about backward waves (n < 0).

38 Travelling waves structures RF power is introduced via the input coupler. Part of RF power is dissipated in the structure, part is taken by the beam (beam loading) and the rest is absorbed in a matched load at the end of the structure. Usually, structure length is such that ~30% of power goes to the load. The traveling wave structure is the standard linac for electrons from β~1.

39 Constant impedance In a purely periodic structure, made by a sequence of identical cells (also called constant impedance structure ), both the RF power flux and the intensity of the accelerating field decay exponentially along the structure

40 Constant gradient It is possible to demonstrate that, in order to keep the accelerating field constant along the structure, the iris apertures have to decrease along the structure in such a way that the field attenuation is compensated by the increase of the stored energy (with consequent decrease of the group velocity). In general the constant gradient structures are more efficient than constant impedance ones, because of the more uniform distribution of the RF power along them.

41 Standing wave A direct and a reflected sinusoidally varying wave, travelling with the same velocity but in opposite directions, combine to create a standing-wave pattern At the points where the direct (solid line) and reflected (dotted line) space harmonics join, they have the same phase velocity, and if this velocity is synchronous with the particle, both harmonics contribute to the acceleration.

42 Dispersion for standing wave Due to the boundary conditions only certain modes with distinct frequencies are possible in this resonator. The number of modes will be identical to the number of cells (N cells N modes) The mode names (0,..,π/2,.., π) correspond to the phase difference between the modes.

43 SW vs TW TW structures are filled with power in space : the power fills one cell after another with typically 1-3% of c (<μs, depending on f). SW structures are filled in time : the reflected waves build up in time until the final standing wave pattern is achieved at the desired amplitude: (~10 μs range for NC, depending on f). For very short beam pulses (< μs), there is a clear power efficiency advantage for TW structures, for longer pulses (μs range) both structure types can be optimized to similar efficiencies and cost. Due to the extremely short RF pulse lengths, TW can typically sustain much higher peak fields than any SW structure

44 Comparison Long pulses Used for Ions and electrons at all energies Superconductive structures Short pulses, High frequency. Used for Electrons at v~c

45 Normal Conducting Accelerator Structures E acc limited by breakdown RF-field > MV/m in X band Higher gradients than SCRF cavities, but requires very high frequency: >10 GHz very short pulse lengths: < 1μs high ohmic losses

46 High Frequency structures 11.4 GHz structure (NLC) 1 cm 30 GHz structure (CLIC)

47 Superconducting SW structure

48 Choice of the frequency Some basic considerations for the choice include: higher power efficiency at higher frequencies because of the 1/2 dependence of shunt impedance tighter beam-positioning tolerances at higher frequencies because of smaller apertures. Other considerations are often equally important. For applications requiring acceleration of very short, intense bunches of electrons, it is desirable to provide large stored energy per unit length, which scales as -2 favoring lower frequencies

49 At the beginning Many particle sources, be it for electrons, protons, or ions, produce a continuous stream of particles at modest energies limited by electrostatic acceleration between two electrodes. Not all particles of such a beam will be accelerated because of the oscillatory nature of the accelerating field. For this reason and also in the case short bunches or a small energy spread at the end of the linac is desired, the particles should be concentrated at a particular phase. This concentration of particles in the vicinity of an optimum phase maximizes the particle intensity in the bunch

50 Prebuncher A bunched beam can be obtained from a continuous stream of nonrelativistic particles by the use of a prebuncher. The basic components of a prebuncher is an rf-cavity followed by a drift space. As a continuous stream of particles passes through the prebuncher, some particles get accelerated and some are decelerated. Particles before acceleration (a) and right after (b). A distance L downstream of the buncher cavity (c) the phase space distribution shows strong bunching

51 Chopper There are still particles between the bunches which could either be eliminated by an rf-chopper or let go to be lost in the linear accelerator because they are mainly distributed over the decelerating field period in the linac This is a chopper which consists of an rf-cavity excited similar to the prebuncher cavity but with the beam port offset by a distance r from the cavity axis. In this case the same rf source as for the prebuncher or main accelerator can be used and the deflection of particles is effected by the azimuthal magnetic field in the cavity To produce a single pulse, the chopper system may consist of a permanent magnet and a fast pulsed magnet. The permanent magnet deflects the beam into an absorber while the pulsed magnet deflects the beam away from the absorber across a small slit

52 Klystron Cavities installed in circular accelerators and linac structures require RF power of at least a few tens of kw and as much as several MW in large high-energy accelerators or linacs. The klystron has proven to be the most effective power generator for accelerator applications

53 How the klystron works Electrons are emitted from a round cathode with large surface area, and are accelerated by a voltage of a few tens of kv. This yields a round beam, with a current ranging from a few amperes up to more than 10 A. Suitably shaped electrodes near the cathode are used to focus the beam Sometimes several solenoids are added along the tube to ensure good collimation of the beam. The beam comes out of the cathode with a very well-defined particle velocity and passes through a first cylindrical cavity. A wave is excited in this resonator by an external preamplifier with a power output of a few tens of watts, which, depending on the phase, will accelerate, brake, or simply not influence the particles in the beam. The velocity of the particles through the cavity is thus modulated, with a frequency exactly equal to the resonant frequency. In the zero-field drift section which follows, the faster particles move ahead, while the slower ones lag behind.

54 Cont. After a certain distance bunches of particles are formed, with a separation given by the wavelength of the driving wave. The continuous current from the cathode thus becomes a pulsed current, with a frequency equal to the frequency of the coupled driving supply A second cavity mounted at this point is resonantly excited by the pulsed current, and the RF wave generated in this cavity can then be coupled out. In an optimal design the beam is almost completely stopped by the RF field it excites in the second cavity, leaving just a small residual energy, i.e. the kinetic energy stored in the beam is transformed into RF energy

55 Klystron efficiency Here U 0 is the supply voltage of the klystron, I beam the beam current and the efficiency of the klystron, which nowadays ranges from 45% and 65%. In particular, klystrons used to drive linac structures operating in the S-band at the commonly-used frequency of GHz require voltages between 250 and 300 kv, traversed by beam currents of around 250 A. For an efficiency of = 45% this yields a power output of P kl = MW. Naturally such high power can only be handled in pulsed operation. The pulse length in linacs is a few s with a repetition rate of a few hundred Hz. This yields an average power output of a few tens of kw, which is relatively easy to manage