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

Electrostatic Acceleration + - - +

Van-de Graaff - 1930s 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).

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.

Early Linear Accelerators (Drift Tube) Proposed by Ising (1925) First built by Wideröe (1928) 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.

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

Cavity Quality Factor An important definition is the cavity Q factor, given by Where U is the stored energy given by, The Q factor is 2p times the number of rf cycles it takes to dissipate the energy stored in the cavity. The Q factor determines the maximum energy the cavity can fill to with a given input power.

Cavities (w/c)2=(mp/a)2+ (np/b)2+ (pp/L)2 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 BT 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. a L The resonant frequency of a rectangular cavity can be given by (w/c)2=(mp/a)2+ (np/b)2+ (pp/L)2 Where a, b and L are the width, height and length of the cavity and m, n and p are integers

Pillbox Cavities Transverse Electric (TE) modes Wave equation in cylindrical co-ordinates Transverse Electric (TE) modes Transverse Magnetic (TM) modes Solution to the wave equation

Bessel Function Jm(kTr) m=0 Ez (TM) and Hz (TE) vary as Bessel functions in pill box cavities. All functions have zero at the centre except the 0th order Bessel functions. m=1 m=2 m=3 kTr First four 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 1st order Bessel functions

Cavity Modes r TE2,1 TE1,1 TE0,1 TM0,1 TEr,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/(2p) * (z/r) Where z is the nth root of the mth bessel function for TE modes or the nth root of the derivative of the mth bessel function for TE modes or

TM010 Accelerating mode Electric Fields Almost every RF cavity operates using the TM010 accelerating mode. 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. Magnetic Fields

TM010 Monopole Mode H E Beam Z0=377 Ohms

A standing wave cavity What is the purpose The maximum volts per metre determines the acceleration of the electrons

Accelerating Voltage Ez, at t=0 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 Ez, at t=z/v Position, z 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 Position, z

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, To receive the maximum kick with multiple cells the particle should traverse the cavity in a half RF period (see end of lecture).

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, Where T is the transit time factor given by For a gap length, g. For a given Voltage (=E0L) it is clear that we get maximum energy gain for a small gap.

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

For TM010 mode Ez, at t=z/v Position, z Hence voltage is maximised when L=c/2f This is often approximated as Where L=c/2f, T=2/p

Peak Surface Fields The accelerating gradient is the average gradient seen by an electron bunch, 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. For a pillbox Electric Field Magnitude

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. Current Density, J. However if the conductivity is finite the fields will not be completely shielded at the surface and the field will penetrate into the surface. x . 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 The surface resistance is defined as For copper 1/s = 1.7 x 10-8 Wm

Power Dissipation The power lost in the cavity walls due to ohmic heating is given by, Rsurface is the surface resistance 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.

Capacitor E-Field The electric field of the TM010 mode is contained between two metal plates – This is identical to a capacitor. This means the end plates accumulate charge and a current will flow around the edges Surface Current

Inductor – B-Field Surface Current The surface current travels round the outside of the cavity giving rise to a magnetic field and the cavity has some inductance. –

Resistor Finally, if the cavity has a finite conductivity, the surface current will flow in the skin depth causing ohmic heating and hence power loss. Surface Current This can be accounted for by placing a resistor in the circuit. In this model we assume the voltage across the resistor is the cavity voltage. Hence R takes the value of the cavity shunt impedance (not Rsurface).

Equivalent circuits To increase the frequency the inductance and capacitance has to be increased. The stored energy is just the stored energy in the capacitor. The voltage given by the equivalent circuit does not contain the transit time factor, T. So remember Vc=V0 T

Shunt Impedance Another useful definition is the shunt impedance, 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)

TM010 Shunt Impedance

Geometric shunt impedance, R/Q If we divide the shunt impedance by the Q factor we obtain, This is very useful as it relates the accelerating voltage to the stored energy. Also like the geometry constant this parameter is independent of frequency and cavity material.

TM010 R/Q

Geometry Constant It is also useful to use the geometry constant This allows different cavities to be compared independent of size (frequency) or material, as it depends only on the cavity shape. The Q factor is frequency dependant as Rs is frequency dependant.

Q factor Pillbox

Equivalent circuits These simple circuit equations can now be used to calculate the cavity parameters such as Q and R/Q. In fact equivalent circuits have been proven to accurately model couplers, cavity coupling, microphonics, beam loading and field amplitudes in multicell cavities.

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

Frequency Scaling Rsurf ~ f0.5 normal conducting Rsurf ~ f2 superconducting Qo ~ f-0.5 normal conducting Qo ~ f-2 superconducting Rs ~ f-0.5 normal conducting Rs ~ f-2 superconducting R/Q ~ f0 normal conducting R/Q ~ f0 superconducting

Multicell It takes x4 power to double the voltage in one cavity but only x2 to use two cavities/cells to achieve the same voltage (Rs ~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 This however adds complexity in tuning, wakefields and the gradient of all cells is limited by the worst cell.

Synchronous particle Imagine we have a series of gaps. The phase change between two gaps when the beam arrives is given by Where ja is the phase advance, (the phase difference between adjacent coupled cavities) Hence the distance between cells should be In a linac we choose a synchronous phase fs and design the lengths so that the synchronous particle sees the desired phase (not always constant) For a standing wave structure the synchronous phase occurs when the cavity is half a free space wavelength long.