Lecture 1 Introduction to RF for Accelerators. Dr G Burt Lancaster University Engineering

Transcription

1 Lecture 1 Introduction to RF for Accelerators Dr G Burt Lancaster University Engineering

2 Electrostatic Acceleration

3 Van-de Graaff s A standard electrostatic accelerator is a Van de Graaf These devices are limited to about 30 MV by the voltage hold off across ceramic insulators used to generate the high voltages (dielectric breakdown).

4 RF Acceleration By switching the charge on the plates in phase with the particle motion we can cause the particles to always see an acceleration You only need to hold off the voltage between two plates not the full accelerating voltage of the accelerator. We cannot use smooth wall waveguide to contain rf in order to accelerate a beam as the phase velocity is faster than the speed of light, hence we cannot keep a bunch in phase with the wave.

5 Early Linear Accelerators (Drift Tube) Proposed by Ising (195) First built by Wideröe (198) Alvarez version (1955) Replace static fields by time-varying fields by only exposing the bunch to the wave at certain selected points. Long drift tubes shield the electric field for at least half the RF cycle. The gaps increase length with distance.

6 Cavity Linacs These devices store large amounts of energy at a specific frequency allowing low power sources to reach high fields.

7 Cavity Quality Factor An important definition is the cavity Q factor, given by Q = ωu Where U is the stored energy given by, U = 0 1 µ 0 The Q factor is π times the number of rf cycles it takes to dissipate the energy stored in the cavity. U The Q factor determines the maximum energy the cavity can fill to with a given input power. P c H dv ωt = U0 exp Q0

8 Cavities If we place metal walls at each end of the waveguide we create a cavity. The waves are reflected at both walls creating a standing wave. If we superimpose a number of plane waves by reflection inside a cavities surface we can get cancellation of E and B T at the cavity walls. The boundary conditions must also be met on these walls. These are met at discrete frequencies only when there is an integer number of half wavelengths in all directions. L The resonant frequency of a rectangular cavity can be given by (ω/c) =(mπ/a) + (nπ/b) + (pπ/l) Where a, b and L are the width, height and length of the cavity and m, n and p are integers a

9 Pillbox Cavities Wave equation in cylindrical co-ordinates 1 1 r + + µεω k z ψ r r r r ϕ = 0 Solution to the wave equation ψ = A J ( k r) e 1 m t ± imϕ Transverse Electric (TE) modes H Transverse Magnetic (TM) modes E z z ς ' a z, 1 m Ht = th z ς ' m, n m, n ± imϕ ( r ϕ) = A J e ς r a z, 1 m Et = te z ς m, n m, n ± imϕ ( r ϕ) = A J e r ik ik a a E t H iµωa = ˆ ς t ' m, n iεωa = ˆ ς m, n ( z H ) t ( z E ) t z z

10 Bessel Function J m (k T r) m=0 m=1 m= m= First four Bessel functions. k T r E z (TM) and H z (TE) vary as Bessel functions in pill box cavities. All functions have zero at the centre except the 0th order Bessel functions. One of the transverse fields varies with the differential of the Bessel function J All J are zero in the centre except the 1 st order Bessel functions

11 Cavity Modes r θ TE 1,1 TE 0,1 TM 0,1 TE,1 TE r,θ Cylindrical (or pillbox) cavities are more common than rectangular cavities. The indices here are m = number of full wave variations around theta n = number of half wave variations along the diameter P = number of half wave variations along the length The frequencies of these cavities are given by f = c/(π) * (ζ/r) Where ζ is the n th root of the m th bessel function for TE modes or the n th root of the derivative of the m th bessel function for TE modes or

12 TM 010 Accelerating mode Electric Fields Almost every RF cavity operates using the TM 010 accelerating mode. Magnetic Fields This mode has a longitudinal electric field in the centre of the cavity which accelerates the electrons. The magnetic field loops around this and caused ohmic heating.

13 TM 010 Monopole Mode.405r Ez = EJ 0 0 e R H = 0 H z r = 0 i.405r H EJ e E E ϕ = 0 1 Z0 R ϕ r = 0 = 0 iωt iωt Z 0 =377 Ohms H E Beam

14 A standing wave cavity

15 Accelerating Voltage Ez, at t=0 Ez, at t=z/v Position, z Position, z Normally voltage is the potential difference between two points but an electron can never see this voltage as it has a finite velocity (ie the field varies in the time it takes the electron to cross the cavity The voltage now depends on what phase the electron enters the cavity at. If we calculate the voltage at two phases 90 degrees apart we get real and imaginary components

16 Accelerating voltage An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude, + L / iω z/ c E evb e Ez( z, t) e dz L / = = To receive the maximum kick with multiple cells the particle should traverse the cavity in a half RF period (see end of lecture). L = c f

17 Transit time factor An electron travelling close to the speed of light traverses through a cavity. During its transit it sees a time varying electric field. If we use the voltage as complex, the maximum possible energy gain is given by the magnitude, E ev + L / iω z/ c e Ez ( z, t) e dz E0LT L / = = = Where T is the transit time factor given by T + L / i z/ c Ez ( z, t) e dz sin L / = = + L / L / E z (, ) z t dz ( g ) ω π βλ π g βλ For a gap length, g. For a given Voltage (=E 0 L) it is clear that we get maximum energy gain for a small gap. Transit time factor, T g/βλ

18 Overvoltage To provide a stable bunch you often will accelerate off crest. This means the particles do not experience the maximum beam energy. V b =V c cos(φ s ) = V c q Where V c is the cavity voltage and V b is the voltage experienced by the particle, φ is the phase shift and q is known as the overvoltage. V V p Stable region φ s φ Phase stability is given by off-crest acceleration

19 For TM010 mode = R + L / iω z/ c V Ez ( z, t) e dz L / = E + L / 0 L / E ( ω ) ( ωz c) = 0 = E 0 cos z / c dz sin / ω / c ( ωl c) sin / ω / c + L / L / This is often approximated as Where L=c/f, T=/π Ez, at t=z/v Position, z Hence voltage is maximised when L=c/f V = E LT z0 cos ( ϕ ) V = E0 cos( ϕ ) L π

20 Does this mean we don t get breakdown in vacuum? Gas Breakdown If we apply a high voltage across a gap we can ionise the molecules in the intervening gas. At high pressure the mean free path is too low to gain enough momentum At low pressure there are not enough molecules to ionise.

21 Field Emission High electric fields can lead to electrons quantum tunnelling out of the structure creating a field emitted current. Once emitted this field emitted current can interact with the cavity fields. Although initially low energy, the electrons can potentially be accelerated to close to the speed of light with the main electron beam, if the fields are high enough. This is known as dark current trapping.

22 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. b E local =β E 0 h Beta h/b

23 Vacuum 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 vaporises and the gas is then ionised by field emission. The exact mechanisms are still not well understood. When this occurs all the incoming RF is reflected back up the coupler. This is the major limitation to gradient in most pulsed RF cavities and can permanently damage the structure.

24 Kilpatrick Limits A rough empirical formula for the peak surface electric field is It is not clear why the field strength decreases with frequency. It is also noted that breakdown is mitigated slightly by going to lower group velocity structures. The maximum field strength also varies with pulse length as t -0.5 (only true for a limited number of pulse lengths) As a SCRF cavity would quench long before breakdown, we only see breakdown in normal conducting structures.

25 Dark Current Trapping When we looked at beam dynamics we saw that we could inject a low energy bunch in a beta=v/c=1 structure and it could be accelerated to the speed of light and arrive on crest. If we have field emitted electrons in the structure these could also be capture and can travel with the main beam. The gradient at which this occurs is given by

26 Surface Resistance As we have seen when a time varying magnetic field impinges on a conducting surface current flows in the conductor to shield the fields inside the conductor. However if the conductivity is finite the fields will not be completely shielded at the surface and the δ field will penetrate into the surface. This causes currents to flow and hence power is absorbed in the surface which is converted to heat. Skin depth is the distance in the surface that the current has reduced to 1/e of the value at the surface, denoted by. δ = σωµ The surface resistance is defined as R surf = 1 δσ For copper 1/σ = 1.7 x 10-8 Ωm x Current Density, J.

27 Power Dissipation The power lost in the cavity walls due to ohmic heating is given by, R surface is the surface resistance 1 Pc = Rsurface H ds This is important as all power lost in the cavity must be replaced by an rf source. A significant amount of power is dissipated in cavity walls and hence the cavities are heated, this must be water cooled in warm cavities and cooled by liquid helium in superconducting cavities.

28 Pulsed Heating Pulsed RF however has problems due to heat diffusion effects. Over short timescales (<10ms) the heat doesn t diffuse far enough into the material to reach the water cooling. This means that all the heat is deposited in a small volume with no cooling. Cyclic heating can lead to surface damage if the temperature rise creates thermal stress (~40 K). The power deposited is P = d RH s max T = P d R s t = pulse πρκc e µω 0 σ And the temperature on the surface is ρ is density, κ is thermal conductivity and c e is specific heat

29 Peak Surface Fields The accelerating gradient is the average gradient seen by an electron bunch, Vc Eacc = L The limit to the energy in the cavity is often given by the peak surface electric and magnetic fields. Thus, it is useful to introduce the ratio between the peak surface electric field and the accelerating gradient, and the ratio between the peak surface magnetic field and the accelerating gradient. E E max acc = π For a pillbox H E max / acc A m = 430 MV / m Electric Field Magnitude

30 Maximum Gradient Limits All the limiting factors scale differently with frequency. They also mostly vary with pulse length. The limiting factor tends to be different from cavity to cavity. For a CW machine the gradient is limited by average heating instead. Also need to think about the electricity bill as 1 MW is 00 per day.

31 Average Heating In normal conducting cavities, the RF deposits large amounts of power as heat in the cavity walls. This heat is removed by flushing cooling water through special copper cooling channels in the cavity. The faster the water flows (and the cooler), the more heat is removed. For CW cavities, the cavity temperature reaches steady state when the water cooling removes as much power as is deposited in the RF structure. (Limit is ~ 1 MW but 500 kw is safer) This usually is required to be calculated in a Finite Element code to determine temperature rises. Temperature rises can cause surface deformation, surface cracking, outgassing or even melting. By pulsing the RF we can reach much higher gradients as the average power flow is much less than the peak power flow.

32 Q factor Pillbox E Pc = π R R+ L Rsurface J Z 0 ( ) (.405) 0 1 U πε E = Q RLJ ( ) ωµ RL 453 L/ R R+ L R R 1 + L/ R 0 = = ( ) ( ) ( ) surface 453 L/ R G = = L/ R surface

33 Shunt Impedance Another useful definition is the shunt impedance, 1 Vc Rs = Pc This quantity is useful for equivalent circuits as it relates the voltage in the circuit (cavity) to the power dissipated in the resistor (cavity walls). Shunt Impedance is also important as it is related to the power induced in the mode by the beam (important for unwanted cavity modes)

34 TM010 Shunt Impedance V c H = EL 0 π i.405r ϕ = EJ 0 1 Z0 R 1 Pc = Rsurface H ds E.405r P = R r J dr R 0 c, ends surface 1 Z π 0 E P RL R J 0 c, walls = π Z0 surface 1 E Pc = π R R+ L Rsurface J Z (.405) 0 ( ) (.405) 0 1 R s ( ZL 0 ) ( + ) (.405) 5x10 = = 3 π R R L R J R surface 1 4 surface

35 Similarly larger apertures lead to higher peak fields. Using thicker walls has a similar effect. Higher frequencies need smaller apertures as well Cavity geometry Figures borrowed from Sami Tantawi The shunt impedance is strongly dependant on aperture

36 Multicell It takes x4 power to double the voltage in one cavity but only x to use two cavities/cells to achieve the same voltage (R s ~number of cells). To make it more efficient we can add either more cavities or more cells. This unfortunately makes it worse for wakefields (see later lectures) and you get less gradient per unit power. In order to make our accelerator more compact and cheaper we can add more cells. We have lots of cavities coupled together so that we only need one coupler. For N cells the shunt impedance is given by R total = NR sin gle This however adds complexity in tuning, wakefields and the gradient of all cells is limited by the worst cell.