Statistical and Collective Effects*

Abstract

Transverse and longitudinal beam dynamics as discussed in earlier chapters is governed by purely single-particle effects where the results do not depend on the presence of other particles or any interactive environment. Space-charge effects were specifically excluded. This restriction is sometimes too extreme and collective effects must be taken into account where significant beam intensities are desired. In most applications high beam intensities are desired and it is therefore prudent to test for the appearance of space charge and other intensity effects.

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Transverse and longitudinal beam dynamics as discussed in earlier chapters is governed by purely single-particle effects where the results do not depend on the presence of other particles or any interactive environment. Space-charge effects were specifically excluded. This restriction is sometimes too extreme and collective effects must be taken into account where significant beam intensities are desired. In most applications high beam intensities are desired and it is therefore prudent to test for the appearance of space charge and other intensity effects.

Collective effects can be divided into two distinct groups according to the physics involved. The compression of a large number of charged particles into a small volume increases the probability for collisions of particles within the same beam. Because particles perform synchrotron and betatron oscillations, statistical collisions occur in longitudinal, as well as transverse phase space often causing a mixing of phase space coordinates. The other group of collective effects includes effects which are associated with electromagnetic fields generated by the collection of all particles in a beam.

The study and detailed understanding of the cause and nature of collective effects or collective instabilities with corrective measures is important for a successful design of the accelerator. Most accelerator design and developments are conducted to eliminate collective effects as much as possible through self-imposed limitation on the performance or installation of feedback systems and other stabilizing control mechanisms. Beyond that, we also must accept limitations in beam performance imposed by nature or lack of understanding and technological limits. Pursuit of accelerator physics is the attempt to explore and push such limits as far as nature and general understanding of the subject allows.

1 Statistical Effects

Coupling of individual particles to the presence of other particles may occur through very short range forces in collisions with each other. In this section, we will discuss statistical effects related to the finite number of particles and from collision processes within a particle bunch.

1.1 Schottky Noise

Electrical current is established by moving charged particles. The finite electrical charge and finite number of particles gives rise to statistical variations of the electrical current. This phenomenon has been observed and analyzed by Schottky [1] and we will discuss this Schottky noise in the realm of particle dynamics in circular accelerators. The information included in the Schottky noise is of great diagnostic importance for the nondestructive determination of particle beam parameters, a technique which has been developed at the CERN Intersecting Storage Ring (ISR) [2] and has become a standard tool of beam diagnostics.

We consider a particle k with charge q orbiting in an accelerator with the angular revolution frequency ω_k and define a particle line density by $2\pi \overline{R}\lambda (t) = 1$ where $2\pi \overline{R}$ is the circumference of the ring. On the other hand, we may describe the orbiting particle by delta functions

$$\displaystyle{ q = q\int \nolimits _{0}^{2\pi }\sum \limits _{ m=-\infty }^{+\infty }\delta (\omega _{ k}t +\theta _{k} - 2\pi m)\,\mathrm{d}\theta \,, }$$

where ω_k is the angular revolution frequency of the particle k and θ_k its phase at time t = 0. The delta function can be expressed by a Fourier series and the line-charge density at time t becomes

$$\displaystyle{ q\lambda _{k}(t) = \frac{q} {2\pi \overline{R}}\,\left [1 + 2\sum \limits _{n=0}^{\infty }\cos (n\omega _{ k}t + n\theta _{k})\right ]\,. }$$

(21.1)

From a pick up electrode close to the circulating particle, we would obtain a signal with a frequency line spectrum ω = n ω_k where n is an integer. In a real particle beam there are many particles with a finite spread of revolution frequencies ω_k and therefore the harmonic lines n ω_k spread out proportionally to n. For not too high harmonic numbers the frequency spreads do not yet overlap and we are able to measure the distribution of revolution frequencies. Tuning the spectrum analyzer to ω, we observe a signal with an amplitude proportional to $N(\omega /n) \frac{\delta \omega }{n}$ where N(ω∕n) is the particle distribution in frequency space and δ ω the frequency resolution of the spectrum analyzer. The signal from the pick up electrode is proportional to the line-charge density which is at the frequency ω from (21.1) 

$$\displaystyle{ q\lambda _{\mathrm{rms}}(\omega ) = \frac{\sqrt{2}\,q} {2\pi \overline{R}} \,\sqrt{N(\omega /n)\, \frac{\delta \omega } {n}} }$$

(21.2)

and has been derived first by Schottky for a variety of current sources [1]. The spread in the revolution frequency originates from a momentum spread in the beam and measuring the Schottky spectrum allows its nondestructive determination.

Individual particles orbiting in an accelerator perform transverse betatron oscillations which we describe, for example, in the vertical plane by

$$\displaystyle{ y_{k}(t) = a_{k}\cos (\nu _{k}\omega _{k}t +\psi _{k})\,, }$$

(21.3)

where a_k is the amplitude and ψ_k the phase of the betatron oscillation for the particle k at time t = 0. The difference signal from two pick up electrodes above and below the particle beam is, in linear approximation, proportional to the product of the betatron amplitude (21.3) and the line-charge density (21.1) and of the form

$$\displaystyle\begin{array}{rcl} D_{k}(t)& =& A_{k}\sum \limits _{n=0}^{\infty }\cos [(n -\nu _{ k})(\omega _{k}t +\phi _{k})] \\ & & \quad +A_{k}\sum \limits _{n=0}^{\infty }\cos [(n +\nu _{ k})(\omega _{k}t +\varphi _{k})]\,,{}\end{array}$$

(21.4)

where we have ignored terms at frequencies n ω_k. The transverse Schottky signal is composed of two side bands for each harmonic at frequencies

$$\displaystyle{ \omega = (n \pm \nu _{k})\,\omega _{k}\,. }$$

(21.5)

which are also called the fast wave for ω = (n +ν_k)ω_k and the slow wave for ω = (n −ν_k)ω_k.

The longitudinal Schottky noise depends on the rms contribution of all particles which are spread over a range of revolution frequencies due to a momentum spread and over betatron frequencies by virtue of the chromaticity. For $\varDelta \omega _{\mathrm{rms}} =\eta _{\mathrm{c}}\omega _{0}\delta _{\mathrm{rms}}$ and $\varDelta \nu _{\mathrm{rms}} =\xi _{y}\delta _{\mathrm{rms}}$ where ω₀ is the revolution frequency of the bunch center, δ_rms = Δ p_rms∕p₀ the rms relative momentum error, η_c the momentum compaction and ξ_y the vertical chromaticity, the frequency distribution of the signal from the pick up is

$$\displaystyle\begin{array}{rcl} \omega & =& [n \pm (\nu _{y0} +\xi _{y}\delta _{k})]\,(\omega _{0} +\eta _{\mathrm{c}}\omega _{0}\delta _{k}) \\ & =& (n \pm \nu _{y0})\,\omega _{0} + [(n \pm \nu _{y0})\,\eta _{\mathrm{c}} \pm \xi _{y}]\,\omega _{0}\,\delta _{k} + O(\delta ^{2})\,.{}\end{array}$$

(21.6)

The momentum spread δ_k causes a frequency spread which is different for the slow and fast wave. For example, for positive chromaticity above transition, η_c < 0 and the frequency spreads add up for the slow wave and cancel partially for the fast wave. This has been verified experimentally for a coasting proton beam in the ISR [2].

A transverse Schottky scan may exhibit the existence of weak resonances which may dilute the particle density, specifically in a coasting proton or ion beam. To control coasting beam instabilities, it is desirable to make use of Landau damping by introducing a large momentum and tune spread. This tune spread, however, can be sufficiently large to spread over higher order resonances and blow up that part of the beam which oscillates at those resonance frequencies. A Schottky scan can clearly identify such a situation as reported in [2].

In this text we are able to touch only the very basics of Schottky noise and the interested reader is referred to references [3–6] for more detailed discussions on the theory and experimental techniques to obtain Schottky scans and how to interpret the signals.

1.2 Stochastic Cooling

The “noise” signal from a circulating particle beam includes information which can be used to drive a feedback system in such a way as to reduce the beam emittance, longitudinal as well as transverse. Due to the finite number of particles in a realistic particle beam, the instantaneous center of a beam at the location of a pick up electrode exhibits statistical variations. This statistical displacement of a slice of beam converts to a statistical slope a quarter betatron wavelength downstream. The signal from the small statistical displacement of the beam at the pick up electrode can be amplified and fed back to the beam through a kicker magnet located an odd number of quarter wavelength downstream, assuming that the statistical variations do not smear out between pick up electrode and kicker. Van der Meer [7] proposed this approach to reduce the transverse proton beam emittance in ISR for increased luminosity and the process is now known as stochastic cooling.

This process of correction is not a statistical process and we must ask ourselves if this is an attempt to circumvent Liouville’s theorem. It is not. Due to the finite number of particles in the beam, the phase space is not uniformly covered by particles but rather exhibits many holes. The method of stochastic cooling detects the moment one of these holes appears on one or the other side of the beam in phase space. At the same moment, the whole emittance is slightly shifted with respect to the center of the phase space and this shift can be both detected and corrected. The whole process of stochastic cooling therefore only squeezes the “air” out of the particle distribution in phase space. The most prominent application of this method occurs in the cooling of an antiproton beam to reach a manageable beam emittance for injection into high energy proton antiproton colliders. To discuss this process in more detail, theoretically as well as technically, would exceed the scope of this text and the interested reader is referred to a series of articles published in [8].

1.3 Touschek Effect

The concentration of many particles into small bunches increases the probability for elastic collisions between particles. This probability is further enhanced considering that particles perform transverse betatron as well as longitudinal synchrotron oscillations. In each degree of freedom, we have acceptance limits and if a particle’s oscillation amplitude exceeds such limits due, for example, to a collision with another particle one or both particles can get lost. In this section, we discuss the process of single collisions where the momentum transfer is large enough to lead to the loss of both particles involved in the collision and postpone the discussion of multiple collisions with small momentum transfer to the next section.

We may consider two collision processes which could lead to beam loss. First, we observe two particles performing synchrotron oscillations and colliding head-on in such a way that they transfer their longitudinal momentum into transverse momentum. This collision process is insignificant in particle accelerators because the longitudinal motion includes not enough momentum to increase the betatron oscillation amplitude enough for particle loss. On the other hand, transverse oscillations of particles represent large momenta and a transfer into longitudinal momenta can lead to the loss of both particles. This effect was discovered on the first electron storage ring ever constructed [9, 10] and we therefore call this the Touschek effect.

In this text, we will not pursue a detailed derivation of the collision process and refer the interested reader to references [11–13]. Of particular interest is the expression for the beam lifetime as a result of particle losses due to a momentum transfer into the longitudinal phase space exceeding the rf-bucket acceptance of Δ p∕p₀ | _rf. Whenever such a transfer occurs both particles involved in the collision are lost. The beam decay rate is proportional to the number of particles in the bunch and the beam current therefore decays exponentially. Last, but not least, a loss occurs only if there is sufficient momentum in the transverse motion to exceed the rf-momentum acceptance. We assume the momentum acceptance to be limited by the rf-voltage and combining these parameters in a collision theory results in a beam lifetime for a Gaussian particle distribution given by

$$\displaystyle{ \frac{1} {\tau } = - \frac{1} {N_{\mathrm{b}}} \frac{\mathrm{d}N_{\mathrm{b}}} {\mathrm{d}t} = \frac{r_{\mathrm{c}}^{2}\,c\,N_{\mathrm{b}}} {8\pi \,\sigma _{x}\,\sigma _{y}\,\sigma _{\ell}} \frac{\lambda ^{3}} {\gamma ^{2}}D(\epsilon ), }$$

(21.7)

where r_c is the classical particle radius, $\sigma _{x},\sigma _{y},\sigma _{\ell}$ are the standard values of the Gaussian bunch width, height and length, respectively, and $\lambda ^{-1} =\varDelta p/p_{0}\vert _{\mathrm{rf}}$ the momentum acceptance parameter. The function D(ε) (Fig. 21.1) is defined by [13]

$$\displaystyle\begin{array}{rcl} D(\epsilon )& =& \sqrt{\epsilon }\left [-\tfrac{3} {2}\mathrm{e}^{-\epsilon } + \frac{\epsilon } {2}\int \nolimits _{\epsilon }^{\infty }\frac{\ln u} {u}\mathrm{e}^{-u}\,\mathrm{d}u\right. \\ & & \qquad \quad \left.+\tfrac{1} {2}(3\epsilon -\epsilon \,\ln \epsilon +2)\int \nolimits _{\epsilon }^{\infty }\frac{\mathrm{e}^{-u}} {u} \,\mathrm{d}u\right ]\,,{}\end{array}$$

(21.8)

where the argument is

$$\displaystyle{ \epsilon = \left (\frac{\varDelta p_{\mathrm{rf}}} {\gamma \,\sigma _{p}} \right )^{2}\qquad \mathrm{with}\qquad \sigma _{\mathrm{ p}} = \frac{mc\gamma \,\sigma _{x}} {\beta _{x}} \,. }$$

(21.9)

Fig. 21.1

Touschek lifetime function $D\left (\epsilon \right )$

Particle losses due to the Touschek effect is particularly effective at low energies and where the rf-acceptance is small. For high particle densities $N_{\mathrm{b}}/(\sigma _{x}\,\sigma _{y}\,\sigma _{\ell})$ the rf-acceptance should therefore be maximized. This seems to be the wrong thing to do because the bunch length is reduced at the same time and the particle density becomes even higher but a closer look at (21.7) shows us that the Touschek lifetime increases faster with rf-acceptance than it decreases with bunch length.

1.4 Intra-Beam Scattering

The Touschek effect describes collision processes which lead to immediate loss of both colliding particles. In reality, however, there are many other collisions with only small exchanges of momentum. While these collisions do not lead to immediate particle loss, there might be sufficiently many during a damping time in electron storage rings or during the storage time for proton and ion beams to cause a significant increase in the bunch volume, or in the case of a coasting beam an increase in beam cross section. During the discussion of the Touschek effect we neglected the transfer from the longitudinal momentum space into transverse momentum space because the transverse momentum acceptance is larger than the longitudinal acceptance and particles are generally not lost during such an exchange. This is not appropriate any more for the multiple Touschek effect or intra-beam scattering where we are interested in all collisions.

The multiple Touschek effect was observed in the first ever constructed storage ring, AdA (Anello di Accumulatione) in Frascati, Italy. The Touschek effect had been expected and analyzed before but did give too pessimistic beam lifetimes compared to those observed in AdA. A longer beam lifetime had been obtained because of multiple elastic scattering between particles increasing the bunch volume and thereby reducing the Touschek effect [10].

During the exchange of momentum as a consequence of collisions between particles within the same bunch or beam, each degree of freedom can increase its energy or temperature because the beam is able to absorb any amount of energy from the rf- system. We are particularly interested in the growth times of transverse and longitudinal emittances to asses the long-term integrity of the particle beam. The multiple Touschek effect or intra-beam scattering has been studied extensively [14, 15] and we will not repeat here the derivations but merely recount the results.

The growth time of the beam emittances for Gaussian particle distributions are for the longitudinal phase space or momentum and bunch distribution [14, 15]

$$\displaystyle{ \tau _{p}^{-1} = \frac{1} {2\sigma _{p}^{2}} \frac{\mathrm{d}\sigma _{p}^{2}} {\mathrm{d}t} = A\,\frac{\sigma _{h}^{2}} {\sigma _{p}^{2}} f(a,b,c)\,, }$$

(21.10)

where the particle bunch density is expressed by

$$\displaystyle{ A = \frac{r_{\mathrm{c}}^{2}cN_{\mathrm{b}}} {64\pi ^{2}\sigma _{z}\,\sigma _{p}\,\sigma _{x}\,\sigma _{y}\,\sigma _{x^{{\prime}}}\,\sigma _{y^{{\prime}}}\,\beta ^{3}\,\gamma ^{4}} }$$

(21.11)

with the standard dimensions of a Gaussian distribution for the bunch length σ_z, the relative momentum spread σ_p, horizontal and vertical betatron amplitudes $\left (\sigma _{x},\sigma _{y}\right )$ and divergences $\left (\sigma _{x^{{\prime}}},\sigma _{y^{{\prime}}}\right )$ and number of particles per bunch N_b. The constants r_c and β = v∕c, finally, are the classical particle radius and velocity in units of the velocity of light.

The function

$$\displaystyle{ f(a,b,c) = 8\pi \int \nolimits _{0}^{1}\left \{\ln \left [\frac{c^{2}} {2} \left ( \frac{1} {\sqrt{p}} + \frac{1} {\sqrt{q}}\right )\right ] - 0.577..\right \}\frac{1 - 3x^{2}} {\sqrt{pq}} \,\mathrm{d}x }$$

(21.12)

where

$$\displaystyle{ \begin{array}{lll} p = a^{2} + x^{2}(1 - a^{2})\,,&\qquad &q = b^{2} + x^{2}(1 - b^{2})\,, \\ a = \frac{\sigma _{h}} {\gamma \sigma _{x^{{\prime}}}}\,, &&b = \frac{\sigma _{h}} {\gamma \sigma _{y^{{\prime}}}}\,, \\ \sigma _{h}^{2} = \frac{\sigma _{p}^{2}\,\sigma _{ x}^{2}} {\sigma _{x}^{2}+\eta ^{2}\,\sigma _{p}^{2}} \,, &&c^{2} =\beta ^{2}\sigma _{h}^{2}\,\frac{\sqrt{2\pi }\sigma _{y}} {r_{\mathrm{c}}} \,. \end{array} }$$

The transverse emittance growth times are similarly given by

$$\displaystyle{ \tau _{x}^{-1} = \frac{1} {2\sigma _{x}^{2}} \frac{\mathrm{d}\sigma _{x}^{2}} {\mathrm{d}t} = A\left [f\,\left (\frac{1} {a}, \frac{b} {a}, \frac{c} {a}\right ) + \frac{\eta ^{2}\sigma _{p}^{2}} {\sigma _{x}} f\,(a,b,c)\right ]\,, }$$

(21.13)

and

$$\displaystyle{ \tau _{y}^{-1} = \frac{1} {2\sigma _{y}^{2}} \frac{\mathrm{d}\sigma _{y}^{2}} {\mathrm{d}t} = A\,f\,\left (\frac{1} {b}, \frac{a} {b}, \frac{c} {b}\right )\,. }$$

(21.14)

These expressions allow the calculations of the emittance growth rate, which for most electron accelerators is small compared to radiation damping but become significant in proton and ion storage rings where high particle densities and long storage times are desired. Progress in the design of modern synchrotron radiation facilities allow ever smaller emittances which have reached a level where intra-beam scattering is significant again. From the density factor A it is apparent that high particle density in six-dimensional phase space increases the growth rates while this effect is greatly reduced at higher beam energies.

2 Collective Self Fields

The electric charges of a particle beam can become a major contribution to the forces encountered by individual particles while travelling along a beam transport line or orbiting in a circular accelerator. These forces may act directly from beam to particle or may originate from electromagnetic fields being excited by the beam interaction with its surrounding vacuum chamber. In this section, we will derive expressions for the fields from a collection of particles and determine the force due to these fields on an individual test particle. We use the particle charge q rather than the elementary charge e to cover particles with multiple charges like ions for which q = eZ. For all cases to be correct, we should distinguish between the electrical charge of particles in the beam and that of the individual test particle. This, however, would significantly complicate the expressions and we use therefore the same charge for both the beam and test particle. In a particular situation whenever particles of different charges are considered, the sign and value of the charge factors in the formulas must be reconsidered.

Individual particles in an intense beam are under the influence of strong repelling electrostatic forces creating the possibility of severe stability problems. Particle beam transport over long distances could be greatly restricted unless these space-charge forces can be kept under control. First, it is interesting to calculate the magnitude of the problem.

If all particles would be at rest within a small volume, we would clearly expect the particles to quickly diverge from the center of charge under the influence of the repelling forces from the other particles. This situation may be significantly different in a particle beam where all particles propagate in the same direction.

In Sect. 1.5.10 we obtained the encouraging result that at least relativistic particle beams become stable under the influence of their own fields. For lower particle energies, however, significant diverging forces must be expected and adequate focusing measures must be applied. The physics of such space charge dominated beams is beyond the scope of this book and is treated elsewhere, for example in considerable detail in [16].

2.1 Self Field for Elliptical Particle Beams

The self fields of a beam depend on beam parameters like particle type, particle distribution, bunching, and energy of the particle. Here, we will derive the nature and effect of these self fields in a more restricted way for common particle beam cases in accelerators.

To determine self fields, we consider a continuous beam of particles with a line charge λ, or a volume charge ρ(x, y). The electric fields within a beam are derived from a potential V defined by

$$\displaystyle{ \bigtriangleup V = -\frac{1} {\epsilon _{0}} \rho (x,y)\,, }$$

(21.15)

where ρ, being the electric charge density in the beam, is finite within and zero outside the beam. Similarly, the magnetic vector potential is defined by

$$\displaystyle{ \boldsymbol{\varDelta A} = -\frac{1} {\epsilon _{0}} \boldsymbol{v}\rho (x,y)\,. }$$

(21.16)

For a particle beam, we may set $\boldsymbol{v} \approx (0,0,v)$ and the vector potential therefore contains only a longitudinal component $\boldsymbol{A} = (0,0,A_{z})$.

In Sect. 1.5.10 we discussed the self fields of a round beam. Generally, however, particle beams have an elliptical cross section and the solution to (21.15) for such a beam with constant charge density $\left (\rho = \text{const}\right )$ has been derived by Teng [17, 18]. Within the elliptical beam cross section, where x ≤ a and y ≤ b, the electric potential is

$$\displaystyle{ V (x,y) = -\frac{1} {2\epsilon _{0}}\rho \frac{ab} {a + b}\left [\frac{x^{2}} {a} + \frac{y^{2}} {b} \right ] }$$

(21.17)

and a, b are the horizontal and vertical half axis respectively. The vector potential for the magnetic field is from the discussions above

$$\displaystyle{ A_{z}(x,y) = -\frac{1} {2\epsilon _{0}}\rho \frac{v} {c} \frac{ab} {a + b}\left [\frac{x^{2}} {a} + \frac{y^{2}} {b} \right ] }$$

(21.18)

and both the electric and magnetic field can be derived by simple differentiations

$$\displaystyle{ \mathbf{E} = -\nabla V \qquad \text{and}\qquad \mathbf{B} = \nabla \times \mathbf{A} }$$

(21.19)

for

$$\displaystyle{ E_{x} = \frac{1} {4\pi \epsilon _{0}} \frac{4q\lambda } {a(a + b)}x\,,\qquad \quad E_{y} = \frac{1} {4\pi \epsilon _{0}} \frac{4q\lambda } {b(a + b)}y\,, }$$

(21.20)

and

$$\displaystyle{ B_{x} = -\frac{c\mu _{0}} {4\pi } \frac{4q\lambda \beta } {b(a + b)}y\,,\qquad \quad B_{y} = \frac{c\mu _{0}} {4\pi } \frac{4q\lambda \beta } {a(a + b)}x\,, }$$

(21.21)

where β = v∕c and the linear charge density λ is defined by

$$\displaystyle{ \lambda =\pi ab\,\rho (x,y)\,. }$$

(21.22)

Comparing (21.20) and (21.21) reveals the relationship between electric and magnetic self fields of the beam to be (21.22)

$$\displaystyle{ cB_{x} = -\beta E_{y}\,,\qquad \quad cB_{y} = +\beta E_{x}\,. }$$

(21.23)

The electric as well as the magnetic field scales linearly with distance from the beam center and therefore both cause focusing and a tune shift in a circular accelerator.

In many applications it is not acceptable to assume a uniform transverse charge distribution. Most particle beams either have a bell shaped particle distribution or a Gaussian distribution as is specially the case for electrons in circular accelerators. We therefore use in the transverse plane a Gaussian charge distribution given by

$$\displaystyle{ \rho (x,y) = \frac{\lambda } {2\pi \sigma _{x}\sigma _{y}}\exp \left [-\frac{x^{2}} {2\sigma _{x}^{2}} - \frac{y^{2}} {2\sigma _{y}^{2}}\right ]\,, }$$

(21.24)

which also well describes a beam with bell shaped distribution. Although many particle beams, but specifically electron beams, come in bunches with a Gaussian distribution in all degrees of freedom, we will only introduce a bunching factor for the longitudinal particle distribution and refer the interested reader for the study of a fully six dimensional Gaussian charge distribution to reference [19].

The potential for a transverse bi-Gaussian charge distribution (21.24) can be expressed by [18]

$$\displaystyle{ V (x,y) = -\frac{e} {4\pi \epsilon _{0}}\,\lambda \int \limits _{0}^{\infty }\frac{1 -\exp \left [- \frac{x^{2}} {2(\sigma _{x}^{2}+t)} - \frac{y^{2}} {2(\sigma _{y}^{2}+t)}\right ]} {\sqrt{(\sigma _{x }^{2 } + t)(\sigma _{y }^{2 } + t)}} \mathrm{d}t }$$

(21.25)

Equation (21.25) can be verified by back insertion into (21.15). From this potential we obtain for example the vertical electric field component by differentiation

$$\displaystyle{ E_{y} = -\frac{\partial V (x,y)} {\partial y} = \frac{e} {4\pi \epsilon _{0}}\,\lambda \,y\int \limits _{0}^{\infty } \frac{\exp \left [- \frac{x^{2}} {2(\sigma _{x}^{2}+t)} - \frac{y^{2}} {2(\sigma _{y}^{2}+t)}\right ]} {(\sigma _{y}^{2} + t)\sqrt{(\sigma _{x }^{2 } + t)(\sigma _{y }^{2 } + t)}}\,\mathrm{d}t\,. }$$

(21.26)

No closed analytical expression exists for these integrals unless we restrict ourselves to a symmetry plane with x = 0 or y = 0 and small amplitudes y ≪ σ_y or x ≪ σ_x respectively. These assumptions are appropriate for most space-charge effects and the potential in the vertical midplane becomes

$$\displaystyle{ V (x = 0,y \ll \sigma _{y}) = -\frac{1} {4\pi \epsilon _{0}} \frac{\lambda } {\sigma _{y}(\sigma _{x} +\sigma _{y})}\,y^{2}\,. }$$

(21.27)

For reasons of symmetry a similar expression can be derived for the horizontal mid plane by merely interchanging x and y in (21.27). The associated electric fields are for x = 0 and y ≪ σ_y

$$\displaystyle{ E_{x} = \frac{1} {4\pi \epsilon _{0}} \frac{2\lambda } {\sigma _{x}(\sigma _{x} +\sigma _{y})}x,\qquad \quad E_{y} = \frac{1} {4\pi \epsilon _{0}} \frac{2\lambda } {\sigma _{y}(\sigma _{x} +\sigma _{y})}y, }$$

(21.28)

and the magnetic fields according to (21.23) are from (21.28) 

$$\displaystyle{ B_{x} = -\frac{c\mu _{0}} {4\pi } \frac{2\lambda \beta } {\sigma _{y}(\sigma _{x} +\sigma _{y})}y,\qquad \quad B_{y} = +\frac{c\mu _{0}} {4\pi } \frac{2\lambda \beta } {\sigma _{x}(\sigma _{x} +\sigma _{y})}x. }$$

(21.29)

All fields increase linearly with amplitude and we note that the field components in the horizontal midplane are generally much smaller compared to those in the vertical midplane because most particle beams in circular accelerators are flat and σ_y ≪ σ_x.

2.1.1 Forces from Space-Charge Fields

The electromagnetic self fields generated by the collection of all particles within a beam exert forces on individual particles of the same beam or of another beam. The Lorentz force due to these fields can be expressed by

$$\displaystyle{ \boldsymbol{F} = e\boldsymbol{E}f_{\text{e}} + e[\boldsymbol{v} \times \boldsymbol{ B}]\,f_{\text{e}}f_{\text{v}}\,, }$$

(21.30)

where we have added to the usual expression for the Lorentz force the factors f_e and f_v. Because the fields act differently depending on the relative directions and charge of beam and individual particle distinct combinations occur. We set f_e = 1 if both the beam particles and the test particle have the same sign of their charge and f_e = −1 if their charges are of opposite sign. Similarly we set f_v = 1 or f_v = −1 depending on whether the beam and test particle have the same or opposite direction of movement with respect to each other.

The vertical force from the self field, for example, of a proton beam on an individual proton within the same beam moving with the same velocity is from (21.30)

$$\displaystyle{ F_{y}(_{\uparrow \uparrow,++}) = +e(1 -\beta ^{2})E_{ y}\,. }$$

(21.31)

An antiproton moving in the opposite direction through a proton beam would feel the vertical force

$$\displaystyle{ F_{y}(_{\uparrow \downarrow,+-}) = -e(1 +\beta ^{2})E_{ y}\,. }$$

(21.32)

Expansion to other combinations of particles and directions of velocities are straightforward. For ions the charge multiplicity Z must be added to the fields or the individual particle or both depending on the case. The possible combinations of the force factors ± (1 ±β²) are summarized in Table 21.1.

Table 21.1 Self field force factors

The ±-signs in Table 21.1 indicate the charge polarity of beam and test particle and the arrows the relative direction. We note a great difference between the case, where particles move in the same direction, and the case of beams colliding head on.

2.2 Beam–Beam Effect

In colliding beam facilities two counter rotating beams within one storage ring or counter rotating beams from two intersecting storage rings are brought into collision to create a high center of mass energy at the collision point which transforms into known or unknown particles to be studied by high energy experimentalists. The event rate is given by the product of the cross section for the particular event and the luminosity which is determined by storage ring operating conditions. By definition, the luminosity is the density of collision centers in the target multiplied by the number of particles colliding with this target per unit time. In the case of a colliding beam facility a bunch of one beam is the target for the other beam. For simplicity we assume here that both beams have the same cross section. We also assume that each beam consists of n_b bunches. In this case the luminosity is

$$\displaystyle{ \mathcal{L} = \frac{N_{1}} {n_{\text{b}}\,A}N_{2}\nu _{\mathrm{rev}}\,, }$$

(21.33)

where N₁ and N₂ are the total number of particles in each beam, A the cross section of the beams, and ν_rev the revolution frequency in the storage ring. In most storage rings the transverse particle distribution is Gaussian or bell shaped and since only the core of the beam contributes significantly to the luminosity we may define standard beam sizes for all kinds of particles. For a Gaussian particle distribution the effective beam cross section is

$$\displaystyle{ A_{\text{g}} = 4\pi \sigma _{x}\sigma _{y} }$$

(21.34)

and the luminosity

$$\displaystyle{ \mathcal{L} = \frac{N_{1}} {4\pi \sigma _{x}\sigma _{y}\,B}N_{2}\nu _{\mathrm{rev}}\,. }$$

(21.35)

The recipe for high luminosity is clearly to maximize the beam intensity and to minimize the beam cross section. This approach, however, fails because of the beam-beam effect which, due to electromagnetic fields created by the beams themselves, causes a tune shift and therefore limits the amount of beam that can be brought into collision in a storage ring. The beam-beam effect has first been recognized and analyzed by Amman and Ritson [20].

In case of counter rotating beams colliding at particular interaction points in a colliding-beam facility, we always have f_v = −1 but the colliding particles still may be of equal or opposite charge. In addition, there is no contribution from magnetic image fields since collisions do not occur within magnets. Even image fields from vacuum chambers are neglected because the beam-beam interaction happens only over a very short distance. A particle in one beam will feel the field from the other beam only during the time it travels through the other beam which is equal to the time it takes the particle to travel half the effective length of the oncoming bunch. With these considerations in mind, we obtain for the beam-beam tune shift in the vertical plane from (21.63) with f_corr = 1 and assuming head on collisions of particle-antiparticle beams (f_e = −1)

$$\displaystyle{ \varDelta \nu _{y,\mathrm{bb}} = \frac{r_{\mathrm{c}}\,N_{\mathrm{tot}}} {2\pi B\gamma } \frac{\beta _{y}^{{\ast}}} {\sigma _{y}^{{\ast}}(\sigma _{x}^{{\ast}} +\sigma _{ y}^{{\ast}})} }$$

(21.36)

and in the horizontal plane

$$\displaystyle{ \varDelta \nu _{x,\mathrm{bb}} = \frac{r_{\mathrm{c}}\,N_{\mathrm{tot}}} {2\pi B\gamma } \frac{\beta _{x}^{{\ast}}} {\sigma _{x}^{{\ast}}(\sigma _{x}^{{\ast}} +\sigma _{ y}^{{\ast}})},\ }$$

(21.37)

where^∗ indicates that the quantities be taken at the interaction point. In cases where other particle combinations are brought into collision or when both beams cross under an angle these equations must be appropriately modified to accurately describe the actual situation.

From (21.32) and (21.28)we find for two counter rotating beams of particle and antiparticle a vertical beam-beam force of

$$\displaystyle{ F_{y} = -\frac{1} {4\pi \epsilon _{0}} \frac{e(1 +\beta ^{2})2\lambda } {\sigma _{y}(\sigma _{x} +\sigma _{y})} y\,. }$$

(21.38)

This force is attractive and therefore focusing, equivalent to that of a quadrupole of strength

$$\displaystyle{ k = -\frac{F_{y}/y} {c^{2}\beta ^{2}\gamma m} }$$

(21.39)

causing a vertical tune shift of

$$\displaystyle{ \delta \nu _{y} = \frac{1} {4\pi }\int \limits _{\mathrm{coll}}\beta _{y}\,k\,\mathrm{d}z\,. }$$

(21.40)

Integrating over the collision length which is equal to half the bunch length ℓ because colliding beams move in opposite directions, we note that the linear charge density is $\lambda = eN/\left (B\ell\right )$, where N is the total number of particles per beam and B the number of bunches per beam. With these replacements the beam tune shift becomes finally

$$\displaystyle{ \delta \nu _{y} = \frac{r_{\text{c}}N\beta _{y}} {2\pi B\gamma \sigma _{y}(\sigma _{x} +\sigma _{y})}\,, }$$

(21.41)

where r_c is the classical particle radius of the particle which is being disturbed. Obviously, the tune shift scales linear with particle intensity or particle beam current and inversely with the beam cross section. Upon discovery of this effect it was thought that the particle beam intensity is limited when the tune shift is of the order of ≈ 0. 15 – 0. 2 which is the typical distance to the next resonance. Experimentally, however, it was found that the limit is much more restrictive with maximum tune shift values of ≈ 0. 04 – 0. 06 for electrons [20–23] and less for proton beams [24].

A definitive quantitative description of the actual beam-beam effect has not been possible yet due to its highly nonlinear nature. Only particles with very small betatron oscillation amplitudes will experience the linear tune shift derived above. For betatron oscillations larger than one σ, however, the field becomes very nonlinear turning over to the well known 1∕r-law at large distances from the beam center.

In spite of the inability to quantitatively describe the beam-beam effect by the linear tune shift it is generally accepted practice to quantify the beam-beam limit by the value of the linear tune shift. This is justified since the nonlinear fields of a particle beam are strictly proportional to the linear field and therefore the linear tune shift is a good measure for the amount of nonlinear fields involved.

2.3 Transverse Self Fields

Expressions for space-charge fields originating from a beam of charged particles have been derived earlier and we obtained for a Gaussian transverse distribution of particles with charge q the electric fields in (21.28) and the magnetic fields in (21.29).

The local linear particle density λ is defined by

$$\displaystyle{ \lambda (z) =\int \int \rho (x,y,z)\mathrm{d}x\,\mathrm{d}y\,, }$$

(21.42)

where ρ(x, y, z) is the local particle density normalized to the total number of particles in the beam $\int _{-\infty }^{\infty }\lambda (z)\mathrm{d}z = N_{\mathrm{p}}$. With these fields and the Lorentz equation, we formulate the transverse force acting on a single particle within the same particle beam. Since both expressions for the electrical and magnetic field differ only by the factor β we may, for example, derive from the Lorentz equation the vertical force on a particle with charge q

$$\displaystyle{ F_{y} = q(1 -\beta ^{2})E_{ y} = \frac{1} {4\pi \epsilon _{0}} \frac{2q\lambda } {\gamma ^{2}\sigma _{y}(\sigma _{x} +\sigma _{y})}y\,. }$$

(21.43)

The space-charge force appears at its strongest for nonrelativistic particles and diminishes quickly like 1∕γ² for relativistic particles. In accelerator physics, however, particle beams are carried from low to high energies and therefore space-charge effects may become important during some or all phases of acceleration. This is specifically true for heavy particles like protons and ions for which the relativistic parameter γ is rather low for most any practically achievable particle energies.

2.4 Fields from Image Charges

Discussing space charges, we ignored so far the effect of metallic and magnetic surfaces close to the beam. The electromagnetic self fields of the beam circulating in a metallic vacuum chamber and between ferromagnetic poles of magnets must meet certain boundary conditions on such surfaces. Laslett [25] derived appropriate corrections to free space electromagnetic fields by adding the electromagnetic fields from all image charges to the fields of the particle beam itself.

Following his reasoning, we consider a particle beam with metallic and ferromagnetic boundaries as shown in Fig. 21.2. For full generality, let the elliptical particle beam be displaced in the vertical plane by $\bar{y}$ from the midplane, the metallic vacuum chamber and magnet pole are simulated as pairs of infinitely wide parallel surfaces at ± b and ± g, respectively, and the observation point of the fields be at y.

Fig. 21.2

Particle beam with metallic and ferromagnetic boundaries

The linear particle density is

$$\displaystyle{ \lambda = \frac{N_{\mathrm{tot}}} {n_{\mathrm{b}}\,\ell_{\mathrm{b}}} = \frac{N_{\mathrm{tot}}} {n_{\mathrm{b}}\sqrt{2\pi }\sigma _{\ell}}, }$$

(21.44)

where N_tot is the total number of particle in the circulating beam, n_b the number of bunches, $\ell_{\mathrm{b}} = \sqrt{2\pi }\sigma _{\ell}$ the effective bunch length and σ_ℓ the standard bunch length for a Gaussian distribution.

The locations and strength of the electrical images of a line current in the configuration of Fig. 21.2 are shown in Fig. 21.3. The boundary condition for electric fields is E_z(b) = 0 on the surface of the metallic vacuum chamber and is satisfied if the image charges change sign from image to image. To calculate the electrical field E_y(y), with (21.28) we add the contributions from all image fields in the infinite series

$$\displaystyle\begin{array}{rcl} E_{y,\mathrm{image}}(y)& =& \frac{1} {4\pi \epsilon _{0}}2\lambda \\ & & \quad \times \left ( \frac{1} {2b -\bar{y} - y} - \frac{1} {2b + \bar{y} + y} - \frac{1} {4b + \bar{y} - y} + \frac{1} {4b -\bar{y} + y}\right. \\ & & \quad + \frac{1} {6b -\bar{y} - y} - \frac{1} {6b + \bar{y} + y} - \frac{1} {8b + \bar{y} - y} + \frac{1} {8b -\bar{y} + y} \\ & & \quad \left.+ \frac{1} {10b -\bar{y} - y} - \frac{1} {10b + \bar{y} + y}-\ldots \right )\,. {}\end{array}$$

(21.45)

Fig. 21.3

Location and source of image fields

These image fields must be added to the direct field of the line charge to meet the boundary condition that the electric field enter metallic surfaces perpendicular. Equation (21.45) can be split into two series with factors $(\bar{y} + y)$ and $(\bar{y} - y)$ in the numerator. We get after some manipulations with $\bar{y} + y \ll b$ and $\bar{y} - y \ll b$

$$\displaystyle\begin{array}{rcl} E_{y,\mathrm{image}}(y)& =& \frac{1} {4\pi \epsilon _{0}} \frac{\lambda } {b^{2}}\left [\sum _{m=1}^{\infty } \frac{\bar{y} + y} {(2m - 1)^{2}} +\sum _{ m=1}^{\infty }\frac{\bar{y} - y} {4m^{2}} \right ], \\ & =& \frac{1} {4\pi \epsilon _{0}} \frac{\lambda } {b^{2}}\left [(\bar{y} + y)\frac{\pi ^{2}} {8} + (\bar{y} - y) \frac{\pi ^{2}} {24}\right ], \\ & =& \frac{1} {4\pi \epsilon _{0}} \frac{\lambda } {b^{2}} \frac{\pi ^{2}} {12}(2\bar{y} + y) = \frac{1} {4\pi \epsilon _{0}} \frac{4q\,\lambda } {b^{2}} \epsilon _{1}(2\bar{y} + y).{}\end{array}$$

(21.46)

The electric image fields depend linearly on the deviations $\bar{y}$ and y from the axis of bunch center and test particle, respectively, and act therefore like a quadrupole causing a tune shift.

A similar derivation is used to get the magnetic image fields due to ferromagnetic surfaces at ± g above and below the midplane. The magnetic field lines must enter the magnetic pole faces perpendicular and the image currents therefore flow in the same direction as the line current causing a magnetic force on the test particle which is opposed to that by the magnetic field of the beam itself.

Bunched beams generate high frequency electromagnetic fields which do not reach ferromagnetic surfaces because of eddy current shielding by the metallic vacuum chamber. For magnetic image fields we distinguish therefore between dc and ac image fields. The dc Fourier component of a bunched beam current is equal to twice the average beam current c β λ B, where the Laslett bunching factor B is the bunch occupation along the ring circumference defined by

$$\displaystyle{ B = \frac{\overline{\lambda }} {\lambda } = \frac{n_{\mathrm{b}}\ell_{\mathrm{b}}} {2\pi R}. }$$

(21.47)

The dc magnetic image fields are derived similar to electric image fields with $B_{\varphi } = -2\lambda \beta /r$ from (21.29) and are with (21.47)

$$\displaystyle\begin{array}{rcl} B_{x,\mathrm{image,dc}}(y)& =& \frac{c\mu _{0}} {4\pi } \frac{2\lambda \beta } {g^{2}}B\left [\sum _{m=1}^{\infty } \frac{\bar{y} + y} {(2m - 1)^{2}} +\sum _{ m=1}^{\infty }\frac{\bar{y} - y} {4m^{2}} \right ] \\ & =& \frac{c\mu _{0}} {4\pi } \frac{\lambda \beta } {g^{2}}B\left [(\bar{y} + y)\frac{\pi ^{2}} {8} + (\bar{y} - y) \frac{\pi ^{2}} {24}\right ] \\ & =& \frac{c\mu _{0}} {4\pi } \frac{4\lambda \beta } {g^{2}}B\epsilon _{2}(2\bar{y} + y). {}\end{array}$$

(21.48)

The magnetic image fields must penetrate the metallic vacuum chamber to reach ferromagnetic poles. This is no problem for dc or low frequency field components but in case of bunched beams relevant frequencies are rather high and eddy current shielding of the vacuum chamber for ac magnetic fields must be taken into account. In most cases we may assume that they do not penetrate the thick metallic vacuum chamber. Consequently, we ignore here the effect of ferromagnetic poles and consider only the contribution of magnetic ac image fields due to eddy currents in vacuum chamber walls. Similar to electric image fields, the magnetic image fields are in analogy to (21.46)

$$\displaystyle\begin{array}{rcl} B_{x,\mathrm{image,ac}}(y)& =& -\frac{c\mu _{0}} {4\pi } \frac{\lambda \beta } {b^{2}}(1 - B) \frac{\pi ^{2}} {12}(2\bar{y} + y), \\ & =& -\frac{c\mu _{0}} {4\pi } \frac{4\lambda \beta } {b^{2}}(1 - B)\epsilon _{1}(2\bar{y} + y),{}\end{array}$$

(21.49)

where the factor (1 − B) accounts for the subtraction of the dc component β λ B. Similar to the electric image fields, the magnetic image fields must be added to the direct magnet fields (21.29) from the beam current to meet the boundary condition of normal field components at ferromagnetic surfaces. The coefficients ε₁ and ε₂ are the Laslett form factors which are for infinite parallel plate vacuum chambers and magnetic poles

$$\displaystyle{ \epsilon _{1} = \frac{\pi ^{2}} {48}\qquad \text{and}\qquad \epsilon _{2} = \frac{\pi ^{2}} {24}. }$$

(21.50)

The vacuum chamber and ferromagnetic poles are similar to infinitely wide surfaces. While this is a sufficiently accurate approximation for the magnet poles, corrections must be applied for circular or elliptical vacuum chambers. Laslett [25] has derived what we call now Laslett form factors for vacuum chambers with elliptical cross sections and variable aspect ratios which are compiled in Table 21.2.

Table 21.2 Laslett incoherent tune shift form factors for elliptical vacuum chambers

All relevant field components have been identified and we collect these fields first for $\bar{y} = 0$ and obtain from (21.28), (21.46) for the electric field in the vertical mid plane

$$\displaystyle{ E_{y}(y) = \frac{c^{2}\mu _{0}} {4\pi } \frac{2\lambda } {\sigma _{y}(\sigma _{x} +\sigma _{y})}\left [1 + \frac{2\sigma _{y}(\sigma _{x} +\sigma _{y})} {b^{2}} \epsilon _{1}\right ]y\,. }$$

(21.51)

From (21.29), (21.48) the dc magnetic field is

$$\displaystyle{ B_{x,\mathrm{dc}} = -\frac{c\mu _{0}} {4\pi } \frac{2\lambda \beta B} {\sigma _{y}\,(\sigma _{x} +\sigma _{y})}\left [1 -\frac{2\sigma _{y}(\sigma _{x} +\sigma _{y})} {g^{2}} \,\epsilon _{2}\right ]y }$$

(21.52)

and from (21.49) the ac magnetic field

$$\displaystyle{ B_{x,\mathrm{ac}} = -\frac{c\mu _{0}} {4\pi } \frac{2\,\lambda \,\beta } {\sigma _{y}(\sigma _{x} +\sigma _{y})}\left [1 + \frac{2\sigma _{y}(\sigma _{x} +\sigma _{y}} {b^{2}} \epsilon _{1}\right ](1 - B)\,y. }$$

(21.53)

Tacitly, we have assumed that the transverse particle distribution is Gaussian which is a true representation of an electron beam but may not be correct for proton or ion beams. The standard deviations σ of a Gaussian distribution are very well defined and can therefore be replaced by other quantities like the full-width half maximum or as the particle distribution may require.

The electromagnetic force due to space charge on individual particles in a beam has been derived and it became obvious that image field effects can play a significant role in the perturbation of the beam. The fields scale linear with amplitude for very small amplitudes and act therefore like focusing quadrupoles. At larger amplitudes, however, the fields reach a maximum and then evanesce like 1∕r. Consequently, the field gradient is negative decaying quickly with amplitude.

A complete set of direct and image fields have been derived which must be considered to account for space-charge effects. Similar derivations lead to other field components necessary to determine horizontal space-charge forces. In most accelerators, however, the beam cross section is flat and so is the vacuum chamber and the magnet pole aperture. As a consequence, we expect the space-charge forces to be larger in the vertical plane than in the horizontal plane.

2.5 Space-Charge Effects

The Lorentz force on individual particles can be calculated from the space-charge fields and we get

$$\displaystyle{ F_{y} = \frac{1} {4\pi \epsilon _{0}} \frac{2\,f_{\mathrm{p}}\lambda (1 -\beta ^{2}f_{\mathrm{v}})} {\sigma _{y}(\sigma _{x} +\sigma _{y})} f_{\mathrm{corr}}\,y = q\mathcal{F}y, }$$

(21.54)

where the correction factor due to image fields is with $\beta ^{2}\gamma ^{2} =\gamma ^{2} - 1,$

$$\displaystyle{ f_{\mathrm{corr}} = 1 + \frac{2\sigma _{y}(\sigma _{x} +\sigma _{y})} {b^{2}} \epsilon _{1}[1 + (\gamma ^{2} - 1)B] +\epsilon _{ 2}(\gamma ^{2} - 1)\frac{b^{2}} {g^{2}}B }$$

(21.55)

and

$$\displaystyle{ \mathcal{F} = \frac{1} {4\pi \epsilon _{0}} \frac{2\,\,f_{\mathrm{p}}\lambda \,(1 -\beta ^{2}f_{\mathrm{v}})} {\sigma _{y}(\sigma _{x} +\sigma _{y})} f_{\mathrm{corr}}\,, }$$

(21.56)

The factors f_p and f_v determine signs depending on the kind of particles interacting and the direction of travel with respect to each other. Specifically, f_p = sign(q q_b) where q is the charge of a test particle and q_b the charge of the field creating particles, e.g. the charge of a bunch. Similarly, $f_{\mathrm{v}} =\mathrm{ sign}(\boldsymbol{vv}_{\mathrm{b}})$ where $\boldsymbol{v}$ is the direction of travel for the test particle and v_b the direction of travel of the bunch. To calculate the space-charge force of head-on colliding proton and antiproton beams, for example, we would set f_p = −1 and f_v = −1.

There is a significant cancellation of two strong terms, the repulsive electrical field and the focusing magnetic field, expressed by the factor 1 −β² for space-charge forces within a highly relativistic beam. This cancellation can be greatly upset if particle beams become partially neutralized by collecting other particles of opposite charge within the beams potential well. For example, proton beams can trap electrons in the positive potential well as can electron beams trap positive ions in the negative potential well. To avoid such partial neutralization and appearance of unnecessarily strong space-charge effects, clearing electrodes must be installed over much of the ring circumference to extract with electrostatic fields low energy electrons or ions from the particle beam.

The electromagnetic space-charge force on an individual particle within a particle beam increases linearly with its distance from the axis. A similar force occurs for the horizontal plane and both fields therefore act like a quadrupole causing a tune shift. This has been recognized and analyzed early by Kerst [26] and Blewett [27]. A complete treatment of space charge dominated beams can be found in [16]. The equation of motion under the influence of space charge forces can be written in the form

$$\displaystyle{ m\gamma \,\ddot{u} + Du = \frac{\partial F_{u}} {\partial u} u\quad \text{with}\quad u = (x,y)\,. }$$

(21.57)

We get the regular form $u^{{\prime\prime}} + (k_{0} +\varDelta k)\,u = 0$ with $\ddot{u} = u^{{\prime\prime}}\left (c\beta \right )^{2}$ and f_v = 1, where k₀ describes the quadrupole strength and the space-charge strength is expressed by

$$\displaystyle{ \varDelta k = \frac{1} {mc^{2}\gamma \beta ^{2}} \frac{\partial F_{u}} {\partial u} = -\frac{2\,r_{\mathrm{c}}} {\beta ^{2}\gamma ^{3}} \frac{\lambda \,} {\sigma _{y}(\sigma _{x} +\sigma _{y})}f_{\mathrm{corr}} }$$

(21.58)

where r_c is the classical particle radius. For ions with charge multiplicity Z and atomic number A the classical particle radius is r_ion = r_p Z²∕A.

2.5.1 Space Charge Dominated Beams

So far, space-charge effects or space-charge focusing has been consistently neglected in the discussions on transverse beam dynamics. In cases of low beam energy and high particle densities, it might become necessary to include space-charge effects. They are defocusing in both planes and compensation therefore requires additional focusing in both planes. However, it should be noted that particles closer to the beam surface will not experience the same linear space-charge defocusing as those near the axis and therefore a compensation of space-charge focusing works only for part of the beam. Here, we will not get involved with the dynamics of heavily space charge dominated particle beams but try to derive a criterion by which we can decide whether or not space-charge forces are significant in transverse particle beam optics.

This distinction becomes obvious from the equation of motion including space charges. From (21.57), (21.58) we get the equation of motion

$$\displaystyle{ u^{{\prime\prime}} + \left [k_{ 0} -\frac{2r_{\mathrm{c}}} {\beta ^{2}\gamma ^{3}} \frac{\lambda \,} {\sigma _{y}(\sigma _{x} +\sigma _{y})}f_{\mathrm{corr}}\right ]u = 0, }$$

(21.59)

where we ignored the image current corrections. Space-charge forces can be neglected if the integral of the space-charge force over a length L which is characteristic for the average distance between quadrupoles in the beam line is small compared to the typical integrated quadrupole length k₀ℓ_qor if

$$\displaystyle{ \frac{2r_{\mathrm{c}}} {\beta ^{2}\gamma ^{3}} \int \nolimits _{L} \frac{\lambda \,f_{\mathrm{corr}}} {\sigma _{y}(\sigma _{x} +\sigma _{y})}\mathrm{d}z \ll k_{0}\ell_{\mathrm{q}}\,. }$$

(21.60)

The effect of space-charge focusing is most severe where the beam cross section is smallest and (21.60) should therefore be applied specifically to such sections of the beam transport line. Obviously, the application of this formula requires some subjective judgement as to how much smaller space-charge effects should be. To aid this judgement, one might also calculate the average betatron phase shift caused by space-charge forces and compare it with the total phase advance along the beam line under investigation. In this case we look for

$$\displaystyle{ \frac{2r_{\mathrm{c}}} {\beta ^{2}\gamma ^{3}} \int \nolimits _{L} \frac{\beta _{u}\lambda \,f_{\mathrm{corr}}} {\sigma _{y}(\sigma _{x} +\sigma _{y})}\,\mathrm{d}z \ll \psi _{0}(L) }$$

(21.61)

to determine the severity of space-charge effects. The nominal phase advance ψ_0, u(L) is defined such that ψ_0, u(0) = 0 at the beginning of the beam line.

2.5.2 Space-Charge Tune Shift

Space-charge focusing may not significantly perturb the lattice functions but may cause a big enough tune shift in a circular accelerator moving the beam onto a resonance. The beam current is therefore limited by the maximum allowable tune shift in the accelerator which is for a linear focusing force F(z) given by

$$\displaystyle{ \varDelta \nu _{u} = -\frac{1} {4\pi } \frac{r_{\mathrm{c}}} {\beta ^{2}\gamma } \int \nolimits _{0}^{L_{\mathrm{int}} }F(z)\beta _{u}\,\mathrm{d}z\,. }$$

(21.62)

The integration in (21.62) is taken over that part of the path in each revolution where the force is effective. For the effect on particles within the same beam this is the circumference and for the beam-beam effect it is the total length of all head on collisions per turn.

The tune shifts are not the same for all particles due to the nonuniform charge distribution within a beam. Only particles close to the beam center suffer the maximum tune shift while particles with increasing betatron oscillation amplitudes are less affected. The effect of space charge therefore introduces a tune spread rather than a specific tune shift and we refer to this effect as the incoherent space-charge tune shift.

As a particular case, consider the space-charge tune shift of a particle within a beam of equal species particles. Applying the Lorentz force (21.54) with (21.56) the space-charge tune shift becomes from (21.62)

$$\displaystyle{ \varDelta \nu _{u,\mathrm{sc}} = -\frac{r_{\mathrm{c}}\lambda } {2\pi } \frac{f_{\mathrm{p}}(1 -\beta ^{2}f_{\mathrm{v}})} {\beta ^{2}\gamma } \int \frac{\beta _{u}} {\sigma _{u}(\sigma _{x} +\sigma _{y})}f_{\mathrm{corr}}\,\mathrm{d}z\,, }$$

(21.63)

where the local linear particle density λ is defined by (21.44).

The maximum incoherent space-charge tune shift is from (21.63) with f_p = 1, f_v = 1,  (1 −β²) = 1∕γ² and (21.56)

$$\displaystyle\begin{array}{rcl} & & \varDelta \nu _{u,\mathrm{sc,incoh}} = -\frac{r_{\mathrm{c}}\,\lambda } {2\pi \beta ^{2}\gamma ^{3}}\left [\int \nolimits _{0}^{2\pi \bar{R}} \frac{\beta _{u}} {\sigma _{u}(\sigma _{x} +\sigma _{y})}\,\mathrm{d}z\right. \\ & & \qquad \qquad \quad \left.+2(1 +\beta ^{2}\gamma ^{2}B)\int \nolimits _{ 0}^{L_{\mathrm{vac}} } \frac{\beta _{u}\epsilon _{1}} {b^{2}} \mathrm{d}z + 2\beta ^{2}\gamma ^{2}B\int \nolimits _{ 0}^{L_{\mathrm{mag}} } \frac{\beta _{u}\,\epsilon _{2}} {g^{2}}\,\mathrm{d}z\right ]\,,{}\end{array}$$

(21.64)

where the integration length L_vac is equal to the total length of the vacuum chamber and L_mag is the total length of magnets along the ring circumference. Note, however, that this last term appears only at low frequencies because of eddy-current shielding in the vacuum chamber at high frequencies. Observing the tune on a betatron side band at a high harmonic of the revolution frequency may not exhibit a tune shift due to this term while one might have a contribution at low frequencies.

A coherent space-charge tune shift can be identified by setting $y = \bar{y}$ in the field expressions (21.46), (21.48), (21.49) to determine the fields at the bunch center. The calculation is similar to that for the incoherent space-charge tune shift except that we define new Laslett form factors for this case

$$\displaystyle{ \xi _{2} = \frac{\pi ^{2}} {16} }$$

(21.65)

for the image fields from the magnetic pole and form factors ξ₁ which depend on the aspect ratio of an elliptical vacuum chamber (Table 21.3).

Table 21.3 Laslett coherent tune shift form factors for elliptical vacuum chambers

21.66

The coherent space-charge tune shift is analogous to (21.64) 

$$\displaystyle\begin{array}{rcl} \varDelta \nu _{u,\mathrm{sc,coh}}& =& -\,\frac{r_{\mathrm{c}}\,\lambda } {2\pi \,\beta ^{2}\,\gamma ^{3}}\left [\int \nolimits _{0}^{2\pi \bar{R}} \frac{\beta _{u}} {\sigma _{u}(\sigma _{x} +\sigma _{y})}\,\mathrm{d}z\right. \\ & & \quad \left.+2(1 +\beta ^{2}\gamma ^{2}B)\int \nolimits _{ 0}^{L_{\mathrm{vac}} } \frac{\beta _{u}\,\xi _{1}} {b^{2}} \mathrm{d}z + 2\beta ^{2}\gamma ^{2}B\int \nolimits _{ 0}^{L_{\mathrm{mag}} } \frac{\beta _{u}b^{2}\xi _{2}} {g^{2}} \mathrm{d}z\right ]\,.{}\end{array}$$

(21.67)

In both cases, we may simplify the expressions significantly for an approximate calculation by applying smooth approximation $\overline{\beta _{u}} \approx \overline{R}/\nu _{0u}$ and assuming a uniform vacuum chamber and magnet pole gaps. With these approximations, (21.63) becomes

$$\displaystyle{ \varDelta \nu _{u,\mathrm{sc}} = -\,\frac{r_{\mathrm{c}}\,N_{\mathrm{tot}}\,\overline{R}} {2\pi \nu _{0u}\,B} \frac{f_{\mathrm{p}}\,(1 -\beta ^{2}f_{\mathrm{v}})} {\beta ^{2}\gamma } \frac{\langle \,f_{\mathrm{corr}}\rangle } {\bar{\sigma }_{u}(\bar{\sigma }_{x} + \bar{\sigma }_{y})}, }$$

(21.68)

where

$$\displaystyle{ \langle f_{\mathrm{corr}}\rangle = 1 + \frac{\bar{\sigma }_{u}\,(\bar{\sigma }_{x} + \bar{\sigma }_{y})} {\bar{b^{2}}} \left [\epsilon _{1}(1 +\beta ^{2}\gamma ^{2}B) +\epsilon _{ 2}\beta ^{2}\gamma ^{2} \frac{\bar{b}^{2}} {\bar{g}^{2}}B\right ]. }$$

(21.69)

Symbols with an overbar are the values of quantities averaged over the circumference of the ring and ν_0u is the unperturbed tune in the plane (x, y). The incoherent tune shift (21.64) becomes then

$$\displaystyle\begin{array}{rcl} & & \varDelta \nu _{u,\mathrm{sc,\ incoh}}\approx -\frac{r_{\mathrm{c}}\,N_{\mathrm{tot}}\,\overline{R}} {2\pi \,\nu _{0u}\,B\,\beta ^{2}\gamma ^{3}} \left [ \frac{1} {\bar{\sigma }_{u}(\bar{\sigma }_{x} + \bar{\sigma }_{y})}\right.\ \\ & & \qquad \qquad \quad \ \left.+2\ (1 +\beta ^{2}\gamma ^{2}B) \frac{\epsilon _{1}} {\bar{b}^{2}} + 2\ \beta ^{2}\gamma ^{2}B \frac{\epsilon _{2}} {\bar{g}^{2}}\eta _{\mathrm{b}}\right ],{}\end{array}$$

(21.70)

where $\eta _{\mathrm{b}} = L_{\mathrm{mag}}/(2\pi \bar{R})$ is the magnet fill factor and the coherent tune shift (21.67) becomes

$$\displaystyle\begin{array}{rcl} & & \ \varDelta \nu _{u,\mathrm{sc,\ coh}}\approx -\frac{r_{\mathrm{c}}\,N_{\mathrm{tot}}\,\overline{R}} {2\pi \nu _{0u}B\,\beta ^{2}\gamma ^{3}} \left [ \frac{1} {\bar{\sigma }_{u}\,(\bar{\sigma }_{x} + \bar{\sigma }_{y})}\right.\ \\ & & \qquad \qquad \quad \left.+\frac{2(1 +\beta ^{2}\gamma ^{2}B)} {\bar{b}^{2}} \xi _{1} + \frac{2\beta ^{2}\gamma ^{2}B} {\bar{g}^{2}} \xi _{2}\eta _{\mathrm{b}}\right ].{}\end{array}$$

(21.71)

 The tune shift diminishes proportional to the third power of the particle energy. As a matter of fact in electron machines of the order of 1 GeV or more, space-charge tune shifts are generally negligible. For low energy protons and ions, however, this tune shift is of great importance and must be closely controlled to avoid beam loss due to nearby resonances. While a maximum allowable tune shift of 0.15–0.25 seems reasonable to avoid crossing a strong third order or half-integer resonance, practically realized tune shifts can be significantly larger of the order 0.5–0.6 [28–30]. Independent of the maximum tune shift actually achieved in a particular ring, space charge forces ultimately lead to a limitation of the beam current.

2.6 Longitudinal Space-Charge Field

Within a continuous particle beam travelling along a uniform vacuum chamber we do not expect longitudinal fields to arise. We must, however, consider what happens if the longitudinal charge density is not uniform since this is a more realistic assumption. For the case of a round beam of radius r₀ in a circular vacuum tube of radius r_w (Fig. 21.4), the fields can be derived by integrating Maxwell’s equation $\nabla \times \boldsymbol{ E} = -\,\frac{\partial \boldsymbol{B}} {\partial t}$ and with Stoke’s law

$$\displaystyle{ \oint \boldsymbol{E}\,\mathrm{d}\boldsymbol{s} = -\frac{\partial } {\partial t}\int \boldsymbol{B}\,\mathrm{d}\boldsymbol{A}\,, }$$

(21.72)

where dA is an element of the area enclosed by the integration path s. The integration path shown in Fig. 21.4 leads to the determination of the electrical field E_z0 in the center of the beam.

Fig. 21.4

Space-charge fields due to a particle beam travelling inside a circular metallic vacuum chamber

Integrating the l.h.s. of (21.72) along the integration path we get with (21.28) for a round beam $\left (r =\sigma \right )$

$$\displaystyle\begin{array}{rcl} & & E_{z0}\varDelta z +\int \nolimits _{ 0}^{r_{\mathrm{w}} }E_{r}(z +\varDelta z)\mathrm{d}r - E_{z\mathrm{w}}\varDelta z -\int \nolimits _{0}^{r_{\mathrm{w}} }E_{r}(z)\mathrm{d}r \\ & & = (E_{z0} - E_{z\mathrm{w}})\varDelta z + \frac{q} {4\pi \epsilon _{0}}\left (1 + 2\ln \frac{r_{\mathrm{w}}} {r_{\mathrm{0}}} \right ) \frac{\partial \lambda } {\partial z}\varDelta z\,, {}\end{array}$$

(21.73)

where a Taylor’s expansion was applied to the linear particle density λ(z +Δ z) and only linear terms were retained. E_zw is the longitudinal electrical field on the vacuum chamber wall.

For the r.h.s. of (21.72) we use the expressions for the magnetic field (21.29) and get with $\int B_{\varphi }\mathrm{d}A =\varDelta z\int B_{\varphi }\mathrm{d}r$

$$\displaystyle{ -\frac{\beta } {c}q\left (1 + 2\ln \frac{r_{\mathrm{w}}} {r_{0}} \right ) \frac{\partial \lambda } {\partial t}\,\varDelta z =\beta ^{2}q\left (1 + 2\ln \frac{r_{\mathrm{w}}} {r_{0}} \right ) \frac{\partial \lambda } {\partial z}\,\varDelta z }$$

(21.74)

while using the continuity equation

$$\displaystyle{ \frac{\partial \lambda } {\partial t} +\beta c \frac{\partial \lambda } {\partial z} = 0\,. }$$

(21.75)

The longitudinal space-charge field is therefore

$$\displaystyle{ E_{z0} = E_{z\mathrm{w}} - \frac{q} {4\pi \epsilon _{0}} \frac{1} {\gamma ^{2}} \left (1 + 2\ln \frac{r_{\mathrm{w}}} {r_{0}} \right ) \frac{\partial \lambda } {\partial z} }$$

(21.76)

and vanishes indeed for a uniform charge distribution because E_zw = 0 for a dc current. However, variations in the charge distribution cause a longitudinal field which together with the associated ac field in the vacuum chamber wall, acts on individual particles.

The perturbation of a uniform particle distribution in a circular accelerator is periodic with the circumference of the ring and we may set for the longitudinal particle distribution keeping only the nth harmonic for simplicity

$$\displaystyle{ \lambda =\lambda _{0} +\lambda _{n}\,\mathrm{e}^{\mathrm{i}(n\theta -\omega _{n}t)}\,, }$$

(21.77)

where ω_n is the nth harmonic of the perturbation $\left (\omega _{n} = n\omega _{0}\right )$. Of course a real beam may have many modes and we need therefore to sum over all modes n. In case of instability, it is clear that the whole beam is unstable if one mode is unstable.

With the derivative dλ∕dz, smooth approximation and $\theta = z/\bar{R}$ with $\bar{R}$ the average ring radius an integration of (21.76) around the circular accelerator gives the total induced voltage due to space-charge fields

$$\displaystyle{ V _{z0} = 2\pi \bar{R}\,E_{z\mathrm{w}} -\mathrm{ i}\frac{I_{n}} {4\pi \epsilon _{0}} \frac{2\pi n} {\beta c\gamma ^{2}} \left (1 + 2\ln \frac{r_{\mathrm{w}}} {r_{0}} \right )\,\mathrm{e}^{\mathrm{i}(n\theta -\omega _{n}t)}\,. }$$

(21.78)

In this expression we have also introduced the nth harmonic of the beam-current perturbation I_n = β c q λ_n. Equation (21.78) exhibits a relation of the induced voltage to the beam current. Borrowing from the theory of electrical currents, it is customary to introduce here the concept of a frequency dependent impedance which will become a powerful tool to describe the otherwise complicated coupling between beam current and induced voltage. We will return to this point in Chap. 22

3 Beam-Current Spectrum

In the last section a beam stability issue appeared based on instantaneous current variations. This is particularly true in circular accelerators where the particle distribution is periodic with the circumference of the ring. On one hand, we have an orbiting particle beam which constitutes a harmonic oscillator with many eigen-frequencies and harmonics thereof and on the other hand, there is an environment with a frequency dependent response to electromagnetic excitation. Depending on the coupling of the beam to its environment at a particular frequency, periodic excitations occur which can create perturbations of particle and beam dynamics. This interaction is the subject of this discussion. In this text, we will concentrate in Chap. 22 on the discussion of basic phenomena of beam-environment interactions or beam instabilities. For a more detailed introduction into the field of beam instabilities, the interested reader is referred to the general references for this chapter. In this discussion, we will follow mainly the theories as formulated by Chao [31], Laclare [32], Sacherer [33] and Zotter [34].

Since the coupling of the beam to its environment depends greatly on the frequency involved, it seems appropriate to discuss first the frequency spectrum of a circulating particle beam.

3.1 Longitudinal Beam Spectrum

In case of a single circulating particle of charge q in each of n_b equidistant bunches, a pick up electrode located at azimuth $\varphi$ would produce a signal proportional to the single-particle beam current which is composed of a series of delta function signals

$$\displaystyle{ i_{\parallel }(t,\varphi ) = \frac{q} {T_{0}}\sum \limits _{k=-\infty }^{+\infty }\delta (t - \frac{\varphi } {2\pi }T_{0} - k\frac{T_{0}} {n_{\mathrm{b}}} -\tau )\,, }$$

(21.79)

where τ is the longitudinal offset of the particle from the reference point, n_b the number of equidistant bunches and T₀ the revolution time (Fig. 21.4).

With the revolution frequency $\omega _{0} = 2\pi /T_{0}$, we use the mathematical relations $2\pi \sum _{k=-\infty }^{+\infty }\delta (y - 2\pi k) =\sum _{ p=-\infty }^{+\infty }\mathrm{e}^{\mathrm{i}py}$ and | c | δ(cy) = δ(y) for

$$\displaystyle{ \sum \limits _{k=-\infty }^{+\infty }\delta \left (x - \frac{2\pi k} {n_{\mathrm{b}}\omega _{0}}\right ) = \frac{n_{\mathrm{b}}\omega _{0}} {2\pi } \sum \limits _{p=-\infty }^{+\infty }\mathrm{e}^{\mathrm{i}pn_{\mathrm{b}}\omega _{0}\,x}, }$$

(21.80)

where $x = t - \frac{\varphi } {\omega _{ 0}} -\tau,$ to replace the delta functions. We also replace the exponential function

$$\displaystyle{ \mathrm{e}^{\mathrm{i}y\sin \psi } =\sum \limits _{ n=-\infty }^{+\infty }J_{ n}(y)\,\mathrm{e}^{\mathrm{i}n\psi } }$$

(21.81)

and replace τ by the synchrotron oscillation $\tau = \hat{\tau }\cos [(m +\nu _{\mathrm{s}})\omega _{0}t +\zeta _{i}]$ where ν_s is the synchrotron oscillation tune. The term m ω₀t reflects the mode of the longitudinal particle distribution in all buckets. This distribution is periodic with the periodicity of the circumference and the modes are the harmonics of the distribution in terms of the revolution frequency (Fig. 21.5).

Fig. 21.5

Particle distribution along the circumference of a circular accelerator and definition of parameters

Inserting (21.80) on the r.h.s. of (21.79) and replacing the term $\mathrm{e}^{-\mathrm{i}pn_{\mathrm{b}}\omega _{0}\tau }$ with (21.81) one gets

$$\displaystyle\begin{array}{rcl} i_{\parallel }(t,\varphi )& =& \frac{qn_{\mathrm{b}}\omega _{0}} {2\pi } \sum \limits _{p=-\infty }^{+\infty }\sum \limits _{ n=-\infty }^{+\infty }\text{i}^{-n}J_{ n}(qn_{\mathrm{b}}\omega _{0}\hat{\tau })\, \\ & & \qquad \qquad \qquad \quad \times \mathrm{ e}^{\mathrm{i}\left [\left (pn_{\mathrm{b}}+nm+n\nu _{\mathrm{s}}\right )\omega _{0}t-pn_{\text{b}}\varphi +n\zeta _{i}\right ] }, {}\end{array}$$

(21.82)

Performing a Fourier transform

$$\displaystyle{ i_{\parallel }(\omega,\varphi ) = \frac{1} {2\pi }\int \limits _{-\infty }^{+\infty }i(t,\varphi )\,\mathrm{\ e}^{-\mathrm{i}\omega t}\,\mathrm{d}t }$$

(21.83)

we get instead of (21.82) the single particle longitudinal current spectrum

$$\displaystyle{ i_{\parallel }(\omega,\varphi ) = \frac{qn_{\mathrm{b}}\omega _{0}} {2\pi } \sum \limits _{\begin{array}{c}p= \\ -\infty \end{array}}^{+\infty }\sum \limits _{ \begin{array}{c}n=\\ -\infty \end{array}}^{+\infty }\mathrm{i}^{-n}J_{ n}(pn_{\mathrm{b}}\omega _{0}\hat{\tau })\,\mathrm{e}^{-\mathrm{i}(pn_{\mathrm{b}}\varphi -n\zeta _{i})}\,\delta (\varOmega )\,, }$$

(21.84)

where $\varOmega =\omega -(pn_{\mathrm{b}} + nm + n\nu _{\mathrm{s}})\omega _{0}$ and making use of the identity $\int \mathrm{e}^{-\mathrm{i}\omega t}\,\mathrm{d}t = 2\pi \delta (\omega )$. This spectrum is a line spectrum with harmonics of the revolution frequency separated by n_bω₀. Each of these main harmonics is accompanied on both sides with satellites separated by Ω_s = ν_sω₀. Schematically, some of the more important lines of this spectrum are shown in Fig. 21.6 for a single particle.

Fig. 21.6

Current spectrum of a single particle orbiting in a circular accelerator and executing synchrotron oscillations

In the approximation of small synchrotron oscillation amplitudes, one may neglect all terms with | n | > 1 and the particle beam includes only the frequencies ω = [p n_b ± (m +ν_s)]ω₀. In Sect. 19.5.1 the interaction of this spectrum for p = h with the narrow-band impedance of a resonant cavity was discussed in connection with Robinson damping.

A real particle beam consists of many particles which are distributed in initial phase ζ_i as well as in oscillation amplitudes $\hat{\tau }$. Assuming the simple case of equal and equidistant bunches with uniform particle distributions in synchrotron phase ζ_i we may set n = 0. The time independent particle distribution is then $\varPhi _{0}(t,\hat{\tau }) =\phi _{0}(\hat{\tau })$ which is normalized to unity and the total beam-current spectrum is given by

$$\displaystyle{ I_{\parallel }(\omega,\varphi ) = I_{\mathrm{b}}\sum \limits _{p=-\infty }^{+\infty }\delta (\omega -\varOmega _{ 0})\,\mathrm{e}^{-\mathrm{i}p\varphi }\int \nolimits _{ -\infty }^{+\infty }J_{ 0}(pn_{\mathrm{b}}\omega _{0}\hat{\tau })\,\phi _{0}(\hat{\tau })\,\mathrm{d}\hat{\tau }\,, }$$

(21.85)

where I_b = q∕T₀ is the bunch current and Ω₀ = pn_bω₀. All synchrotron satellites vanished because of the uniform distribution of synchrotron phases and lack of coherent bunch oscillations. Observation of synchrotron satellites, therefore, indicates a perturbation from this condition either by coherent oscillations of one or more bunches ($n\neq 0,\hat{\tau }\neq 0$) or coherent density oscillations within a bunch $\varPhi _{0}(t,\hat{\tau }) = f\,(\zeta _{i})$.

The infinite sum over p represents the periodic bunch distribution along the circumference over many revolutions whether it be single or multiple bunches. The beam-current spectrum is expected to interact with the impedance spectrum of the environment and this interaction may result in a significant alteration of the particle distribution $\varPhi (t,\hat{\tau })$. As an example for what could happen, the two lowest order modes of bunch oscillations are shown in Fig. 21.7.

Fig. 21.7

Dipole mode oscillation (a) and quadrupole bunch shape oscillations (b)

In lowest order a collection of particles contained in a bunch may perform dipole mode oscillations where all particles and the bunch center oscillate coherently (Fig. 21.7a). In the next higher mode, the bunch center does not move but particles at the head or tail of the bunch oscillate 180^∘ out of phase. This bunch shape oscillation is in its lowest order a quadrupole mode oscillation as shown in Fig. 21.7b. Similarly, higher order mode bunch shape oscillations can be defined.

3.2 Transverse Beam Spectrum

Single particles and a collection of particles in a bunch may also perform transverse betatron oscillations constituting a transverse beam current which can interact with its environment. Again, we observe first only a single particle performing betatron oscillations

$$\displaystyle{ u = \hat{u}\cos \psi (t)\,, }$$

(21.86)

where u = x or y, ψ(t) is the betatron phase, and the transverse current is

$$\displaystyle{ i_{\perp }(t,\varphi ) = i_{\parallel }(t,\varphi )\,\hat{u}\cos \psi (t)\,. }$$

(21.87)

Note that the transverse current has the dimension of a current moment represented by the same spectrum as the longitudinal current plus additional spectral lines due to betatron oscillations. The betatron phase is a function of time and depends on the revolution frequency and the chromaticity, which both depend on the momentum of the particle. From the definition of the momentum compaction dω∕ω₀ = η_c δ, chromaticity ξ_u = dν∕δ and relative momentum deviation δ = dp∕p₀, the variation of the betatron phase with time is

$$\displaystyle\begin{array}{rcl} \dot{\psi }(t)& =& \omega _{u} =\nu _{0}\left (1 + \frac{\xi _{u}} {\nu _{0}} \delta \right )\omega _{0}\left (1 +\eta _{\mathrm{c}}\,\delta \right )\,, \\ & \approx & \nu _{0}\omega _{0} + \left (\nu _{0} + \frac{\xi _{u}} {\eta _{\mathrm{c}}} \right )\omega _{0}\,\dot{\tau }\,, {}\end{array}$$

(21.88)

where we have kept only linear terms in δ and used $\dot{\tau } = -\eta _{\mathrm{c}}\delta$. Equation (21.88) can be integrated for

$$\displaystyle{ \psi (t) =\nu _{0}\omega _{0}(t-\tau ) -\omega _{0}\frac{\xi _{u}} {\eta _{\mathrm{c}}} \tau +\psi _{0} }$$

(21.89)

and (21.87) becomes with (21.74), (21.81), (21.84)

$$\displaystyle\begin{array}{rcl} i_{\perp }(t,\varphi )& =& i_{\parallel }(t,\varphi )\hat{u}\cos \psi (t)\, \\ & =& q\hat{u}\cos \psi (t)\sum \limits _{m=-\infty }^{+\infty }\delta \left (t - \frac{\varphi } {2\pi }\,T_{0} - m\frac{T_{0}} {n_{\mathrm{b}}} \right )\, \\ & =& q\hat{u}\frac{\mathrm{e}^{\mathrm{i}\psi (t)} +\mathrm{ e}^{-\mathrm{i}\psi (t)}} {2} \frac{n_{\mathrm{b}}} {T_{0}}\sum \limits _{p=-\infty }^{+\infty }\mathrm{e}^{\mathrm{i}[pn_{\mathrm{b}}\omega _{0}(t-\tau )-p\varphi ]}\,.{}\end{array}$$

(21.90)

Following the derivation for the longitudinal current and performing a Fourier transform we get the transverse beam spectrum

$$\displaystyle\begin{array}{rcl} i_{\perp }(\omega,\varphi )& =& \frac{q} {2T_{\mathrm{0}}}\hat{u}\,\text{e}^{\mathrm{i}\psi _{\mathrm{0}} }\sum \limits _{p=-\infty }^{+\infty }\sum \limits _{ n=-\infty }^{+\infty }\text{i}^{-n}J_{ n}\left \{\left [(p +\nu _{\mathrm{0}})n_{\mathrm{b}}\omega _{\mathrm{0}} -\frac{\xi _{u}} {\eta _{\mathrm{c}}} \right ]\hat{\tau }\right \} \\ & & \qquad \quad \times e^{-\mathrm{i}(p\varphi -n\zeta _{i})}\,\delta (\varOmega _{ u})\,, {}\end{array}$$

(21.91)

where $\varOmega _{u} =\omega -(p +\nu _{0})n_{\mathrm{b}}\omega _{0} + n\,\varOmega _{\mathrm{s}}$ defines the line spectrum of the transverse single particle current (Fig. 21.8).

Fig. 21.8

Oscillation spectrum of a single particle orbiting in a circular accelerator and executing betatron and synchrotron oscillations

We note that the betatron harmonics $(p +\nu _{0})n_{\mathrm{b}}\omega _{0}$ are surrounded by synchrotron oscillation satellites, however, in such a way that the maximum amplitude is shifted in frequency by ω₀ξ_u∕η_c. It is interesting to note at this point that the integer part of the tune ν₀ cannot be distinguished from the integer p of the same value. This is the reason why a spectrum analyzer shows only the fractional tune Δ ν ω₀.

The transverse current spectrum is now just the sum of all contributions from each individual particles. If we assume a uniform distribution $\varPhi (t,\hat{\tau },\hat{u})$ in betatron phase, we get no transverse coherent signal because $\langle \mathrm{e}^{\mathrm{i}\psi _{0}}\rangle = 0$, although the incoherent space-charge tune shift is effective. Additional coherent signals appear as a result of perturbations of a uniform transverse particle distribution.

Problems

21.1 (S). The linear focusing of the beam-beam effect changes also the betatron function. Derive an expression that relates the change in the value of the betatron function β_y^∗ at the collision point to the beam-beam tune shift δ ν. 

21.2. Verify that (21.17) and (21.25) are indeed solutions of the respective Poisson equation.

21.3. Prove that (21.27) is indeed the potential for small vertical amplitudes and x = 0.

21.4. Calculate the linear beam-beam tune shift for each beam under the following head on colliding beam conditions:

• a) A 250 GeV proton beam colliding with a fully ionized 30 GeV/u Au ion beam. (proton emittance ε_x, y = 20 mm-mrad, gold ion emittance ε_x, y = 33 mm-mrad, β_x, y^∗ = 2. 0 m, proton intensity 10¹¹ p/bunch, a total of 60 bunches per beam, gold ion intensity 10⁹ Au ions/bunch).

• b) A 250 GeV proton beam colliding with a fully ionized 100 GeV/u Au ion beam (parameters same as in a) but gold ion emittance ε_x, y = 10 mm-mrad).

• c) A 30 GeV electron beam colliding with a 820 GeV proton beam. The circumference of the rings is 6336 m, there are 2. 1 × 10¹³ protons and 0. 8 × 10¹³ electrons in 210 bunches and the horizontal and vertical beam sizes at the collision point are σ_x∕y = 0. 29∕0. 07 mm for the proton beam and 0. 26∕0. 02 mm for the electron beam, respectively.

• d) A 1.5 GeV electron beam colliding with a 1.5 GeV positron beam at a collision point with ε_x = 0. 67 mm-mrad, emittance coupling 27.7 %, β_x^∗ = 1. 3 m, β_y^∗ = 0. 1 m and a beam current of 66 mA [35].

21.5. Estimate the strength of the octupole field component of the proton beam in RHIC at the collision point. Would an octupole be technically feasible to compensate for the beam-beam octupole term?

21.6. At the Stanford Linear Collider, SLC, an electron beam collides with a positron beam at up to 50 GeV per beam. Each bunch contains 5 × 10¹¹ particles and is focused to a beam diameter of $2.0\,\upmu \mathrm{m}$ at the collision point where the betatron functions in both planes are β^∗ = 0. 005 m. Calculate the beam-beam tune shift and the focal length of the beam lens for a bunch length of ℓ = 1 mm. Compare with beam-beam limits in storage rings. Why can we tolerate a much greater beam-beam tune shift in a linear collider compared with a storage ring?

21.7. Show that the horizontal damping partition number is negative in a fully combined function FODO lattice as employed in older synchrotron accelerators. Why, if there is horizontal antidamping in such synchrotrons, is it possible to retain beam stability during acceleration? What happens if we accelerate a beam and keep it orbiting in the synchrotron at some higher energy?

21.8. Future colliding beam facilities for high-energy physics experimentation are based on two linear accelerators aimed at each other and producing beams of very high energy for collision. In this arrangement synchrotron radiation is avoided compared to a storage ring. We assume that such beams can be directed to different detectors. Design an S-shaped beam transport system based on a FODO lattice, which would allow the beams to be directed into a detector being displaced by the distance D normal to the linac axis. The beams have an energy of E₀ = 1, 000 GeV and a beam emittance of ε = 1. 0 × 10⁻¹² m which should not be diluted in this beam transport system by more than 10 %. Determine quadrupole and bending magnet parameters.

21.9. Strong focusing is required along a 500 GeV linear accelerator. Misalignments and path correction introduce dipole fields which are the source of synchrotron radiation and quantum excitation. Assume a normalized emittance of γ ε = 10⁻⁶ m and an initial beam energy of 1 GeV at the entrance to the linac. The high-energy linac has a circular aperture of 3 mm diameter. Design a FODO cell with sufficient focusing to contain this beam within a radius of 0.5 mm leaving the rest for path distortions. The distance between quadrupoles increases linearly with energy. Determine with statistical methods the number and strength of the quadrupoles for an acceleration of 100 MeV/m. Determine the alignment tolerances for these quadrupoles to keep the emittance increase due to quantum excitation in the dipole field from misaligned quadrupoles and due to correctors to 10 %.

21.10. Consider the FODO lattice along the linear accelerator in Problem 21.9 and estimate the increase in beam energy spread due to synchrotron radiation from the finite beam size in quadrupoles.

21.11. Consider an electron beam in a 6 GeV storage ring with a bending radius of ρ = 20 m in the bending magnets. Calculate the rms energy spread σ_ε∕E₀ and the damping time τ. What is the probability for a particle to emit a photon with an energy of σ_ε and 2σ_ε. How likely is it that this particle emits another such photon within a damping time? In evaluating the particle distribution, do we need to consider multiple photon emissions?

21.12. Consider one of the storage rings in Table 10.1 and calculate the equilibrium beam emittance and energy spread. To manipulate the beam emittance we vary the rf-frequency. Determine the maximum variation possible with this method.

21.13. A large hadron collider LHC operates in the LEP tunnel of 28 km circumference at CERN in Geneva. The maximum proton energy is 15 TeV. Determine the magnetic bending field required if 80 % of the circumference can be used for bending magnets. Calculate the synchrotron radiation power for a circulating proton current of 200 mA, damping times, equilibrium beam emittance and energy spread.

21.14. Determine basic FODO lattice parameters for a 2 GeV e⁺∕e⁻-colliding beam storage ring with two collision points to reach a design luminosity of $\mathcal{L}_{\text{e}} = 10^{31}$ cm⁻²s⁻¹. The betatron functions at the collision point be β_y^∗ = 5 cm and β_x^∗ = 1. 3 m and the emittance coupling 10 %. Calculate beam sizes in the arc, aperture requirements, circumference and beam current. What is the total synchrotron radiation power? Adjust, if necessary, your design to keep the maximum synchrotron radiation power at the vacuum chamber wall below a practical limit of 5 kW/m.