The Rapid Cycling Synchrotron (RCS), whose beam energy ranges from 400 MeV to 3 GeV and which is located in the Japan Proton Accelerator Research Complex, is a kicker-impedance-dominated machine, which violates the impedance budget from a classical viewpoint. Contrary to conventional understanding, we have succeeded in accelerating a 1 MW equivalent beam. The machine has some interesting features: e.g., the beam tends to be unstable for the smaller transverse beam size and the beam is stabilized by increasing the peak current. Space charge effects play an important role in the beam instability at the RCS. In this study, a new theory has been developed to calculate the beam growth rate with the head-tail and coupled-bunch modes $(m,\mu )$ while taking space charge effects into account. The theory sufficiently explains the distinctive features of the beam instabilities at the RCS.

## 1. Introduction

The 3 GeV Rapid Cycling Synchrotron (RCS) at the Japan Proton Accelerator Research Complex [1] aims to achieve a megawatt-class beam. Two bunched beams ($4.15\xd71013$ particles per bunch) are accelerated from 400 MeV to 3 GeV with a repetition rate of 25 Hz. To avoid the effects of eddy currents on metal chambers [2,3], ceramic chambers are adopted instead [4,5]. Accordingly, the resistive wall impedance [5–10] is negligible in the RCS [11–13].

However, there has been some concern that the kicker impedance limits the beam intensity of the RCS [14] by exciting beam instabilities [6]. Precise offline and online measurements of the impedance show that the kicker at the RCS has a huge impedance [15]. The offline measurement is conducted using the standard wire method [7], while the online one is conducted by observing the beam induced-voltage at the end of the power cable [15]. The results of these two independent measurements agree with each other. Finally, we demonstrate that the RCS is a kicker-impedance-dominated machine; we show this by suppressing the beam growth rate in accordance with the reduction of the kicker impedance [12,13].

In general, when we design accelerators, a lower impedance source along the rings is preferable for achieving a high intensity beam. Concretely, ceramic chambers are adopted, the chambers are connected as smoothly as possible over the rings [16], and significant efforts are made to lower the kicker impedances [12,17].

The conventional Sacherer formula [18,19] estimates the beam growth rate by using the impedances as an input parameter. However, such estimation differs significantly from the measured results at the RCS. We suspected the main reason for this is that the formula neglects space charge effects. Because the RCS covers the intermediate energy region (from 400 MeV to 3 GeV), space charge should have an effect on the beam instability.

Other theories exist to assess beam instability that includes space charge effects [20–22]. However, those theories assume simple forms of impedance, e.g., resistive wall impedance, resonator type impedance with a single resonance frequency, and constant wakes. Moreover, the theories do not include coupled-bunch-type instabilities.^{1}

In this paper, we develop a new theory that includes coupled-bunch and head-tail instabilities with space charge effects based on the Vlasov equation [6,23]. Using this theory, we try to understand the parameter dependence (the transverse emittance dependence, the beam peak current dependence, the tune dependence, etc.) of the beam instability observed at the RCS.

In Sect. 2, we start with the Hamiltonian and construct the Vlasov equation [6]. In Sect. 2.1, we derive a dispersion relation with the head-tail and coupled-bunch modes $(m,\mu )$ that includes space charge effects. In Sect. 2.2, we reproduce the previous Sacherer formula by neglecting the space charge effects.

In Sect. 3, typical parameters at the RCS are shown, and we show that the observed beam instability cannot be explained at all using the classical theory, i.e., Sacherer's theory [18,19], where space charge effects are neglected.

In Sect. 4, the beam instability observed at the RCS is analyzed using our new theory. In Sect. 4.1, the space charge effects on the beam instability are investigated by comparing the measurements with the theoretical results. In Sect. 4.2, tune manipulations are discussed from both the theoretical and the experimental viewpoints. The paper is summarized in Sect. 5.

In Appendix A, the scalar potential describing the space charge effect is calculated by solving the Poisson equation with the boundary condition of being surrounded by a perfectly conductive cylindrical chamber. In Appendix B, we explain canonical transformations to derive the Hamiltonian describing nonlinear betatron oscillation by using action-angle variables from the original Hamiltonian given by Eq. (1) in the next section.

## 2. Linearized Vlasov approach

The linearized Vlasov approach is a standard theoretical method to analyze beam instabilities [6]. In Sect. 2.1, the linearized Vlasov equation converts to a dispersion relation as a powerful tool to discuss the space charge effect on the beam instability. In Sect. 2.2, the classical Sacherer formula is reproduced, based on the linearized Vlasov equation.

### 2.1. Dispersion relation with the head-tail and coupled-bunch modes that include space charge effects

Here, we present the dispersion relation, by which we calculated the beam growth rate; this method takes into account the coupled mode $\mu $, head-tail mode $m$, and space charge effects.

The original Hamiltonian is given by [24,25]

Successive canonical transformations convert Eq. (1) to the Hamiltonian with action-angle variables, to describe the nonlinear betatron motions. The derivation is explained in detail in Appendix B. From now on, we consider only the horizontal and the longitudinal motions of the beam, for simplicity. Finally, the Hamiltonian is given by

Here, let us consider the situation that $M$ buckets are filled with $M$ bunches in a ring. We denote by $\Psi n(\theta ,Jx,\psi x,JL,\varphi L)$ the phase space distribution function of the $n$th bunch among $M$ bunches.

The Vlasov equation is expressed as

The distribution function $\Psi n$ is decoupled into an unperturbed part $F0(Jx)G0(JL)$ and a perturbed part $f1(Jx,\psi x)g1(JL,\varphi L)$ as

Substituting Eqs. (15) and (17) into Eq. (13) and using Poisson's sum rule [6],

Second, we introduce the Fourier transforms of the perturbed parts $f1$ and $g1$ as

Substituting Eqs. (22)–(23) into Eq. (15), the dipole current $Dp$ is rewritten as

The equations of motion are given by

Here, let us substitute Eqs. (22)–(23) into Eq. (33) before it is multiplied by $exp(jq\u2032\psi x+jm\u2032\varphi L)$. By integrating the result over $\psi x$ and $\varphi L$, we obtain approximately

Here, let us introduce the function $Dm$ as

If we choose the functions $Dm(JL)$, $G0(JL)$, and $F0(JL)$ as

The beam growth rate and the coherent tune affected by the wake force, which are given by the real parts of $j\omega 0\nu $ and of $\nu $, respectively, are solved by Eq. (48) as a function of nominal tune $Qx$. The differences among $Qx$, $\nu X0$, and $\u211c[\nu ]$ are tiny ($\u22720.01$) in a practical situation.

### 2.2. The Sacherer formula

Here, we reproduce the classical Sacherer formula [18,19,23], where the space charge effect on the beam oscillations is neglected. In this case, the $Jx$ and $JL$ dependence of the tunes $\nu L$ and $\nu x$ vanishes. If we confine ourselves to the case, the $Jx$-integration in Eq. (43) can be performed for the distribution Eq. (46). Consequently, Eq. (43) is simplified as

Let us expand the function $Dm(JL(rs))$ using a complete set of orthogonal functions $fk(m)(rs)$ as

Let us introduce the particle distribution function $\rho m,l(\tau )$ with head-tail mode $m$ and radial mode $l$ in real space as

Substituting Eq. (64) into Eq. (65), Eq. (65) is written as

By substituting Eq. (58) into Eq. (57), in combination with Eqs. (60) and (66), Eq. (57) is solved as

The constants $C$ and $Cml$ are determined as follows. If we impose the condition $\rho m,l(\xb1\tau 0s)=0$ on the distribution function, the function $\rho m,l(\tau )$ should be written as

Here, let us focus on the lowest-order term for the radial mode $l=1$. The factor

By summarizing all these results (by substituting Eqs. (89) and (90) into Eq. (70), and by calculating Eq. (72) with Eq. (87)), we finally derive the conventional Sacherer formula:

## 3. RCS parameters and the beam growth rate estimated by the Sacherer formula

At the RCS, the bunched beams are formed by accumulating the injection beam from the LINAC with a painting scheme [28,29]. They are accelerated from 400 MeV to 3 GeV over 20 ms. Figure 2 shows the typical patterns of the acceleration voltage $Vrf$ (red), and of the synchronous phase $\phi s$ (blue) in that period. Table 1 shows typical machine and beam parameters for the RCS, which were used in this paper's calculations. The average chamber radius $a$ around the ring is determined to be 145 mm, in order that the coherent betatron tune shift reproduces the measured date for a 400 MeV beam.

$T$ (kinetic energy, GeV) | 0.4 | 3 |

$f0$ (revolution frequency, MHz) | 0.61 | 0.84 |

$\eta $ (slippage factor) | -0.478 | -0.047 |

$Ic$ (average current, A) | 8.1 | 11.2 |

$\nu s0$ (synchrotron tune) | 0.0053 | 0.0005 |

$\u2329\beta x(s)\u232a$ (m) | 11.6 | |

$\u2329\beta x2(s)\u232a$ (m^{2}) | 172.3 | |

$\u2329D2(s)\u232a$ (m)^{2} | 3.46 | |

$\u03f5x,rms$ (mmrad) | $100/6\beta s\gamma s$ | |

$JL0$ (eV $\u22c5$ s) | $0.1645$ |

$T$ (kinetic energy, GeV) | 0.4 | 3 |

$f0$ (revolution frequency, MHz) | 0.61 | 0.84 |

$\eta $ (slippage factor) | -0.478 | -0.047 |

$Ic$ (average current, A) | 8.1 | 11.2 |

$\nu s0$ (synchrotron tune) | 0.0053 | 0.0005 |

$\u2329\beta x(s)\u232a$ (m) | 11.6 | |

$\u2329\beta x2(s)\u232a$ (m^{2}) | 172.3 | |

$\u2329D2(s)\u232a$ (m)^{2} | 3.46 | |

$\u03f5x,rms$ (mmrad) | $100/6\beta s\gamma s$ | |

$JL0$ (eV $\u22c5$ s) | $0.1645$ |

(Circumference $C=348.333\u2002m$, harmonic number $h=2$, repetition rate = 25 Hz, particles per bunch $Nb=4.15\xd71013$, and the average chamber radius $a=145\u2002mm$).

Here, $\beta x(s)$ and $D(s)$ are the $\beta $-function and the dispersion function, respectively; $\u03f5x,rms$ and $JL0$ are the root mean square (rms) horizontal and the longitudinal emittances, respectively; $\u2329\cdots \u232a$ denotes the average value around the ring; and $\beta s$ and $\gamma s$ are the Lorentz-$\beta $ and Lorentz-$\gamma $ on the designed particle.

Eight kickers are installed in the RCS. The real and the imaginary parts of the horizontal impedance $ZT(\omega )$ for one kicker are shown in the left and the middle panels of Fig. 3, respectively. The red and blue lines show the impedances at $\beta s=0.7$ and $\beta s=1$, respectively. The impedance is roughly proportional to the Lorentz-$\beta $ [15]. The corresponding wake function $WT(\omega 0t)$ calculated by Eq. (17) is denoted by the same color in the right-hand figure. The reflection wave excited at the end of the power cable of the kicker creates the spike structure of the kicker impedance.

As shown in the left and middle panels, the impedance is very large indeed. We have demonstrated that the RCS is a kicker-impedance-dominated machine by stabilizing unstable beams by temporarily reducing the impedance [12,13]. For simplicity, we assume in this paper that the only source of impedance in the RCS is kicker impedance.

Mostly (except the discussion about chromaticity dependence of beam growth rates shown in Figs. 16 and 17), let us consider a case in which the chromaticity $\xi Qx$ is activated by a DC-power supply at the injection energy. In this case the chromaticity approaches the natural chromaticity $(\xi Qx=\u221210.3)$ [30] as the beam energy is increased, as shown in Fig. 4.

We have observed beam instabilities at the J-PARC RCS, where the chromaticity was fully corrected only at the injection energy. The blue line of Fig. 5 shows an example of the results of the horizontal beam position for a 750 kW equivalent beam ($3.10\xd71013$ particles per bunch). For reference, the green line shows the results where only one bucket among the two is filled with one bunched beam. Since no instability occurs on the green line, we have judged that the instabilities on the blue line are the coupled-bunch instabilities.

Figure 6 shows the measured results for a 1 MW equivalent beam ($4.15\xd71013$ particles per bunch), where the chromaticity was fully corrected only at the injection energy. Both results for 750 kW equivalent and 1 MW equivalent beams have demonstrated that the beam is stable at low energies, while they tend to be unstable at high energies.

Here, let us investigate whether the conventional Sacherer formula Eq. (92) can explain the measured beam behavior. From now on, we assume that the maximum number of the head-tail mode $m$ is 5, and that the coupled mode $\mu $ runs from 0 to 1. Figure 7 shows the theoretical results for the case. The results predict that the beam is unstable at low energies, while it is stable at high energies. These results suggest that a partial chromaticity correction at low energies should enhance the beam instability at low energies. However, these theoretical results (Fig. 7) differ significantly from the measured data (Figs. 5 and 6).

The measurement results indicate that space charge stabilizes the beam instability at low energies. Note that Eq. (92) is derived by neglecting this effect. In the next section, let us theoretically examine the space charge effect on the beam instability at the RCS.

## 4. Investigation of the beam instability at the RCS

### 4.1. Space charge effects on the beam growth rate

The Landau damping caused by the space charge effect appears in Eq. (51). Because this equation depends only on the longitudinal emittance $JL0$ in our model, only the longitudinal size of the beam is likely to affect the effect, significantly. However, the true space charge effect is revealed in Eq. (48) after integration with respect to $Jx$ according to Eq. (43). In particular, the damping effect is neglected for a beam with infinitesimal transverse beam size, and Eq. (48) is sufficiently well approximated by the analytical formula

Figure 8 shows the maximum beam growth rate among different modes $(m,\mu )$ estimated according to Eq. (99). As in the results obtained using the conventional formula (shown in Fig. 7), these results show that the beam is unstable at low energies. However, this result successfully explains the beam instability of the measured results at high energies, which the conventional formula does not explain. To understand the beam stabilization at low energies, the Landau damping effects owing to space charge must be taken into account.

Here, let us investigate the effect more closely. First, we present the theoretical results of taking the space charge effect into account for the maximum beam growth rate by solving Eq. (48). The results are shown in Fig. 9. Comparing the results shown in Fig. 8 with the present results, we find that the beam is stabilized at low energies and that the theoretical results explain well the characteristic of the measurement ones (shown in Figs. 5 and 6). We can see a sharp rise at $t=13\u2002ms$ only in the measured data of the 1 MW-equivalent beam (Fig. 6). The space charge damping effect seems to be drastically reduced for a beam with larger oscillation amplitudes. If this is a kind of nonlinear phenomenon, our theory, based on the linearized Vlasov equation, has a limit to explain it.

Figure 10 shows the theoretical results of the transverse beam emittance dependence of the beam growth rate. The red, black, and purple lines are the beam growth rates excited by $(m=0,\mu =1)$, $(m=2,\mu =1)$, and $(m=4,\mu =1)$ modes, respectively (the growth rate excited by the other modes is negligibly low.). As already explained, the Landau damping effect becomes ineffective for all modes, as the transverse emittance decreases. Figure 11 illustrates the measured beam positions for different transverse emittances. The red, blue, black, and yellow lines show the results for the cases that the injection painting areas are $0\pi $ (center injection), $100\pi $, $150\pi $, and $200\pi $ mmrad, respectively [28,29]. The emittance dependence is clearly observable in the results. As the painting area is larger at the injection period, the beam tends to be more stabilized at high energies.

Thus, we find that the Landau damping effect owing to the space charge (depending on the longitudinal beam size) is enhanced by enlarging the transverse beam size. From a phenomenological point of view, the space charge damping effect is easily activated for the lower-energy beam, as a result of the larger transverse beam emittance.

Now, let us closely investigate the bunching factor $Bf$ (longitudinal beam size) dependence of the beam growth rate for different head-tail and coupled-bunch modes $(m,\mu )$. Figure 12 shows the theoretical results of the beam growth rate at 15 ms, where the bunching factor $Bf$ is evaluated by using

The conventional Sacherer formula (92) indicates that the beam growth rate without space charge is roughly inversely proportional to the bunching factor $Bf$. The left and middle panels demonstrate that the overall behavior of the beam growth rate including space charge effect is also roughly inversely proportional to the bunching factor $Bf$. However, the beam is ultimately stabilized in the extremely compressed beam (with the extremely small bunching factor). In this case, the Landau damping due to the space charge force absolutely stabilizes the beam instability.

The beam growth rates for the different modes ($m,\mu $) in all panels of Fig. 12 reveal the respective comb-like structures along the bunching factor. The behavior originates from the head-tail motion of the beam, as shown in the form factor $Fm(JL,x)$ in Eq. (99). Thus, when we fix a mode, the beam growth rate for the mode follows the characteristic comb-like behavior, even in the results without space charge effect (right). However, because the growth rate patterns are overlapped for the different modes ($m,\mu $) in the results without space charge effect, it is hard to specify the optimized point along the bunching factor from the viewpoint of beam instability. Thus, we reach the conventional conclusion that the larger bunching factor (smaller peak current) is preferable for beam stabilization, when the space charge effect is neglected.

Contrary to the such conventional understanding, beam stabilized regions emerge along the bunching factor in the results with the space charge effects for $a=145\u2002mm$ (e.g., around the area $A$). Comparing both the results for $a=145\u2002mm$ (left) and for $a=160\u2002mm$ (middle), we find that the bandwidth of the stabilized region caused by the space charge effect significantly depends on the chamber radius $a$. Though the difference between the chamber radii is only 15 mm, the beam stabilization area $A$ in the results for $a=145\u2002mm$ (left) disappears in the results for $a=160\u2002mm$ (middle). In conclusion, the smaller chamber radius is preferable in view of the beam stabilization to make maximum use of the space charge damping effect.

The existence of such a beam stabilization region, stemming from the space charge effects, along the bunching factor can be demonstrated at a low-energy proton ring like the RCS. At the RCS, the bunching factor can be changed by changing the momentum spread of the injection beam from the LINAC. We can prepare two types of injection beams: $dp/p=0.08%$ and $dp/p=0.18%$. The injection beam with the smaller momentum spread creates an accumulated beam with a smaller bunching factor. The measurement results for the beam positions and their corresponding bunching factors are illustrated in Fig. 13 with the same colors, where the number of particles per bunch is $3.10\xd71013$. It is observable that the beam can be more stabilized with the smaller bunching factor, contrary to conventional understanding. Theoretically, this stabilization is caused by the dip around the area $A$ in Fig. 12.

### 4.2. The effects of tune manipulation on beam growth rate

Here, let us illustrate the tune dependence of the beam growth rate. The measurement results are shown in the left panel of Fig. 14. The tracking pattern of the tune during the acceleration period is shown in the right panel of Fig. 14 using the same color. The results represented by the red line correspond to the highest beam growth rate case. The second highest case is represented by the yellow line. The most stable case is indicated by the black line, which is sandwiched by these two unstable cases (the red and the yellow lines). Figure 15 shows the theoretical results of the beam growth rate at 15 ms, which are obtained by solving Eq. (48). The red, black, and purple lines are the beam growth rates excited by the $m=0$, $m=2$, and $m=4$ modes, respectively. The solid and dashed lines show the $\mu =1$ and $\mu =0$ modes, respectively (the other head-tail modes do not excite the beam instabilities). The theoretical calculation explains the characteristic of the tune dependence of the beam growth rate sufficiently well, as revealed by the measured results (Fig. 14). The tune dependence of the beam growth rate originates from the spike structure of the kicker impedance (see Fig. 3).

Finally, we illustrate the chromaticity dependence of the beam growth rate. Figure 16 shows the theoretical results of the beam growth rate at 15 ms for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,\mu =1)$, $(m=1,\mu =1)$, $(m=2,\mu =1)$, $(m=3,\mu =1)$, $(m=4,\mu =1)$, and $(m=5,\mu =1)$ modes, respectively (the other modes do not excite the beam instabilities). We expect that the beam growth rate will be drastically suppressed, as the chromaticity correction is weakened.

The measured results are shown in Fig. 17. To clearly observe the chromaticity dependence of the beam growth rate, let us study the highest growth rate case (the tracking pattern of the tune is designated by the red line in the right panel of Fig. 14). The red, blue, and black lines in Fig. 17 show, respectively, the results for which the chromaticity was fully corrected only at the injection energy by the DC-power supply, half corrected compared to the full correction, and quarter corrected in the same manner. Concretely, the chromaticity values at 15 ms are $\u22127.46$ for the red line, $\u22128.92$ for the blue line, and $\u22129.64$ for the black line. As expected, the beam is drastically suppressed by an increase in chromaticity in the negative direction.

## 5. Summary

The RCS in J-PARC, where kicker impedance dominates, is a special machine from an impedance viewpoint, which means that the RCS violates the impedance budget from a classical viewpoint [6,18,19]. Nevertheless, we have successfully accelerated a 1 MW equivalent beam ($4.15\xd71013$ particles per bunch). The RCS is an accelerator covering the intermediate beam energy region (from 400 MeV to 3 GeV). Thus, it is pertinent to study the space charge effects on the beam instability.

The machine has some interesting characteristics: e.g., the beam can be stabilized by reducing the bunching factor (increasing the peak current) and the beam tends to be unstable when reducing the transverse beam size. The classical theory, i.e., Sacherer's theory, fails to explain these characteristics by neglecting the space charge effects.

Recently, there has been a significant development in the field of computer technologies. Numerical computer simulations are powerful tools to quantitatively estimate the beam behavior associated with space charge effects [32–34]. It may seem that a numerical simulation study is sufficient to accelerate beams from a practical viewpoint.

However, such simulations take excessive CPU time and memory for one set of fixed parameters. If we theoretically understand what conditions (parameters sets) excite beam instabilities in combination with space charge effects in advance, numerical studies are more efficiently performed by selecting the appropriate parameters sets, based on the theoretical comprehension. Consequently, we can focus on the quantitative discussion about the issues concerning beam commissioning (beam loss, beam halo, etc.). Moreover, the theoretical study is vital to understand the nature of the phenomena concerning beams in accelerators.

In this paper, we try to understand the beam instabilities associated with the space charge effects by developing a new theory. And, we have clarified the parameters (such as the transverse emittance, the bunching factor, etc.) dependence on the beam growth rate.

The space charge damping effect is significant at low energies, not only due to the smaller Lorentz-$\gamma $ but also due to the larger transverse beam size. The large transverse emittance is essential to activate the Landau damping owing to the space charge effect.

It is of interest that the beam growth rate is suppressed by increasing the peak current (shortening the bunch length, or reducing the bunching factor) at the RCS. Theoretically, the beam growth rate for different modes $(m,\mu )$ follows different characteristic comb-like structures along the bunching factor. The dependence of the beam growth rate on the bunching factor originates from the head-tail motion of the beam. Thus, even in the case without the space charge effect, the beam growth rate for one fixed mode can be suppressed by increasing the peak current (shortening the bunch length, or reducing the bunching factor).

However, the beam growth rates excited by different modes $(m,\mu )$ are sufficiently overlapped along the bunching factor in the case. Finally, the theory reproduces the conventional conclusion that the maximum beam growth rate among different modes $(m,\mu )$ is reduced by increasing the bunch length (reducing the peak current or increasing the bunching factor) when the space charge effect is neglected.

On the contrary, if we take the space charge effect into consideration, the overlap of the beam growth rates for different modes ($m,\mu $) is separated over the axis of the bunching factor, and some beam stabilization regions emerge on the axis. The optimization of bunching factor enables the beam to be stabilized, regardless of the amount of the bunching factor, in a lower-energy proton synchrotron like the RCS.

The space charge damping effect is quite sensitive to the chamber radius. Consequently, a smaller radius chamber is preferable from a beam instability point of view. As the beam energy becomes higher, the space charge damping effect becomes less effective, and the beam stabilization region diminishes along the bunching factor.

In a low-energy proton machine, such as the RCS, the violation of the impedance budget from a classical viewpoint is not vital to achieve high intensity beams. They can be realized by optimizing the machine's (beam) parameters, i.e., the bunching factor, transverse emittance, tune, chromaticity, etc.

## Acknowledgement

The authors would like to thank Kazuhito Ohmi, Jie Wei, Katsunobu Oide, Yoshishige Yamazaki, Tadashi Koseki, Kazuo Hasegawa, and Michikazu Kinsho for fruitful discussions. They also would like to thank all members of the J-PARC Accelerator Technical Advisory Committee, which was led by Steve Holmes until 2009, and has been led by Thomas Roser since 2010. The authors would also like to thank all the members of the J-PARC project at JAEA/KEK.

#### Appendix A. A solution of the Poisson equation with cylindrical chamber

In this section, we show how to solve the Poisson equation for an axisymmetric beam that is surrounded by a perfectly conductive cylindrical chamber with radius $a$. The Poisson equation in the rest frame of the beam $(ct\xaf,x,y,z\xaf)$ is described by

When a perfectly conductive chamber with radius $a$ exists, the Green function $G(r\u2192,r\u2192\u2032)$ that satisfies the boundary condition $G=0$ at $\rho =a$, is given by [35]

The potential $\Phi \xafc$ felt at the bunch center is calculated by plugging in $\rho =r0$ and $\phi =\theta 0$ [26]. Figure 18 illustrates typical behavior of the potential $\Phi \xafc(z\xaf=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.

Here, let us expand the result for small $\rho $ around zero. As a result, it is expressed as

The terms $\Phi \xafcoh,2(z\xaf)$ and $\Phi \xafcoh,4(z\xaf)$ contribute to the coherent space charge tune shift, and to the nonlinear motion of the beam, respectively.

The scalar potential $\Phi $ and the vector potential $Az$ in the lab-frame $(ct,x,y,z)$ are given by

#### Appendix B. Derivation of the Hamiltonian with action-angle variables including horizontal wake and space charge effects

In this section, we will obtain the Hamiltonian Eq. (B50) with action-angle variables, by successively canonically transforming Hamiltonians.

The original Hamiltonian in an electromagnetic field is approximately given by [24,25]

The orbit length $s$ is used as an independent variable. The canonical variables are $(x,px),(y,py)$, and $(t,\u2212E)$ for the horizontal, vertical, and the longitudinal directions, respectively. It is noticeable that the contribution from the vector potentials is included in the Hamiltonian, where the contributions from both the scalar and vector potentials are confined to the scalar potential only with Eqs. (A16) and (A17).

Using the generating function $F1$,

The new Hamiltonian $H1$ is obtained as

Next, using the generating function $F2$,

Then, the new Hamiltonian $H2$ is described as

By extracting the Hamiltonian $H3,L$ for the synchrotron oscillation, we obtain

Before describing the Hamiltonian in terms of action-angle variables, let us continue to make the canonical transformations from $(x\u02dc,p\u02dcx),(y\u02dc,p\u02dcy)$ to $(x\xaf\xaf,p\xaf\xafx),(y\xaf\xaf,p\xaf\xafy)$, respectively, which are generated by the function $F3$:

For the longitudinal motion, let us consider the generating function

To extract the Twiss parameters dependence from the transverse variables $(x\xaf\xaf,p\xaf\xafx)$ and $(y\xaf\xaf,p\xaf\xafy)$, we consider the canonical transformations generated by the function $F4$:

Thus, the new Hamiltonian $H4$ is expressed as

The application of the canonical perturbation theory (see, e.g., Ref. [36]) for the Hamiltonian and neglecting the higher-order terms lead to the new Hamiltonian $H$:

Here, we consider a rather nonrelativistic condition, namely, a long bunch beam in the ring with the conditions

## References

^{1}After this paper had been submitted on 22 June 2016, the authors attended the 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams (HB2016: https://hb2016.esss.se/) and found that A. Burov had submitted a document entitled “Coupled-beam and coupled-bunch instabilities” to http://arxiv.org/pdf/1606.07430v1.pdf on 27 June 2016. He discusses the space charge effect on coupled-bunch-type instabilities by another approach.