1. Lecture 10 - Radiofrequency Cavities II Emmanuel Tsesmelis (CERN/Oxford) John Adams Institute for Accelerator Science 13 November 2009

2. Table of Contents II Group Velocity Dispersion Diagramme for Waveguide Resonant Cavities  Rectangular and Cylindrical Cavities Quality Factor of Resonator Shunt Impedance and Energy Gain Transit-Time Factor Iris-loaded Structures Synchronizing Particles with Cavities Accelerating Structures for Linacs Operation of Linac Structure

3. Group Velocity  Energy (and information) travel with wave group velocity.  Interference of two continuous waves of slightly different frequencies described by:

4. Group Velocity  Mean wavenumber & frequency represented by continuous wave  Any given phase in this wave is propagated such that kx – ωt remains constant.  Phase velocity of wave is thus  Envelope of pattern described by  Any point in the envelope propagates such that x dt – t dω remains constant and its velocity, i.e. group velocity, is

5. 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

6. 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 velocity.  At cut-off frequency, no energy flows along the waveguide.  Also Dispersion diagramme for waveguide is the hyperbola

7. Resonant Cavities General solution of wave equation  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  Describes field configuration which has a static amplitude 2Acos(k·r), i.e. a standing wave.

8. Resonant Cavities Resonant Wavelengths  Stable standing wave forms in fully-closed cavity if where l = distance between entrance and exit of waveguide after being closed off by two perpendicular sheets.  only certain well-defined wavelengths λ r are present in the cavity. General resonant condition 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

9. Rectangular Resonant Cavities Inserting into the resonance condition yields 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

10. Cylindrical Resonant Cavities Inserting the expression for cut-off frequency into general resonance condition yields  where x 1 =2.0483 is the first zero of the Bessel function. For the case of q=0, termed the TM 010 mode, the resonant wavelength reduces to

11. Pill-box Cylindrical Cavity Cylindrical pill-box cavity with holes for beam and coupler. Lines of force for the electrical field. TM 010 TM 011  The simplest RF cavity type  The accelerating modes of this cavity are TM 0lm  Indices refer to the polar co-ordinates φ, r and z

12. Pill-box Cylindrical Cavity The modes with no φ variation are: l indicates the radial variation while m controls the number of wavelengths in the z-direction. P 0l is the argument of the Bessel function when it crosses zero for the lth time.  J 0 (P 0l ) = 0 for P 0l = 2.405

13. Pill-box Cylindrical Cavity TM 010 Mode

14. Quality Factor of Resonator, Q  Ratio of stored energy to energy dissipated per cycle divided by 2  W s = stored energy in cavity W d = energy dissipated per cycle divided by 2  P d = power dissipated in cavity walls  = frequency

15. Quality Factor of Resonator, Q  Stored energy over cavity volume is 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.

16. 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 with s = inner surface of conductor

17. Quality Factor of Resonator, Q R surf = surface resistance δ = skin depth For Cu, R surf = 2.61  10 -7 Ω

18. Shunt Impedance - R s  Figure of merit for an accelerating cavity  Relates accelerating voltage to the power P d to be provided to balance the dissipation in the walls.  Voltage along path followed by beam in electric field E z is V =  path | E z (x,y,z) | dl from which (peak-to-peak)

19. Shunt Impedance - R s

20. Energy Gain Energy gain of particle as it travels a distance through linac structure depends only on potential difference crossed by particle:

21. Analogous to Electrical Oscillator  Cavity behaves as an electrical oscillator but with very high quality factor (sharp resonance)  r resonant frequency Δ  = frequency shift at which amplitude is reduced by -3 dB relative to resonance peak On resonance the impedance is Electrical response of cavity described by parallel circuit containing C, L, and R s

22. Transit-Time Factor  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  Makes cavity less efficient and resultant energy gain which is only a fraction of the peak voltage The RF Gap Field is uniform along gap axis and depends sinusoidally on time Phase  refers to particle in middle of gap z=0 at t=0

23. 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:

24. Transit-Time Factor the Transit Gap Factor becomes The Transit Gap Factor is defined as Defining a transit angle with 0 <  < 1

25. 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.

26. The Transit-Time Factor  Compromise cavity design  Increasing ratio of volume/surface area  Reduces ohmic losses  Increases Q factor  Minimise gap factor Field in resonant cavity ‘Nose-cones’

27. Iris-loaded Structures  Accelerating systems for particles travelling close to c consist of series of cavities in single assembly.  Structures consist of  Sequence of pill-boxes.  Cylindrical waveguide, loaded with number of equidistant irises.  Power from amplifier is coupled into cavity at one end and is either absorbed in load at the other end or reflected to set up a standing wave.

28. Iris-loaded Structures  Waveguides cannot be used for sustained acceleration as all points on dispersion curve lie above diagonal in dispersion diagramme.  Phase velocity > c  An iris-loaded structure slows down the phase velocity. Dispersion diagramme for a loaded waveguide The k-value for each space harmonic is By choosing any frequency in dispersion diagramme it will intercept dispersion curve at k values spaced by 2nπ/d First rising slope used for acceleration.

29. Synchronising Particles with Cavities  If accelerator has more than single cavity, particles should be bunched to arrive at the same phase with respect to the voltage at each cavity.  Space cavities by distance L that a particle travels in one RF period

30. Synchronising Particles with Cavities  Alvarez Structure  Increasing L between accelerating gaps along structure.  Snapshot of fields across each gap shows them all exactly in phase.  Particle‘s phase advance between cells is 2π  Wideröe Structure  Alternate drift tubes grounded.  Snapshot shows vector alternating in sign from gap to gap.  In these cases, cells oscillate either in phase or in antiphase.  Difficult for power to propagate along the waveguide and small errors produce serious distortions. Adjacent single-gap cavities in (a) π mode and b) 2π mode Alvarez Cavity Wideröe Cavity

31. Multicell Cavities  Travelling waves can be used instead of standing waves.  Wave travels along a long chain of cavities to be absorbed in a load at the other end.  Structures which repeat every one, two, three or four cavities correspond to phase changes of 0, π, 2π/3 or π/2 per cell, respectively. Modes of a multicell cavity

32. Accelerating Structures for LINACS Acceleration in a waveguide is not possible as the phase velocity of the wave exceeds that of light.  Particles, which are travelling more slowly, 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. Need to modify waveguide to reduce phase velocity to match that of the particle (less than speed of light). Install iris-shaped screens with a constant separation in the waveguide.

33. Accelerating Structures for LINACS Recall that the dispersion relation in a waveguide is With the installation of irises, curve flattens off and crosses boundary at v φ =c at k z =π/2 With suitable choice of iris separation d the phase velocity can be set to any value

34. Operation of LINAC Structure Standard operation of linac structure is in the S-band.  λ=0.100m (f RF =3 GHz) As in radar technology, RF power supplied by pulsed power tubes – klystrons.  Power fed into linac structure by TE 10 wave in rectangular waveguide which is connected perpendicular to cylindrical TM 01 cavity.

35. Operation of LINAC Structure Travelling wave mode, in which an absorber is installed at the end of the structure to prevent reflections, is more commonly used. In a standing wave mode, the energy is reflected virtually without loss. The two modes of operation of a linac structure.

36. Operation of LINAC Structure Irises form a periodic structure within cavity, reflecting the wave as it passes through and causing interference. Loss-free propagation only if wavelength is integer multiple of iris separation d:  Irises only allow certain wavelengths, characterised by number p, to travel in longitudinal direction.  These fixed wave configurations are termed modes.  In principle there are arbitrary such modes but only three used for acceleration.

37. Operation of LINAC Structure π-mode  Takes long time for transient oscillations to die away and a stationary state to be used.  Not suitable for fast-pulsed operation. π/2-mode  Low shunt impedance so for fixed RF power energy gain per structure is small. 2π/3-mode  Best compromise between π-mode & π/2-mode Field configurations of three most important modes in linac structures.