Particle Accelerator Physics pp 641-665 | Cite as

# Beam-Cavity Interaction*

## Abstract

The proper operation of the rf-system in a particle accelerator depends more than any other component on the detailed interaction with the particle beam. This results from the observation that a particle beam can induce fields in the accelerating cavities of significant magnitude compared to the generator produced voltages and we may therefore not neglect the presence of the particle beam.

## Keywords

Coupling Coefficient Particle Beam Phase Oscillation Generator Voltage Loss Parameter

The proper operation of the rf-system in a particle accelerator depends more than any other component on the detailed interaction with the particle beam. This results from the observation that a particle beam can induce fields in the accelerating cavities of significant magnitude compared to the generator produced voltages and we may therefore not neglect the presence of the particle beam. This phenomenon is called beam loading and can place severe restrictions on the beam current that can be accelerated. In this section, main features of such interaction and stability conditions for most efficient and stable particle acceleration will be discussed.

## 19.1 Coupling Between rf-Field and Particles

In our discussions about particle acceleration we have tacitly assumed that particles would gain energy from the fields in accelerating cavities merely by meeting the synchronicity conditions. This is true for a weak particle beam which has no significant effect on the fields within the cavity. As we try, however, to accelerate an intense beam, the actual accelerating fields become modified by the presence of considerable electrical particle beam currents. This beam loading can ultimately limit the maximum beam intensity.

The phenomenon of beam loading will be defined and characterized in this section leading to conditions and parameters to assure positive energy flow from the rf-power source to the beam. Fundamental consideration to this discussion are the principles of energy conservation and linear superposition of fields which allow us to study field components from one source independent of fields generated by other sources. Specifically, we may treat beam induced fields separately from fields generated by rf-power sources.

### 19.1.1 Network Modelling of an Accelerating Cavity

*I*

_{g}from a generator and the particle beam

*I*

_{b}.

*R*

_{s}of an empty cavity as defined in ( 18.74)

*β*is the coupling coefficient still to be defined. This coefficient depends on the actual hardware of the coupling arrangement for the rf-power from the generator at the entrance to the cavity and quantifies the generator impedance as seen from the cavity in units of the cavity shunt impedance

*R*

_{s}(Fig. 19.1). Since this coupling coefficient depends on the hardware, we need to specify the desired operating condition to determine the proper adjustment of the coupling during assembly. This adjustment is done by either rotating a loop coupler with respect to the cavity axis or adjustment of the aperture in case of capacitive coupling through a hole.

*L*and capacitance

*C*form a parallel resonant circuit with the resonant frequency

*Y*

_{L}is the loaded cavity admittance including energy transfer to the beam and

*V*

_{g}is the generator voltage. Unless otherwise noted, the voltages, currents and power used in this section are the amplitudes of otherwise oscillating quantities. At resonance where all reactive power vanishes we use the generator current

*I*

_{g}and network admittance \(Y = Y _{\text{g}} + Y _{\text{L}}\) to replace the generator voltage

*R*=

*R*

_{s}at resonance

*Ψ*becomes from (19.12) with

*ω*≈

*ω*

_{r}

^{∘}, which was introduced here to be consistent with our definition of the synchronous phase

*ψ*

_{s}. The variation of the tuning angle is shown in Fig. 19.2 as a function of the generator frequency. From (19.12), the generator voltage at the cavity is finally

At frequencies below the resonance frequency the tuning angle is positive and therefore the generator current lags the voltage by the phase *Ψ*. This case is also called inductive detuning since the impedance looks mainly inductive. Conversely, the detuning is called capacitive detuning because the impedance looks mostly capacitive for frequencies above resonance frequency.

A bunched particle beam passing through a cavity acts as a current just like the generator current and therefore the same relationships with respect to beam induced voltages exist. In case of capacitive detuning, for example, the beam induced voltage *V*_{b} lags in phase behind the beam current *I*_{b}.

*V*

_{gr}at resonance and without beam loading while voltage and current are in phase. From Fig. 19.1 we get

This is the cavity voltage seen by a negligibly small beam and can be adjusted to meet beam stability requirements by varying the tuning angle *Ψ* and rf-power *P*_{g}.

## 19.2 Beam Loading and Rf-System

*I*

_{h}of the beam current and find for bunches short compared to the rf-wavelength

*I*

_{b}is the average beam current and

*h*the harmonic number. The approximation for short bunches with \(\ell\ll \lambda _{\text{rf-}}\) holds as long as \(\sin k_{\text{rf-}}\ell \approx k_{\text{rf-}}\ell\) with \(k_{\text{rf-}} = 2\pi /\lambda _{\text{rf-}}\). For longer bunches the factor 2 becomes a more complicated formfactor as can be derived from an appropriate Fourier expansion. At the resonance frequency

*ω*

_{r}=

*h ω*

_{0}, the beam induced voltage in the cavity is then with (19.8) from (19.15)

The resulting cavity voltage is the superposition of both voltages, the generator and the induced voltage. This superposition, including appropriate phase factors, is often represented in a phasor diagram. In such a diagram a complex quantity \(\tilde{z}\) is represented by a vector of length \(\left \vert \tilde{z}\right \vert\) with the horizontal and vertical components being the real and imaginary part of \(\tilde{z}\), respectively. The phase of this vector increases counter clockwise and is given by \(\tan \varphi =\) Im\(\left (\tilde{z}\right )/\) Re\(\left (\tilde{z}\right )\). In an application to rf-parameters we represent voltages and currents by vectors with a length equal to the magnitude of voltage or current and a counter clockwise rotation of the vector by the phase angle \(\varphi\).

*Ψ*behind the beam current. The resulting cavity voltage \(\tilde{\boldsymbol{V }}_{\mathrm{cy}}\) is the phasor addition of both voltages \(\tilde{\boldsymbol{V }}_{\mathrm{g}} +\tilde{\boldsymbol{ V }}_{\mathrm{b}}\) as shown in Fig. 19.3.

*ψ*

_{s}the synchronous phase. To maximize the energy flow from the generator to the cavity the load must be matched such that it appears to the generator purely resistive. This is achieved by adjusting the phase

*ψ*

_{g}to get the cavity voltage

*V*

_{cy}and generator current

*I*

_{g}in phase which occurs for

*I*

_{b}, rf–generator power

*P*

_{g}, coupling coefficient

*β*, and shunt impedance

*R*

_{s}to sustain a cavity voltage

*V*

_{cy}. Specifically, considering that the rf-power

*P*

_{g}and coupling coefficient

*β*is fixed by the hardware installed a maximum supportable beam current can be derived as a function of the desired or required cavity voltage. Solving for the cavity voltage, (19.31) becomes after some manipulation

*β*. Optimum coupling can be derived from \(\partial P_{\mathrm{g}}/\partial \beta = 0\) with the solution

*V*

_{cy}sin

*ψ*

_{s}is therefore from (19.31) with (19.35)

Conditions have been derived assuring most efficient power transfer to the beam by proper adjustment of the cavity power input coupler to obtain the optimum coupling coefficient. Of course this coupling coefficient is optimum only for a specific beam current which in most cases is chosen to be the maximum desired beam current.

*P*

_{r}is the reflected power which vanishes for the case of optimum coupling.

## 19.3 Higher-Order Mode Losses in an Rf-Cavity

The importance of beam loading for accurate adjustments of the rf-system has been discussed qualitatively but not yet quantitatively. In this paragraph, quantitative expressions will be derived for beam loading. Accelerating cavities constitute an impedance to a particle current and a bunch of particles with charge *q* passing through a cavity induces electromagnetic fields into a broad frequency spectrum limited at the high frequency end by the bunch length. The magnitude of the excited frequencies in the cavity depends on the frequency dependence of the cavity impedance, which is a function of the particular cavity design and need not be known for this discussion. Fields induced within a cavity are called modes, oscillating at different frequencies with the lowest mode being the fundamental resonant frequency of the cavity. Although cavities are designed primarily for one resonant frequency, many higher-order modes or HOM’scan be excited at higher frequencies. Such modes occur above the fundamental frequency first at distinct well-separated frequencies with increasing spectral densities at higher frequencies.

For a moment we consider here only the fundamental frequency and deal with higher-order modes later. Fields induced by the total bunch charge act back on individual particles modifying the overall accelerating voltage seen by the particle. To quantify this we use the fundamental theorem of beam loading formulated by Wilson [1] which states that each particle within a bunch sees one half of the induced field while passing through the cavity.

*q*passing through a lossless cavity inducing a voltage

*V*

_{i1}in the fundamental mode. This induced field is opposed to the accelerating field since it describes a loss of energy. While the bunch passes through the cavity this field increases from zero reaching a maximum value at the moment the particle bunch leaves the cavity. Each particle will have interacted with this field and the energy loss corresponds to a fraction

*f*of the induced voltage

*V*

_{i, h}, where the index

*h*indicates that we consider only the fundamental mode. The total energy lost by the bunch of charge

*q*is

*c*

_{1}is a constant.

*q*due to its own induced field is therefore

This theorem will be used to determine the energy transfer from cavity fields to a particle beam. To calculate the induced voltages in rf-cavities, or in arbitrarily shaped vacuum chambers providing some impedance for the particle beam can become very complicated. For cylindrically symmetric cavities the induced voltages can be calculated numerically with programs like SUPERFISH [2], URMEL[3] or MAFIA [3].

*k*which can be determined either by electronic measurements or by numerical calculations. This loss parameter for the fundamental mode loss of a bunch with charge

*q*is defined by

*h*indicates that the parameter should be taken at the fundamental frequency. The loss parameter can be expressed in terms of cavity parameters. From the definition of the cavity quality factor ( 18.80) and cavity losses from ( 18.77) we get

*ω*is the frequency and

*W*the stored field energy in the cavity. Applying this to the induced field, we note that

*Δ E*

_{h}is equal to the field energy

*W*

_{h}and combining (19.47), (19.48) the loss parameter to the fundamental mode in a cavity with shunt impedance

*R*

_{h}and quality factor

*Q*

_{h}is

*r*

_{hom}is the ratio of the total energy losses into all cavity modes to the loss into the fundamental mode only. The induced higher order field energy in the cavity is therefore

*k*

_{n}for an arbitrary

*n*th-mode and get analogous to (19.49)

*R*

_{n}and

*Q*

_{n}are the shunt impedance and quality factor for the

*n*th-mode or frequency

*ω*

_{n}, respectively. The total loss parameter due to all modes is by linear superposition

The task to determine the induced voltages has been reduced to the determination of the loss parameters for individual modes or if this is not possible or desirable we may use just the overall loss parameter *k* as may be determined experimentally. This is particularly convenient for cases where it is difficult to calculate the mode losses but much easier to measure the overall losses by electronic measurements.

The higher-order mode losses will become important for discussion of beam stability since these fields will act back on subsequent particles and bunches thus creating a coupling between different parts of one bunch or different bunches.

### 19.3.1 Efficiency of Energy Transfer from Cavity to Beam

Higher-order mode losses affect the efficiency by which energy is transferred to the particle beam. Specifically, since the higher-order mode losses depend on the beam current we must expect some limitation in the current capability of the accelerator.

*V*

_{g}is the generator voltage and

*Ψ*

_{g}the generator voltage phase with respect to the particle beam. To combine the generator voltage with the induced voltage we use phasor diagrams in the complex plane.

*Ψ*

_{g}from the real axis representing the cavity state just before the beam passes. The beam induced voltage is parallel and opposite to the real axis. Both vectors add up to the voltage

*V*just after the beam has left the cavity.

*α*is the proportionality factor between the energy gain

*Δ E*and the square of the voltage defined by \(\alpha =\varDelta E/V ^{2}.\) With (19.45), (19.46), we get from (19.55) the net energy transfer to a particle bunch [4]

*W*

_{cy}=

*α V*

_{cy}

^{2}and the energy transfer efficiency to the beam becomes

## 19.4 Beam Loading

*m*

_{b}of the fundamental rf-wavelength. The induced voltage decays exponentially by a factor e

^{−ρ}between two consecutive bunches with

*t*

_{b}is the time between bunches and

*t*

_{d}the cavity voltage decay time for the fundamental mode. The phase of the induced voltage varies between the passage of two consecutive bunches by

*V*

_{i}.

*V*

_{b}acting back on the bunch is therefore

*V*

_{i, h}can be expressed in more practical units. Considering the damping time ( 18.62) for fields in a cavity we note that two damping times exist, one for the empty unloaded cavity

*t*

_{d0}and a shorter damping time

*t*

_{d}when there is also a beam present. For the unloaded damping time we have from ( 18.62)

*Q*

_{0h}is the unloaded quality factor. From (19.45), (19.47) we get with

*q*=

*I*

_{0}

*t*

_{b}, where

*I*

_{0}is the average beam current,

*β*, we get from (19.9), (19.64)

We are finally in a position to calculate from (19.63), (19.68) the total beam induced cavity voltage *V*_{b} in the fundamental mode for circular accelerators.

## 19.5 Phase Oscillation and Stability

*ψ*

_{s}we find with (19.21) and since

*V*

_{gr}cos

*Ψ*> 0 the condition for phase stability sin(

*ψ*

_{g}−

*Ψ*

_{m}) < 0 or

The stability condition is always met for rf-cavities with optimum coupling *β* = *β*_{opt}.

### 19.5.1 Robinson Damping

Correct tuning of the rf-system is a necessary but not a sufficient condition for stable phase oscillations. In Chap. 12 we found the occurrence of damping or anti-damping due to forces that depend on the energy of the particle. Such a case occurs in the interaction of bunched particle beams with accelerating cavities or vacuum chamber components which act like narrow band resonant cavities. The revolution time of a particle bunch depends on the average energy of particles within a bunch and the Fourier spectrum of the bunch current being made up of harmonics of the revolution frequency is therefore energy dependent. On the other hand by virtue of the frequency dependence of the cavity impedance, the energy loss of a bunch in the cavity due to beam loading depends on the revolution frequency. We have therefore an energy dependent loss mechanism which can lead to damping or worse anti-damping of coherent phase oscillation and we will therefore investigate this phenomenon in more detail. Robinson [6] studied first the dynamics of this effect generally referred to as Robinson damping or Robinson instability.

Here the resonance curve or impedance spectrum is shown for the case of a resonant frequency above the beam frequency *h ω*_{0} in Fig. 19.7a and below the beam frequency in Fig 19.7b. Consistent with the arguments made above we would expect damping in case of Fig. 19.7b for a beam above transition and anti-damping in case of Fig. 19.7a. Adjusting the resonance frequency of the cavity to a value below the beam frequency *h ω*_{0}where *ω*_{0}is the revolution frequency, is called capacitive detuning. Conversely, we would tune the cavity resonance frequency above the beam frequency (*ω*_{r} > *h ω*_{0}) or inductively detune the cavity for damping below transition energy (Fig. 19.7a).

*I*

_{b}is the average circulating beam current and \(\varphi\) a phase shift with respect to the reference particle. The Fourier coefficient for bunches short compared to the wavelength of the harmonic is given by

*h*th harmonic

*Ω*

_{s}the synchrotron oscillation frequency of the phase oscillation. We insert this into (19.75) and get after expanding the trigonometric functions for small oscillation amplitudes \(\varphi _{0} \ll 1\)

*h ω*

_{0}±

*Ω*

_{s}. Folding the expression for the beam current with the cavity impedance defines the energy loss of the particle bunch while passing through the cavity. The cavity impedance is a complex quantity which was derived in (19.11) and its real part is shown together with the beam spectrum in Fig. 19.8. The induced voltage in the cavity by a beam

*I*

_{h}(

*t*) =

*I*

_{h}cos

*h ω*

_{0}

*t*is

*Z*

^{0},

*Z*

^{+}and

*Z*

^{−}are the real r and imaginary i cavity impedances at the frequencies

*h ω*

_{0},

*h ω*

_{0}+

*Ω*

_{s},

*h ω*

_{0}−

*Ω*

_{s}respectively. We make use of the expression for the phase oscillation (19.76) and its derivative

*h ω*

_{0}

*T*

_{0}

*I*

_{b}we get the rate of relative energy loss per unit charge

*T*

_{0}is the revolution time and

*I*

_{b}the average beam current.

*α*

_{s0}is the radiation damping in electron accelerators. The total damping decrement must be positive for beam stability. The interaction of the beam with the accelerating cavity above transition is stable for all values of the beam current if \(Z_{\mathrm{r}}^{+} <Z_{\mathrm{r}}^{-}\) or if the cavity resonant frequency is capacitively detuned. Due to the imaginary part of the impedance the interaction of beam and cavity leads to a synchrotron oscillation frequency shift given by

*h ω*

_{0}and a frequency shift for coherent bunch-phase oscillations due to the imaginary part of the cavity impedances. For small frequency shifts \(\varDelta \varOmega _{\mathrm{s}} =\varOmega _{\mathrm{s}} -\varOmega _{\mathrm{s0}},\) (19.88) can be linearized for

This conclusion may in special circumstances be significantly different due to other passive cavities in the accelerator. The shift in the synchrotron tune is proportional to the beam current and can be used as a diagnostic tool to determine the cavity impedance or its deviation from the ideal model (19.90).

In the preceding discussion it was assumed that only resonant cavities contribute to Robinson damping. This is correct to the extend that other cavity like structures of the vacuum enclosure in a circular accelerator have a low quality factor *Q* for the whole spectrum or at least at multiples of the revolution frequency and therefore do not contribute significantly to this effect through a persistent energy loss over many turns. Later we will see that such low-*Q* structures in the vacuum chamber may lead to other types of beam instability.

### 19.5.2 Potential Well Distortion

The synchrotron frequency is determined by the slope of the rf-voltage at the synchronous phase. In the last subsection the effect of beam loading at the cavity fundamental frequency was discussed demonstrating the need to include the induced voltages in the calculation of the synchrotron oscillation frequency. These induced voltages cause a perturbation of the potential well and as a consequence a change in the bunch length. In this subsection we will therefore also include higher-order interaction of the beam with its environment.

*f*

_{t}, which is determined by vacuum chamber dimensions, the impedance is predominantly inductive and becomes capacitive above the transition frequency. We are looking here only for fields with wavelength longer than the bunch length which may distort the rf-voltage waveform such as to change the slope for the whole bunch. Later we will consider shorter wavelength which give rise to perturbations within the bunch. Because the bunch length is generally of the order of vacuum chamber dimensions we only need to consider the impedance spectrum below transition frequency which is predominantly inductive. To preserve generality, however, we assume a more general but still purely imaginary impedance defined by

Above transition energy *η*_{c}cos*ψ*_{s} > 0 and therefore the frequency shift is positive for \(\mathrm{Im}(Z_{_{\parallel }}/n) <0\) and negative for \(\mathrm{Im}\{Z_{_{\parallel }}/n\}> 0\). We note specifically that the shift depends strongly on the bunch length and increases with decreasing bunch length, a phenomenon we observe in all higher-order mode interactions.

Note that this shift of the synchrotron oscillation frequency does not appear for coherent oscillations since the induced voltage also moves with the bunch oscillation. The bunch center actually sees always the unaltered rf-field and oscillates according to the slope of the unperturbed rf-voltage. The coherent synchrotron oscillation frequency therefore need not be the same as the incoherent frequency. This has some ramification for the experimental determination of the synchrotron oscillation frequency.

*σ*

_{ℓ0}is the unperturbed bunch length.

*ℓ*

*Δ p*will not change due to potential-well distortions. For proton or ion bunches we employ the same derivation for the bunch lengthening but note that the bunch length scales with the energy spread in such a way that the product of bunch length

*ℓ*and momentum spread

*Δ p*remains constant. Therefore \(\ell\propto 1/\sqrt{\varOmega _{\mathrm{s}}}\) and the perturbed bunch length is from (19.99) with \(\ell= (\overline{R}/h)\,\varphi _{\ell}\)

## Problems

**19.1 (S).** Consider an electron storage ring to be used as a damping ring for a linear collider. The energy is *E* = 1. 21 GeV, circumference *C* = 35. 27 m, bending radius *ρ* = 2. 037 m, momentum compaction factor *α*_{c} = 0. 01841, rf harmonic number *h* = 84, cavity shunt impedance of *R*_{cy} = 8. 4 M\(\Omega\). An intense bunch of *N*_{e} = 5 × 10^{10} particles is injected in a single pulse and is stored for only a few msec to damp to a small beam emittance. Specify and optimize a suitable rf-system and calculate the required rf-cavity power, cavity voltage, coupling factor first while ignoring beam loading and then with beam loading. Assume a quantum lifetime of 1 h.

**19.2 (S).** Show that for bunches short compared to the rf-wavelength the harmonic amplitudes are *I*_{h} = 2*I*_{b}.

## References

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**212**, 13 (1983)CrossRefGoogle Scholar - 4.P.L. Morton, V.K. Neil, The interaction of a ring of charge passing through a cylindrical rf cavity. vol. UCRL-18103 (LBNL, Berkeley, 1968), p. 365Google Scholar
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