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 $(m,μ)$ 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 ($4.15×1013$ 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 [510] is negligible in the RCS [1113].

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 [2022]. 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 $(m,μ)$ 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 $μ$, head-tail mode $m$, and space charge effects.

The original Hamiltonian is given by [24,25]

(1)
$Ho=−ps(1+xρ)ΔEpsβsc+ps2γs2(ΔEpsβsc)2+px2+py22ps+ps2Kx(s)(1−ΔEEs)x2 +ps2Ky(s)(1−ΔEEs)y2−psxEsFx+eΦc(x,y,s−cβst)βsγs2c −eVrfωRFδp(s)cos(ωRFt−hsR+φs)+…,$
where $φs$ is the synchronous phase; $ps$ is the constant longitudinal momentum of the synchronous particle; $Es=cps/βs$ is the particle energy on the designed orbit; $βs$ and $γs$ are the Lorentz-$β$ and the Lorentz-$γ$ of the designed particle, respectively; $ΔE$ is given by $ΔE=E−Es$; $Fx$ is the transverse wake force; $δp(s)$ is the periodic $δ$-function; $c$ is the velocity of light; $Kx$ and $Ky$ are the periodic focusing forces in the horizontal and the vertical directions, respectively; $Φc$ is the space charge potential felt by the bunch center [26]; $h$ is harmonic number; $Vrf$ is the amplitude of the radio frequency (RF) voltage; $1/ρ$ is the local curvature around the machine; $R$ is the average radius of the machine; and $ωRF$ is the angular frequency of the RF voltage, which is expressed as
(2)
$ωRF=cβshR.$
The orbit length $s$ is used as an independent variable. The canonical variables are $(x,px)$, $(y,py)$, and $(t,−E)$ for the horizontal, vertical, and longitudinal directions, respectively. The scalar potential $Φc$ is obtained by solving the Poisson equation with the boundary condition that the beam is surrounded by a cylindrical, perfectly conductive chamber with radius $a$. The solution is expressed in Appendix A.

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

(3)
$H≃QxJx+νs0JL+Ux+Y′,$
with the horizontal $(Jx,ψx)$ and the longitudinal action-angle variables $(JL,ϕL)$, and its independent variable $θ=s/R$, where $Qx$ and $νs0$ are the horizontal and the synchrotron tunes, and
(4)
$Ux=−2βsRc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$

(5)
$ϕx(s)=∫sdsβx(s),$

(6)
$Y′=Ycoh,0′(JL)+Y′coh,2(JL)βx(s)Jxps+Y′coh,4(JL)3βx2(s)Jx22,$

(7)
$Ycoh,0′(JL)=eRZ0eNb(γ˜−Ei[−a22σx2]+log[a22σx2])4π2βsγs2σz(π2)1/2exp(−c2JL|η|2Esh2νs0σz2ω0) ×I0(c2JL|η|2Esh2νs0σz2ω0),$

(8)
$Y′coh,2(JL)=−eRZ0eNb8π2βsγs2σx2(π2)1/2exp(−c2JL|η|2Esh2νs0σz2ω0)[−σx2(γ˜−Ei[−a22σx2]+log[a22σx2])2γs2σz5 ×((σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω0) +(1−exp(−a22σx2))I0(c2JL|η|2Esh2νs0σz2ω0)σz]+eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)42ππβsγs2σx4σz ×(σx2+2σx4(exp(−a22σx2)−1)a2)I0(c2JL|η|2Esh2νs0σz2ω0) −eRZ0eNb8π2βsγs2σx2σz(π2)1/2exp(−c2JL|η|2σz2h2ω0Esνs0)I0(c2JL|η|2σz2h2ω0Esνs0),$

(9)
$Y′coh,4(JL) =−eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)16π2σx2βsγs2ps2(π2)1/2{−σx2(γ˜−Ei[−a22σx2]+log[a22σx2])2 ×[(3σz4−6c2JLσz2|η|Esh2νs0ω0+2c4JL2|η|2Es2h4νs02ω02)I0(c2JL|η|2Esh2νs0σz2ω0)−2c2JL|η|(−2σz2+c2JL|η|Esh2νs0ω0)I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω08γs4σz9 −(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω0σx2γs2σz5+I0(c2JL|η|2Esh2νs0σz2ω0)σx4σz] +[1−exp(−a22σx2)−γ˜+Ei[−a22σx2]−log[a22σx2]] ×[(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|Esh2νs0ω0I1(c2JL|η|2Esh2νs0σz2ω0)2γs2σz5−I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz] +(6−(a2σx2+6)exp(−a22σx2)−4γ˜+4Ei[−a22σx2]−4log[a22σx2])I0(c2JL|η|2Esh2νs0σz2ω0)8σx2σz} +eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)8π2σx4βsγs2ps2(π2)1/2 ×{(8σx2−exp(−a22σx2)(a4σx2+4a2+8σx2))I0(c2JL|η|2Esh2νs0σz2ω0)4a2σz +σx2[2σx2−(a2+2σx2)exp(−a22σx2)][(14σx2+1a2)I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz −(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω04a2γs2σz5]+σx2[1−exp(−a22σx2)] ×[−I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz+(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω04γs2σz5]} +eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)32π2σx4βsγs2σzps2(π2)1/2(1−8σx4−exp(−a22σx2)(4a2σx2+8σx4)a4) ×I0(c2JL|η|2Esh2νs0σz2ω0)+eRZ0eNbβsγs2π2σx2ps2(π2)1/2exp(−c2JL|η|2h2ω0Esνs0σz2) ×[(−1128γs2σz3+164σx2σz+c2JL|η|128γs2σz5h2ω0Esνs0)I0(c2JL|η|2h2ω0Esνs0σz2) −c2JL|η|I1(c2JL|η|2h2ω0Esνs0σz2)128γs2σz5h2ω0Esνs0];$
$η$ is slippage factor; $Z0=120π Ω$ is the impedance of free space; $ω0$ is the angular revolution frequency; $βx(s)$ is the Twiss parameter; $D(s)$ is the dispersion function; $In(x)$ is the modified Bessel function; $Ei[z]$ is the exponential integral function [27]; $γ˜$ is Euler-$γ$; and $σx$ and $σz$ are the root mean square (rms) horizontal and longitudinal beam sizes, respectively. The potential functions $Ux$ and $Y′$ originate from the horizontal wake and the space charge forces, respectively.

Here, let us consider the situation that $M$ buckets are filled with $M$ bunches in a ring. We denote by $Ψn(θ,Jx,ψx,JL,ϕL)$ the phase space distribution function of the $n$th bunch among $M$ bunches.

The Vlasov equation is expressed as

(10)
$∂Ψn∂θ+Jx′∂Ψn∂Jx+ψx′∂Ψn∂ψx∂ψy+JL′∂Ψn∂JL+ϕL′∂Ψn∂ϕL=0,$
where the prime denotes differentiation with respect to the variable $θ$.

The distribution function $Ψn$ is decoupled into an unperturbed part $F0(Jx)G0(JL)$ and a perturbed part $f1(Jx,ψx)g1(JL,ϕL)$ as

(11)
$Ψn=F0(Jx)G0(JL)+f1(Jx,ψx)g1(JL,ϕL)exp(jνθ−jν2πnM−jQxξxϕcphη−j2πμnM),$
where $Qxξx$ is the chromaticity in the horizontal direction and $Ψn$ is normalized according to
(12)
$∫0∞dJx∫−ππdψx∫0∞dJL∫−ππdψLΨn(Jx,ψx,JL,ϕL)=1.$
In advance, let us formulate the dipole current $Dp$ of the beam (and its Fourier transform $D˜(p)$) and the horizontal wake force $Fx$ (i.e., the potential $Ux$) not only in the time domain but also in the frequency domain. The horizontal wake force $Fx$ is obtained by the summation of the wake force induced by the previous passage of beams. It is expressed as
(13)
$Fx=e2Nb2πR∫0∞dJ′x∫−ππdψ′x∫0∞dJ′L∫−ππdϕ′Lx′f1(J′x,ψ′x)g1(J′L,ϕ′L) ×∑n′=0M−1∑k=−∞∞exp(jνθ+jν(−2πn′M−2πk)−jQxξxϕcp′hη)exp(−jμ2πn′M) ×WT(ϕcp′h−ϕcph+2πk+2π(n′−n)M),$
where $WT(s)$ is the horizontal wake function, which has the causality condition $WT(s)=0$ for $s≤0$, and $ϕcp$ denotes the longitudinal position of beam, which is related to $JL$ and $ϕL$ as
(14)
$ϕcp=(2JLh2|η|ω0cpsβsνs0)1/2sinϕL,$
(see Eq. (B31)). The effect of the wake excited by all previous revolutions of beams is included as a summation over $k$ in Eq. (13). The dipole current $Dp$, its Fourier transform $D˜(p)$, and the horizontal impedance $ZT(ω)$ are defined as
(15)
$Dp(JL,ϕL)=∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)f1(Jx,ψx)g1(JL,ϕL)βx,$

(16)
$D˜(p)=12π∫0∞dJL∫−ππdϕLDp(JL,ϕL)exp(jpϕcph),$
and
(17)
$WT(ω0t)=−∫−∞∞dω2πjZT(ω)exp(jωω0ω0t),$
respectively, where
(18)
$x(Jx,ψx,JL,ϕL)=(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−Qxθ) −D(s)psc(2νs0ω0EsJL|η|)1/2cosϕL,$
and $j$ is the imaginary unit (the definitions of the causality condition of the wake function and of the impedance in Ref. [6] are different from those in this paper (see Eqs. (13) and (17))).

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

(19)
$∑k=−∞∞exp(jkx)=2π∑p=−∞∞δ(x−2πp),$
where $δ(x)$ is the $δ$-function, the wake force $Fx$ is rewritten as
(20)
$Fx=−je2NbβxT0Rexp(jνθ−jν2πnM)∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη) ×exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(Qx+p))exp(−j(ν+p)ϕcph),$
in frequency domain, where $T0$ is the revolution time of the designed particle. Now, the potential $Ux$ in Eq. (4) can be expressed as
(21)
$Ux=je2Nbβs2βxπRexp(jνθ−jν2πnM) ×((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−Dβs(ω0νs0JL2Es|η|)1/2cosϕL) ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M) ×ZT(ω0(ν+p))exp(−j(ν+p)ϕcph).$

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

(22)
$f1(Jx,ψx)=∑qf˜1,q(Jx)exp(−jqψx),$

(23)
$g1(JL,ϕL)=∑mg˜1,m(JL)exp(−jmϕL).$

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

(24)
$Dp(JL,ϕL) =∑q′,m′∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)f˜1,q′(Jx)g˜1,m′(JL)βxexp(−jq′ψx−jm′ϕL).$
Here, we introduce $Dpq′,l′,m′(JL)$ as
(25)
$Dp(JL,ϕL)=∑q′,m′Dpq′,m′(JL)exp(−jm′ϕL),$
so that
(26)
$Dpq′,m(JL)=∑m′∫−ππdϕL∫0∞dJx∫−ππdψx ×f˜1,q′(Jx)g˜1,m′(JL)x(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL).$
Then, Eq. (16) is expanded by $Dpq,m$ as
(27)
$D˜(p)=12π∫0∞dJL∫−ππdϕLDp(JL,ϕL)exp(jpϕcph)=∑q,m∫0∞dJ′LDpq,m(J′L)Jm*[pωRFh(2J′L|η|cpsβsνs0ω0)1/2] ,$
by using the relation
(28)
$∫−ππexp(−jqϕcpωRF+jmϕL)dϕL=∫−ππexp(−jq(2JL|η|cpsβsνs0ω0)1/2sinϕL+jmϕL)dϕL=2πJm[q(2JL|η|cpsβsνs0ω0)1/2],$
where $∗$ denotes the complex conjugate and $Jm[x]$ is the Bessel function [27].

The equations of motion are given by

(29)
$dJxdθ=−∂H∂ψx,$

(30)
$dψxdθ=∂H∂Jx,$

(31)
$dJLdθ=−∂H∂ϕL,$

(32)
$dϕLdθ=∂H∂JL,$
where the Hamiltonian is given by Eq. (3). By substituting Eqs. (11) and (29)–(32) into Eq. (10), and by retaining only the perturbed parts, the linearized Vlasov equation is obtained as
(33)
$(jν−jQxξxhηνL(Jx,JL)ωRF(2|η|RJLβs3cEsνs0)1/2cosϕL)f1(Jx,ψx)g1(JL,ϕL) +je2Nbβs2βx(s)2psπRJxsin(ψx+ϕx(s)−Qxθ) ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×exp(−j(ν+p)ϕcph)∂F0(Jx)∂JxG0(JL)exp(jQxξxϕcphη+j2πμnM) +νx(Jx,JL)∂f1(Jx,ψx)∂ψxg1(JL,ϕL)+νL(Jx,JL)f1(Jx,ψx)∂g1(JL,ϕL)∂ϕL=0,$
where we assume
(34)
$dG0(JL)dJL≃0,$
and
(35)
$νL(Jx,JL)=νs0+dY′dJL,$

(36)
$νx(Jx,JL)=Qx+dY′dJx.$

Here, let us substitute Eqs. (22)–(23) into Eq. (33) before it is multiplied by $exp(jq′ψx+jm′ϕL)$. By integrating the result over $ψx$ and $ϕL$, we obtain approximately

(37)
$f˜1,q′(Jx)g˜1,m′(JL) ≃je2Nbβs2βx(s)Jx(exp(j(ϕx(s)−Qxθ))δq′,−1−exp(−j(ϕx(s)−Qxθ))δq′,1)exp(j2πμnM)22psπR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×Jm′[((ν+p)ωRFh−QxξxωRFhη)(2JL|η|cpsβsνs0ω0)1/2]∂F0(Jx)∂JxG0(JL),$
under the condition
(38)
$1≫(−2ω0JLν0sβs2Esη)1/2|Qxξx|QxνL(Jx,JL),$
where we use the relation Eq. (28).

Multiplying the factor

(39)
$x(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL)$
by Eq. (37), before integrating the result over $ϕL$, $Jx$, $ψx$, $Jy$, $ψy$, and summing it over $m′$, we derive
(40)
$Dpq′,m(JL) =∑m′∫−ππdϕL∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL) ×je2Nbβs2βx(s)Jx(exp(j(ϕx(s)−Qxθ))δq′,−1−exp( −j(ϕx(s)−Qxθ))δq′,1)exp( j2πμnM )22psπR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×Jm′[((ν+p)ωRFh−QxξxωRFhη)(2JL|η|cpsβsνs0ω0)1/2]∂F0(Jx)∂JxG0(JL).$
Then, by substituting Eqs. (18) and (27) into Eq. (40), Eq. (40) is rewritten as
(41)
$Dpq′,m′(JL) =∑q,m∫0∞dJxje2Nbβs2βx(s)Jx(δq′,−1−δq′,1)exp(j2πμnM)2psR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞∫0∞dJ′LDpq,m(J′L)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×∂F0(Jx)∂JxG0(JL)Jm*[(ν+p−Qxξxη)ωRFh(2J′L|η|cpsβsνs0ω0)1/2] ×Jm′[(ν+p−Qxξxη)ωRFh(2JL|η|cpsβsνs0ω0)1/2] .$

Here, let us introduce the function $Dm$ as

(42)
$Dm(JL)=Dp1,m(JL)+Dp−1,m(JL).$
When we retain only the diagonal terms, Eq. (41) is simplified by using the function $Dm$ as
(43)
$Dm(JL) =−je2NbMβs2βx(s)G0(JL)2psR ×∫0∞dJx[1[ν−mνL(Jx,JL)−νx(Jx,JL)]−1[ν−mνL(Jx,JL)+νx(Jx,JL)]] ×Jx∂F0(Jx)∂Jx∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[(ν+μ+pM−Qxξxη)(2JLω0|η|cpsβsνs0)1/2] ×∫0∞dJ′LDm(J′L)Jm*[(ν+μ+pM−Qxξxη)(2J′Lω0|η|cpsβsνs0)1/2] .$

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

(44)
$Dm(JL)=Bmδ(JL−JL0),$

(45)
$G0(JL)=12πδ(JL−JL0),$

(46)
$F0(Jx)=12πJx0exp(−JxJx0),$
where
(47)
$Jx0=βsEsϵx,rmsc,$
and $ϵx,rms$ is the root mean square (rms) emittance of the beam, and expand Eq. (43) around small $Jx$ before integrating Eq. (43) over $Jx$, we finally obtain the dispersion relation as
(48)
$1≃ −je2NbMπβs2〈βx(s)〉[1+(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0exp(−(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0)Γ[0,−(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0]]8π3Jx0psR(mdνLdJx+dνxdJx) ×∑p=−∞∞|Jm[(ν+μ+pM−Qxξxη)(2ω0JL0|η|cpsβsνs0)1/2]|2ZT(ω0(ℜ[ν]+μ+pM)) ,$
where
(49)
$νL0=νs0+dYcoh,0′(JL)dJL|JL=JL0,$

(50)
$νX0=Qx+〈βx(s)〉psYcoh,2′(JL0),$

(51)
$mdνLdJx+dνxdJx≃mdYcoh,2′(JL)dJL|JL=JL0〈βx(s)〉ps+3〈βx2(s)〉Ycoh,4′(JL0),$
$νL0$ and $νX0$ are the coherent synchrotron and the betatron tunes, respectively, $Γ[0,z]$ is the incomplete $Γ$-function [27], and $〈⋯〉$ denotes the average value around the ring. Here we assume the rms beam sizes $σx$ and $σz$ are given as
(52)
$σx=(〈βx(s)〉ϵx,rms+〈D2(s)〉(Δpp)2)1/2=(〈βx(s)〉cJx0βsEs+〈D2(s)〉2JL0νs0ω0Esβs2|η|)1/2,$

(53)
$σz=cω0(2JL0|η|ω0Esνs0)1/2,$
considering Eqs. (14), (47), (B17), (B18), and (B32). For reference, Fig. 1 illustrates typical behavior of the functions $dY′coh,0(JL0)/dJL0$, $Y′coh,2(JL0)$, $dY′coh,2(JL0)/dJL0$, and $Y′coh,4(JL0)$ in Eq. (48), which are calculated by using the beam parameters at the ramping time 15 ms in the RCS (refer to Sect. 3).
Fig. 1.

Typical behavior of the functions $dY′coh,0(JL0)/dJL0$ (left top), $Y′coh,2(JL0)$ (right top), $dY′coh,2(JL0)/dJL0$ (left bottom), and $Y′coh,4(JL0)$ (right bottom) calculated by using the beam parameters at the ramping time 15 ms in the RCS.

Fig. 1.

Typical behavior of the functions $dY′coh,0(JL0)/dJL0$ (left top), $Y′coh,2(JL0)$ (right top), $dY′coh,2(JL0)/dJL0$ (left bottom), and $Y′coh,4(JL0)$ (right bottom) calculated by using the beam parameters at the ramping time 15 ms in the RCS.

The beam growth rate and the coherent tune affected by the wake force, which are given by the real parts of $jω0ν$ and of $ν$, respectively, are solved by Eq. (48) as a function of nominal tune $Qx$. The differences among $Qx$, $νX0$, and $ℜ[ν]$ are tiny ($≲0.01$) 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 $νL$ and $νx$ 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

(54)
$Dm(JL)=je2NbMβx(s)G0(JL)4πpsR(1ν−mνs0−Qx−1ν−mνs0+Qx) ×∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[(ν+μ+pM−Qxξxη)(2JLω0|η|cpsνs0)1/2] ×∫0∞dJ′LDm(J′L)Jm*[(ν+μ+pM−Qxξxη)(2J′Lω0|η|cpsνs0)1/2],$
for an ultra-relativistic beam ($βs=1$). Here, following the conventional manner, let us replace the action-variable $JL$ with the amplitude-variable $rs$:
(55)
$rs=1ω0(2JLω0|η|cpsνs0)1/2,$
and assume the unperturbed distribution function $G0(JL(rs))$ as
(56)
$G0(JL(rs))=|η|πω0cpsτ0s2νs0Θ(τ0s−rs),$
where $Θ(x)$ is the step function and $τs0$ denotes the half-bunch length. Accordingly, Eq. (54) is rewritten as
(57)
$Dm(JL(rs))=jce2NbM4π2EsQxτ0s2Θ(τ0s−rs)(1ν−mνs0−Qx−1ν−mνs0+Qx) ×∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[ω0(ν+μ+pM−Qxξxη)rs] ×∫0∞dr′sr′sDm(J′L(r′s))Jm*[ω0(ν+μ+pM−Qxξxη)r′s] ,$
where $βx(s)$ is replaced by $R/Qx$.

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

(58)
$Dm(JL(rs))=W˜(rs)∑k=0∞ak(m)fk(m)(rs)≡∑k=0∞ak(m)gm,k(rs),$
where $k$ is the radial mode and $W˜(rs)$ is the weight function. The functions $fk(m)(r)$ and $gm,k(r)$ satisfy the orthogonality relationship
(59)
$∫0∞W˜(rs)fk(m)(rs)fl(m)(rs)rsdrs=δkl,$

(60)
$∫0∞gm,k(rs)gm,l(rs)W˜(rs)rsdrs=δkl,$
respectively. Here, the weight function $W˜(rs)$ is defined as
(61)
$W˜(rs)=CηNbπτ0s2ω0νs0Θ(τ0s−rs),$
where $C$ is a normalization constant. Accordingly, the functions $fk(m)(rs)$ or $gm,l(rs)$ can be revealed as
(62)
$fk(m)(rs)=(2W˜)1/2Jm(μmkrsτ0s)τ0sJm+1(μmk), for rs<τ0s,$

(63)
$gm,l(rs)=(2W˜(rs))1/2Jm(μmlrsτ0s)τ0sJm+1(μml)=(2CηRNbπcνs0τ0s2)1/2Jm(μmlrsτ0s)τ0sJm+1(μml)(1−Θ(rs−τ0s)),$
where $μmk$ is the $k$th zero of $Jm(x)$.

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

(64)
$ρm,l(τ)=−∫−∞∞gm,l(rs)exp(−jmϕL)ωRFdWEs≡∫−∞∞gm,l(rs)exp(−jmϕL)dδ,$
and its Fourier transform
(65)
$ρ˜m,l(k)=∫−∞∞dτ2πexp(jkτ)ρm,l(τ),$
where $τ=ϕcp/ωRF$ and $W$ is the momentum conjugate to $ϕcp$ (see Eq. (B17)).

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

(66)
$ρ˜m,l(k)=∫0∞gm,l(r)ω0νs0ηjmJm(kr)r dr,$
(see Eqs. (55), (B31) and (B32)) by using the relation
(67)
$12π∫02πdφexp(ilφ−ixcosφ)=j−lJl[x],$
while its inverse transform $ρm,l(τ)$ is given by
(68)
$ρm,l(τ)=∫gm,l(r)ω0νs0ηjmJm(kr)exp(−jkτ)r dr dk,$
which is expressed as
(69)
$ρm,l(τ)=(2CRNbνs0πcη)1/2ω0μmljm∫0∞dk[((−1)m+1)coskτ+((−1)m−1)jsinkτ]Jm(kτ0s)(μml2−k2τ0s2),$
for the function given by Eq. (63). Equation (66) satisfies the relationship $ρ˜m,l*(k)=(−1)mρ˜m,l(k)$ for real $gm,l(r)$ (see Eq. (91)).

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

(70)
$νm,l=Qx+mνs0+je2cηCm,l8π2EsQxCω02νs0 ×∑p=−∞∞ZT(ω0(ν+μ+Mp))h′m,l(ω0(Qx+mνs0+μ+Mp−Qxξη))∑p=−∞∞h′m,l(ω0(Qx+mνs0+μ+Mp−Qxξη)) ,$
in the lowest-order approximation, where we define the constant $Cm,l$ and the function $h′m,l(ω)$ as
(71)
$Cm,l=∫−∞∞dτ|ρm,l(τ)|2=2π∫−∞∞dω|ρ˜m,l(ω)|2,$

(72)
$h′m,l(ω)≡|ρ˜m,l(ω)|2,$
respectively. In Eq. (70), the $ω$-integration is approximated by the summation of $p$.

The constants $C$ and $Cml$ are determined as follows. If we impose the condition $ρm,l(±τ0s)=0$ on the distribution function, the function $ρm,l(τ)$ should be written as

(73)
$ρm,l(τ)=∑p=1,3,5,…Dpcosπp2τ0sτ+∑p=2,4,6,…Epsinπp2τ0sτ,$
where $Dp$ and $Ep$ are expansion coefficients. By equating Eq. (69) to Eq. (73), they are expressed as
(74)
$Dp=−pπω0μmlτ0s4(2CRNbνs0πcη)1/2(−1)m/2 (−1)(p+1)/2ℜ[∫−∞∞dkJm(kτ0s)exp(jkτ0s)(k2−μml2τ0s2)(k2−(πp2τ0s)2)],for m=0,2,4,…,$

(75)
$Ep=−pπω0μmlτ0s4(2CRNbνs0πcη)1/2(−1)(m−1)/2 (−1)p/2ℑ[∫−∞∞dkJm(kτ0s)exp(jkτ0s)(k2−μml2τ0s2)(k2−(πp2τ0s)2)],for m=1,3,5,…,$
where we use
(76)
$∫−τ0sτ0scoskτcosπpτ2τ0sdτ=(−1)(p+1)/2pπτ0s(k2−(πp2τ0s)2)coskτ0s, for odd p,$

(77)
$∫−τ0sτ0ssinkτsinπpτ2τ0sdτ=(−1)p/2pπτ0s(k2−(πp2τ0s)2)sinkτ0s, for even p,$

(78)
$Jm(−x)=(−1)mJm(x).$
By picking up the residues, the $k$-integration in Eqs. (15) and (75) is performed. As a result, we obtain
(79)
$Dp=ω0τ0s(πCRNbνs0cη)1/2(−1)m/222μmlJm(πp2)(μml2−(πp2)2), for even m,$

(80)
$Ep=ω0τ0s(πCRNbνs0cη)1/2(−1)(m−1)/222μmlJm(πp2)(μml2−(πp2)2), for odd m.$
Finally, $ρm,l(τ)$ is summarized as
(81)
$ρm,l(τ)=∑pAlpmbp(τ),$

(82)
$bp(τ)={cosπpτ2τ0s,for p=1,3,5,…,sinπpτ2τ0s,for p=2,4,6,…,$

(83)
$Alpm=ω0τ0s(πCRNbνs0cη)1/2Pm,p,l{(−1)m/2,for m=0,2,4,…,(−1)(m−1)/2,for m=1,3,5,…,$

(84)
$Pm,p,l=22μmlJm(πp2)(μml2−π2p24),$
where $p$ runs $1,3,5,…$ for $m=0,2,4,…,$ and $p$ runs $2,4,6,…$ for $m=1,3,5,….$

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

(85)
$8μm12Jm2(π(m+1)2)(μm,12−π2(m+1)24)2,$
which appears in Eq. (70), dominates for the component $m+1=p$. Then, the function $ρm,l=1(τ)$, its Fourier transform $ρ˜m,l=1(ω)$, and the factor $Pm,p=m+1,l=1$ are approximated as
(86)
$ρm,l=1(τ)={(−1)m2cosπ(m+1)τ2τ0s,for m=0,2,4,…,(−1)m−12sinπ(m+1)τ2τ0s,for m=1,3,5,…,$

(87)
$ρ˜m,l=1(ω)={(−1)m/2∫−τ0sτ0sdτ2πexp(jωτ)cosπ(m+1)τ2τ0s=2τ0s(1+m)cosωτ0sπ2[(1+m)2−4ω2τ0s2π2], for m=0,2,4,…,(−1)(m−1)/2∫−τ0sτ0sdτ2πexp(jωτ)sinπ(m+1)τ2τ0s=j(1+m)2τ0ssinωτ0sπ2[(1+m)2−4ω2τ0s2π2], for m=1,3,5,…,$
and
(88)
$Pm,p=m+1,l=1=22μm,1Jm(π(m+1)2)(μm,12−π2(m+1)24)≃16(3+2m)(5+4m)π2m+1∼1m+1,$
respectively. Substituting Eq. (86) into Eq. (71), we finally obtain
(89)
$Cm,l=τ0s.$
The constant $C$ is determined by the condition $Al=1,p=m+1m=1$. As a result, it is calculated as
(90)
$C=ηc(m+1)τ0s2ω02νs0πRNb.$
Accordingly, the function $gm,l(rs)$ is described as
(91)
$gm,l(rs)=2(m+1)|η|Jm(μmlrsτ0s)πω0νs0τ0sJm+1(μml)(1−Θ(rs−τ0s)),$
owing to Eqs. (63) and (90).

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:

(92)
$τm−1=−cIc4πQx(m+1)Es/e∑p=−∞∞ℜ[ZT(ω′p)]F′m(ω′p−ωξ),$
where $τm−1$ is the growth rate,
(93)
$F′m(ω)=h′m(ω)B′f∑p=−∞∞h′m(ω′p−ωξ),$

(94)
$h′m(ω)=(2τ0s)22π4(m+1)2[1+(−1)mcos(ω2τ0s)][(ω2τ0sπ)2−(m+1)2]2,$

(95)
$ω′p=ω0(Qx+mνs0+μ+Mp),$

(96)
$ωξ=ω0Qxξη,$

(97)
$Ic=eMNbT0,$

(98)
$B′f=M2τ0sc2πR.$
In this paper's calculations, the factor $Bf′$ is approximated by the typical bunching factor $Bf$ defined by the average current divided by the peak current (see Eq. (101)).

## 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 $φs$ (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.

Fig. 2.

Typical pattern of the acceleration voltage $Vrf$ (red), and the synchronous phase $φs$ (blue) during the ramping time.

Fig. 2.

Typical pattern of the acceleration voltage $Vrf$ (red), and the synchronous phase $φs$ (blue) during the ramping time.

Table 1.

Typical parameter list

 $T$ (kinetic energy, GeV) 0.4 3 $f0$ (revolution frequency, MHz) 0.61 0.84 $η$ (slippage factor) -0.478 -0.047 $Ic$ (average current, A) 8.1 11.2 $νs0$ (synchrotron tune) 0.0053 0.0005 $〈βx(s)〉$ (m) 11.6 $〈βx2(s)〉$ (m2) 172.3 $〈D2(s)〉$ (m)2 3.46 $ϵx,rms$ (mmrad) $100/6βsγs$ $JL0$ (eV $⋅$ s) $0.1645$
 $T$ (kinetic energy, GeV) 0.4 3 $f0$ (revolution frequency, MHz) 0.61 0.84 $η$ (slippage factor) -0.478 -0.047 $Ic$ (average current, A) 8.1 11.2 $νs0$ (synchrotron tune) 0.0053 0.0005 $〈βx(s)〉$ (m) 11.6 $〈βx2(s)〉$ (m2) 172.3 $〈D2(s)〉$ (m)2 3.46 $ϵx,rms$ (mmrad) $100/6βsγs$ $JL0$ (eV $⋅$ s) $0.1645$

(Circumference $C=348.333 m$, harmonic number $h=2$, repetition rate = 25 Hz, particles per bunch $Nb=4.15×1013$, and the average chamber radius $a=145 mm$).

Here, $βx(s)$ and $D(s)$ are the $β$-function and the dispersion function, respectively; $ϵx,rms$ and $JL0$ are the root mean square (rms) horizontal and the longitudinal emittances, respectively; $〈⋯〉$ denotes the average value around the ring; and $βs$ and $γs$ are the Lorentz-$β$ and Lorentz-$γ$ on the designed particle.

Eight kickers are installed in the RCS. The real and the imaginary parts of the horizontal impedance $ZT(ω)$ 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 $βs=0.7$ and $βs=1$, respectively. The impedance is roughly proportional to the Lorentz-$β$ [15]. The corresponding wake function $WT(ω0t)$ 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.

Fig. 3.

Dependence of the horizontal kicker impedance $ZT(ω)$ (left/middle) and of the wake function $WT(ω0t)$ (right) on the Lorentz-$β$. The red and blue lines show the results at $βs=0.7$ and $βs=1$, respectively. The wave propagation speeds in the kicker magnet and in the power cable are about $0.02×c$ and $0.57×c$, respectively. The magnet length and the cable length are 705 mm and 130 m, respectively.

Fig. 3.

Dependence of the horizontal kicker impedance $ZT(ω)$ (left/middle) and of the wake function $WT(ω0t)$ (right) on the Lorentz-$β$. The red and blue lines show the results at $βs=0.7$ and $βs=1$, respectively. The wave propagation speeds in the kicker magnet and in the power cable are about $0.02×c$ and $0.57×c$, respectively. The magnet length and the cable length are 705 mm and 130 m, respectively.

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 $ξQx$ is activated by a DC-power supply at the injection energy. In this case the chromaticity approaches the natural chromaticity $(ξQx=−10.3)$ [30] as the beam energy is increased, as shown in Fig. 4.

Fig. 4.

Calculated chromaticity $ξQx$ change during the ramping time.

Fig. 4.

Calculated chromaticity $ξQx$ change during the ramping time.

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 ($3.10×1013$ 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.

Fig. 5.

Measured horizontal beam positions for the case of $3.10×1013$ particles per bunch and $Qx=6.45$. The blue line shows the results where two buckets are perfectly filled with two bunches, and the green line shows the results where only one bucket among the two is filled with one bunch. The momentum spread of the injection beam from LINAC is 0.18%.

Fig. 5.

Measured horizontal beam positions for the case of $3.10×1013$ particles per bunch and $Qx=6.45$. The blue line shows the results where two buckets are perfectly filled with two bunches, and the green line shows the results where only one bucket among the two is filled with one bunch. The momentum spread of the injection beam from LINAC is 0.18%.

Figure 6 shows the measured results for a 1 MW equivalent beam ($4.15×1013$ 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.

Fig. 6.

Measured horizontal beam positions for the case of $4.15×1013$ particles per bunch and $Qx=6.45$. The momentum spread of the injection beam is 0.18%.

Fig. 6.

Measured horizontal beam positions for the case of $4.15×1013$ particles per bunch and $Qx=6.45$. The momentum spread of the injection beam is 0.18%.

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 $μ$ 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).

Fig. 7.

The maximum beam growth rate among with different modes $(m,μ)$ estimated by using the Sacherer formula, Eq. (92), for $Qx=6.45$.

Fig. 7.

The maximum beam growth rate among with different modes $(m,μ)$ estimated by using the Sacherer formula, Eq. (92), for $Qx=6.45$.

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

(99)
$ν≃mνL0+νX0+je2NbMπcβs〈βx(s)〉8π3EsR ×∑p=−∞∞Fm(JL0,mνL0+νX0+μ+pM−Qxξxη)ZT(ω0(mνL0+νX0+μ+pM)),$
where
(100)
$Fm(JL0,x)=|Jm[x(2ω0JL0|η|cpsβsνs0)1/2]|2.$

Figure 8 shows the maximum beam growth rate among different modes $(m,μ)$ 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.

Fig. 8.

The maximum beam growth rate among different modes $(m,μ)$ estimated by Eq. (99) for $Qx=6.45$.

Fig. 8.

The maximum beam growth rate among different modes $(m,μ)$ estimated by Eq. (99) for $Qx=6.45$.

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 $t=13 ms$ 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.

Fig. 9.

The maximum beam growth rate among different modes $(m,μ)$ estimated by solving Eq. (48) for $Qx=6.45$.

Fig. 9.

The maximum beam growth rate among different modes $(m,μ)$ estimated by solving Eq. (48) for $Qx=6.45$.

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 $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ 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 $0π$ (center injection), $100π$, $150π$, and $200π$ 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.

Fig. 10.

Theoretical results of the transverse beam emittance $ϵx,rms$ dependence of the beam growth rate at 15 ms for $Qx=6.45$. The red, black, and purple lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ modes, respectively.

Fig. 10.

Theoretical results of the transverse beam emittance $ϵx,rms$ dependence of the beam growth rate at 15 ms for $Qx=6.45$. The red, black, and purple lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ modes, respectively.

Fig. 11.

Measurement results ($Nb=4.15×1013$) of the horizontal beam positions for different transverse painting areas, where the chromaticity was fully corrected only at the injection energy. The red, blue, black, and yellow lines show the results for $0π$ (center injection), $100π$, $150π$, and $200π$ mmrad injection painting schemes, respectively, where the tune $Qx$ changes during the ramping time following the black line in the right panel of Fig. 14. The momentum spread of the injection beam from LINAC is 0.18%.

Fig. 11.

Measurement results ($Nb=4.15×1013$) of the horizontal beam positions for different transverse painting areas, where the chromaticity was fully corrected only at the injection energy. The red, blue, black, and yellow lines show the results for $0π$ (center injection), $100π$, $150π$, and $200π$ mmrad injection painting schemes, respectively, where the tune $Qx$ changes during the ramping time following the black line in the right panel of Fig. 14. The momentum spread of the injection beam from LINAC is 0.18%.

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 $(m,μ)$. Figure 12 shows the theoretical results of the beam growth rate at 15 ms, where the bunching factor $Bf$ is evaluated by using

(101)
$Bf=43(π−ϕe−φs)(2JL,0h2|η|ω0Esβs2νs0)1/2,$
where $ϕe$ is the solution of
(102)
$cosϕe+ϕesinφs+cosφs−(π−φs)sinφs=0,$
which satisfies the condition $−π<ϕe<0$ [31]. The left and the middle panels of Fig. 12 illustrate the beam growth rates with space charge effects for the chamber radii $a=145 mm$ and $a=160 mm$, respectively. The right panel illustrates the beam growth rate without space charge effects calculated by using Eq. (99). The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively (the other modes do not excite the beam instabilities).
Fig. 12.

Theoretical results of the beam growth rate (at 15 ms for $Qx=6.45$) with (left/middle) and without (right) space charge effects, dependence on the bunching factor. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively. The left and the middle panels show the results for the chamber radii $a=145 mm$ and $a=160 mm$, respectively.

Fig. 12.

Theoretical results of the beam growth rate (at 15 ms for $Qx=6.45$) with (left/middle) and without (right) space charge effects, dependence on the bunching factor. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively. The left and the middle panels show the results for the chamber radii $a=145 mm$ and $a=160 mm$, respectively.

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 ($m,μ$) 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 ($m,μ$) 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 $a=145 mm$ (e.g., around the area $A$). Comparing both the results for $a=145 mm$ (left) and for $a=160 mm$ (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 $a=145 mm$ (left) disappears in the results for $a=160 mm$ (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 $3.10×1013$. 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.

Fig. 13.

Beam growth rate (for $3.10×1013$ particles per bunch) dependence on the bunching factor, where the tune is fixed to 6.45. The chromaticity was fully corrected only at the injection energy. The left panel shows the measured beam positions for two different bunching factors. The right panel shows the measured bunching factor. The two lines with the same color in both figures denote identical situations.

Fig. 13.

Beam growth rate (for $3.10×1013$ particles per bunch) dependence on the bunching factor, where the tune is fixed to 6.45. The chromaticity was fully corrected only at the injection energy. The left panel shows the measured beam positions for two different bunching factors. The right panel shows the measured bunching factor. The two lines with the same color in both figures denote identical situations.

### 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 $μ=1$ and $μ=0$ 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).

Fig. 14.

The left panel shows the measured ($4.15×1013$ particles per bunch) beam positions for five different tune tracking patterns, where the chromaticity was fully corrected only at the injection energy. The right panel shows the measured tune tracking patterns under the condition that the space charge effect is negligible. The lines of matching color in each panel denote identical situations. The momentum spread of the injection beam is 0.08%.

Fig. 14.

The left panel shows the measured ($4.15×1013$ particles per bunch) beam positions for five different tune tracking patterns, where the chromaticity was fully corrected only at the injection energy. The right panel shows the measured tune tracking patterns under the condition that the space charge effect is negligible. The lines of matching color in each panel denote identical situations. The momentum spread of the injection beam is 0.08%.

Fig. 15.

Dependence of the theoretical results of the beam growth rate at 15 ms on the tune $Qx$. 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 $μ=1$ and $μ=0$ modes, respectively.

Fig. 15.

Dependence of the theoretical results of the beam growth rate at 15 ms on the tune $Qx$. 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 $μ=1$ and $μ=0$ modes, respectively.

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 $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ 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.

Fig. 16.

Theoretical results of the beam growth rate at 15 ms dependence on the chromaticity $ξQx$ for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively.

Fig. 16.

Theoretical results of the beam growth rate at 15 ms dependence on the chromaticity $ξQx$ for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively.

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 $−7.46$ for the red line, $−8.92$ for the blue line, and $−9.64$ for the black line. As expected, the beam is drastically suppressed by an increase in chromaticity in the negative direction.

Fig. 17.

Measured beam positions ($Nb=4.15×1013$, $dp/p=0.08%$) with different chromaticity, where the tune changes, following the red line in the right panel of Fig. 14. The red, blue, and black lines show the results for which, at the injection energy only, the chromaticity was fully, half, and quarter corrected, respectively.

Fig. 17.

Measured beam positions ($Nb=4.15×1013$, $dp/p=0.08%$) with different chromaticity, where the tune changes, following the red line in the right panel of Fig. 14. The red, blue, and black lines show the results for which, at the injection energy only, the chromaticity was fully, half, and quarter corrected, respectively.

## 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 ($4.15×1013$ 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 [3234]. 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-$γ$ 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 $(m,μ)$ 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 $(m,μ)$ 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 $(m,μ)$ 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 ($m,μ$) 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 $(ct¯,x,y,z¯)$ is described by

(A1)
$∂2Φ¯∂x2+∂2Φ¯∂y2+∂2Φ¯∂z¯2=−cZ0ρ¯p(x,y,z¯),$
with
(A2)
$ρ¯p(x,y,z¯)=eNbρ^(z¯)exp(−(ρcosφ−r0cosθ0)2+(ρsinφ−r0sinθ0)22σx2)2πσx2,$

(A3)
$ρ^(z¯)=exp(−z¯22σ¯z2)2πσ¯z,$

(A4)
$σ¯z=γsσz,$
where $γs$ is the Lorentz-$γ$ of the reference particle, $c$ is light velocity, $Z0$ is the impedance of free space, $σx$ is the rms transverse beam size, and $Nb$ is the number of particle per bunch. Polar coordinates are introduced as
(A5)
$x=ρcosφ,$

(A6)
$y=ρsinφ,$
and the center of the bunch on the horizontal plane is given by $(r0cosθ0,r0sinθ0)$. From now on, the condition $σx≪a$ is assumed.

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

(A7)
$G(r→,r→′)=∑m=0∞ϵm2π2cosm(φ−φ′) ×{∫0∞dλ[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλ(z¯−z¯′),for ρ′>ρ,∫0∞dλ[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλ(z¯−z¯′),for ρ′>ρ,$
where $Im(z)$ and $Km(z)$ are the modified Bessel functions, $r→=(ρ,φ,z¯),r→′=(ρ′,φ′,z¯′)$, $ϵm=2−δm0$ and $δmn$ is the Kronecker-$δ$. By using the Green function, the solution $Φ¯$ is approximated as
(A8)
$Φ¯(ρ,φ,z¯)≃∫0∞dλ∫−∞∞dz¯′∫ρaρ′dρ′∫02πdφ′∑m=0∞cZ0ϵm2π2cosm(φ−φ′) ×[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλ(z¯−z¯′) ×eNbexp(−z¯′22σ¯z2)2πσ¯zexp(−(ρ′cosφ′−r0cosθ0)2+(ρ′sinφ′−r0sinθ0)22σx2)2πσx2 +∫0∞dλ∫−∞∞dz¯′∫0ρρ′dρ′∫02πdφ′∑m=0∞cZ0ϵm2π2cosm(φ−φ′) ×[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλ(z¯−z¯′) ×eNbexp(−z¯′22σ¯z2)2πσ¯zexp(−(ρ′cosφ′−r0cosθ0)2+(ρ′sinφ′−r0sinθ0)22σx2)2πσx2.$

By using the formulae

(A9)
$∫02πdφ′cosm(φ−φ′)exp(ρ′r0cos(φ′−θ0)σx2)=2πIm(ρ′r0σx2)cos(m(φ−θ0)),$

(A10)
$12πσ¯z∫−∞∞dz¯′exp(−z¯′22σ¯z2)cosλ(z¯−z¯′)=exp(−λ2σ¯z22)cosλz¯,$
the integrations in the azimuthal and longitudinal directions in Eq. (A8) are performed, so that we get
(A11)
$Φ¯(ρ,φ,z¯)=∫ρaρ′dρ′∑m=0∞cZ0ϵm2π2eNbexp(−ρ′2+r022σx2)σx2Im(ρ′r0σx2)cos(m(φ−θ0)) ×∫0∞dλ[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλz¯exp(−λ2σ¯z22) +∫0ρρ′dρ′∑m=0∞cZ0ϵm2π2eNbexp(−ρ′2+r022σx2)σx2Im(ρ′r0σx2)cos(m(φ−θ0)) ×∫0∞dλ[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλz¯exp(−λ2σ¯z22).$

The potential $Φ¯c$ felt at the bunch center is calculated by plugging in $ρ=r0$ and $φ=θ0$ [26]. Figure 18 illustrates typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.

Fig. 18.

Typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.

Fig. 18.

Typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.

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

(A12)
$Φ¯c(x,y,z¯)≃Φ¯coh,0(z¯)+Φ¯coh,2(z¯)(x2+y2)+Φ¯coh,4(z¯)(x2+y2)2,$
where
(A13)
$Φ¯coh,0(z¯)=cZ0eNb2π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλexp(−λ2σ¯z22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯,$

(A14)
$Φ¯coh,2(z¯)=cZ0eNb2π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ24+ρ′24σx4−12σx2)exp(−λ2σ¯z22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯ +cZ0eNb4σx4π2∫0adρ′ρ′2exp(−ρ′22σx2)∫0∞dλλexp(−λ2σ¯z22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]cosλz¯−cZ0eNb8π2σx2σ¯z(π2)1/2exp(−z¯22σ¯z2),$

(A15)
$Φ¯coh,4(z¯)=cZ0eNb16π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ48+ρ′2λ22σx4+ρ′48σx8−λ2σx2−ρ′2σx6+1σx4) ×exp(−λ2σ¯z22)[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯ +cZ0eNb8π2σx4∫0adρ′exp(−ρ′22σx2)∫0∞dλ(ρ′4λ4σx4+ρ′2λ34−ρ′2λσx2)exp(−λ2σ¯z22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]cosλz¯ +cZ0eNb64π2σx6∫0adρ′exp(−ρ′22σx2)ρ′3∫0∞dλλ2exp(−λ2σ¯z22) ×[K2(λρ′)−K2(λa)I2(λa)I2(λρ′)]cosλz¯ +cZ0eNbπ2σx2(−(σ¯z2−z¯2)128σ¯z5+164σx2σ¯z)(π2)1/2exp(−z¯22σ¯z2).$

The terms $Φ¯coh,2(z¯)$ and $Φ¯coh,4(z¯)$ contribute to the coherent space charge tune shift, and to the nonlinear motion of the beam, respectively.

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

(A16)
$Φ(x,y,z−βsct)=γsΦ¯(x,y,γs(z−βsct)),$

(A17)
$Az(x,y,z−βsct)=βscγsΦ¯(x,y,γs(z−βsct)),$
respectively, where $βs$ is the Lorentz-$β$ of the reference particle.

#### 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]

(B1)
$Ho=−ps(1+xρ)ΔEpsβsc+ps2γs2(ΔEpsβsc)2+px2+py22ps+ps2Kx(s)(1−ΔEEs)x2 +ps2Ky(s)(1−ΔEEs)y2−psxEsFx+eΦc(x,y,s−cβst)βsγs2c −eVrfωRFδp(s)cos(ωRFt−hsR+φs)+…,$
where $φs$ is the synchronous phase; $ps$ is the constant momentum on the synchronous particle; $Es=cps/βs$ is the particle energy on the designed orbit; $βs$ and $γs$ are the Lorentz-$β$ and $γ$, respectively; $ΔE$ is given by $ΔE=E−Es$; $Fx$ is the horizontal wake force; $δp(s)$ is the periodic $δ$-function; $c$ is the velocity of light; $Kx$ and $Ky$ are the periodic focusing forces in the horizontal and vertical directions, respectively; $Φc$ is the space charge potential; $h$ is harmonic number; $Vrf$ is the amplitude of the radio frequency (RF) voltage; $1/ρ$ is the local curvature around the machine; $R$ is the average radius of the machine; and $ωRF$ is the angular frequency of the RF voltage, which is expressed as
(B2)
$ωRF=cβshR.$

The orbit length $s$ is used as an independent variable. The canonical variables are $(x,px),(y,py)$, and $(t,−E)$ 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 $F1$,

(B3)
$F1(x,p¯x,y,p¯y,t,−ΔE¯)=(x−ΔE¯psβscD)p¯x+yp¯y−t(ΔE¯+Es) +ΔE¯βscdDdsx−ps2(ΔE¯psβsc)2DdDds,$
we make a canonical transformation from the variables $(x,px),(y,py)$, and $(t,−E)$ to $(x¯,p¯x),(y¯,p¯y)$, and $(t¯,−ΔE¯)$, respectively, according to
(B4)
$px=∂F1∂x=p¯x+ΔE¯βscdDds,$

(B5)
$x¯=∂F1∂p¯x=x−ΔE¯psβscD,$

(B6)
$py=p¯y,$

(B7)
$y¯=y,$

(B8)
$−E=∂F1∂t=−ΔE¯−Es,$

(B9)
$t¯=−∂F1∂ΔE¯=Dp¯xpsβsc+t−1βscdDdsx¯,$
where the dispersion function $D(s)$ in the horizontal direction satisfies the relation
(B10)
$d2Dds2+KxD=1ρ.$

The new Hamiltonian $H1$ is obtained as

(B11)
$H1≃p¯x2+p¯y22ps+ps2(1−ΔE¯Es)(Kxx¯2+Kyy¯2)−psEs(x¯+ΔE¯psβscD)Fx +eΦc[x¯+ΔE¯psβscD,y¯,s−cβs(t¯−Dp¯xpsβsc+1βscdDdsx¯)]βsγs2c −ΔE¯βsc+ps2γs2(ΔE¯psβsc)2−ps2(ΔE¯psβsc)2Dρ−KxDpsx¯ΔE¯2c2−12KxD2βsΔE¯3ps2c3 −eVrfωRFδp(s)cos(ωRFt¯−hsR+φs) .$
In the above derivation, the assumption is made that $D(s=0)=dD(s=0)/ds=0$ at the RF cavity.

Next, using the generating function $F2$,

(B12)
$F2(x¯,px,y¯,py,W,t¯)=x¯p˜x+y¯p˜y+W(ωRFt¯−hsR) ,$
we make a canonical transform from $(x¯,p¯x),(y¯,p¯y),(t¯,−ΔE¯)$ to $(x˜,p˜x),(y˜,p˜y),(ϕcp,W)$, respectively, using
(B13)
$p¯x=∂F2∂x¯=p˜x,$

(B14)
$x˜=∂F2∂p˜x=x¯,$

(B15)
$p¯y=p˜y,$

(B16)
$y˜=y¯,$

(B17)
$−ΔE¯=∂F2∂t¯=WωRF,$

(B18)
$ϕcp=∂F2∂W=ωRFt¯−hsR.$

Then, the new Hamiltonian $H2$ is described as

(B19)
$H2≃p˜x2+p˜y22ps+ps2(1+WωRFEs)(Kxx˜2+Kyy˜2) +eΦc(x˜−WhpsRD,y˜,−cβsϕcpωRF+Dp˜xps−dDdsx˜)βsγs2c −psEs(x˜−ωRFWpsβscD)Fx−KxDpsx˜W2ωRF2c2+12KxD2βsW3ωRF3ps2c3 +(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs),$
where Eq. (B2) is used.

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

(B20)
$H3,L≡−ηh22R2psW2+eVrfcosφs4πRωRFϕcp2=−ηh22R2psW2−Esc2βs42ηω0RωRF2(νs0R)2ϕcp2,$
where the synchrotron tune $νs0$ is given by
(B21)
$νs0=1βs(−ηheVrfcosφs2πEs)1/2.$
Accordingly, Eq. (B19) is rewritten as
(B22)
$H2≃H3,L+p˜x2+p˜y22ps+ps2(1+WωRFEs)(Kxx˜2+Kyy˜2) +eΦc(x˜−WhpsRD,y˜,−cβsϕcpωRF+Dp˜xps−dDdsx˜)βsγs2c −psEs(x˜−ωRFWpsβscD)Fx−KxDpsx˜W2ωRF2c2+12KxD2βsW3ωRF3ps2c3 +(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs) +ηh22R2psW2−eVrfcosφs4πRωRFϕcp2.$

Before describing the Hamiltonian in terms of action-angle variables, let us continue to make the canonical transformations from $(x˜,p˜x),(y˜,p˜y)$ to $(x¯¯,p¯¯x),(y¯¯,p¯¯y)$, respectively, which are generated by the function $F3$:

(B23)
$F3(x¯¯,p˜x,y¯¯,p˜y)=−x¯¯p˜xps−y¯¯p˜yps.$
The canonical transformations are expressed as
(B24)
$p¯¯x=p˜xps, x˜=x¯¯ps,$

(B25)
$p¯¯y=p˜yps, y˜=y¯¯ps.$
The new Hamiltonian $H3$ is divided as
(B26)
$H3=H3,L+H3,T+ΔH3,T+ΔH3,L,$
where $H3,L$ is given by Eq. (B20), and
(B27)
$H3,T≡p¯¯x2+p¯¯y22+12(Kxx¯¯2+Kyy¯¯2)−psEs(x¯¯ps−ωRFWpsβscD)Fx,$

(B28)
$ΔH3,T=WωRF2Es(Kxx¯¯2+Kyy¯¯2)+eΦc(x¯¯ps−WhpsRD,y¯¯ps,−cβsϕcpωRF+Dp¯¯xps−dDdsx¯¯ps)βsγs2c −KxDpsx¯¯psW2ωRF2c2+12KxD2βsW3ωRF3ps2c3,$

(B29)
$ΔH3,L=(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs)+ηh22R2psW2−eVrfcosφs4πRωRFϕcp2.$

For the longitudinal motion, let us consider the generating function

(B30)
$F(ϕcp,ϕL,s)=−cpsβsνs02h2|η|ω0ϕcp2tan(ϕL−π2) ,$
which gives
(B31)
$ϕcp=(2JLh2|η|ω0cpsβsνs0)1/2sinϕL,$

(B32)
$W=(2JLνs0cpsβs|η|h2ω0)1/2cosϕL.$
Consequently, the Hamiltonian is written as
(B33)
$H3,L+ΔH3,L=−|η|ηνs0RJL+|η|ηνs0RJLsin2ϕL −eVrfωRFδp(s)cos((2JLh2|η|ω0cpsβsνs0)1/2sinϕL+φs) .$

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

(B34)
$F4(x¯¯,ψx,y¯¯,ψy,s)=−x¯¯22βx(s)[tan(ψx+ϕx(s)−QxRs)+αx(s)] −y¯¯22βy(s)[tan(ψy+ϕy(s)−QyRs)+αy(s)],$

(B35)
$ϕx(s)=∫sdsβx(s),$

(B36)
$ϕy(s)=∫sdsβy(s).$
We obtain the canonical transformations from $(x¯¯,p¯¯x),(y¯¯,p¯¯y)$ to $(Jx,ψx),(Jy,ψy)$, respectively, as
(B37)
$p¯¯x=∂F4∂x¯¯=−x¯¯βx(s)[tan(ψx+ϕx(s)−QxRs)+αx(s)] ,$

(B38)
$Jx=−∂F4∂ψx=x¯¯22βx(s)cos2(ψx+ϕx(s)−QxRs),$

(B39)
$p¯¯y=−y¯¯βy(s)[tan(ψy+ϕy(s)−QyRs)+αy(s)] ,$

(B40)
$Jy=y¯¯22βy(s)cos2(ψy+ϕy(s)−QyRs),$
where the Twiss parameters satisfy
(B41)
$d2ds2βi+Kiβi−1(βi)3=0,$

(B42)
$αi=−12dβids,$

(B43)
$βiγs,i−αi2=1,$
and $i$ denotes $x$ or $y$.

Thus, the new Hamiltonian $H4$ is expressed as

(B44)
$H4≃H4,0+ΔH4,$
where
(B45)
$H4,0=QxJxR+QyJyR−η|η|νs0JLR −2βsc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$

(B46)
$ΔH4=Kxβx(s)Jx(2JLνs0βs2ω0|η|Es)1/2cosϕLcos2(ψx+ϕx(s)−QxRs) +Kyβy(s)Jy(2JLνs0βs2ω0|η|Es)1/2cosϕLcos2(ψy+ϕy(s)−QyRs)+eΦc(X,Y,Z)βsγs2c −KxD(2βx(s)JxcβsEs)1/22JLω0νs0βs|η|cos(ψx+ϕx(s)−QxRs)cos2ϕL +Kx2D2ω03Esc(JLνs0|η|ω0)3/2cos3ϕL +|η|ηνs0RJLsin2ϕL−eVrfωRFδp(s)cos((2JLh2|η|ω0cpsβsνs0)1/2sinϕL+φs),$

(B47)
$X=(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−QxRs)−hpsRD(2JLνs0cpsβs|η|h2ω0)1/2cosϕL,$

(B48)
$Y=(2βy(s)Jyps)1/2cos(ψy+ϕy(s)−QyRs) ,$

(B49)
$Z=cβsωRF(2JLh2|η|ω0cpsβsνs0)1/2sinϕL +Dps(2Jxβx(s))1/2[αx(s)cos(ψx+ϕx(s)−QxRs)+sin(ψx+ϕx(s)−QxRs)] +dDds(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−QxRs) .$

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 $H$:

(B50)
$H≃QxJx+QyJy+JLνs0+Ux+Y′,$
with its independent variable $θ=s/R$, where
(B51)
$Ux=−R2βsc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$

(B52)
$Y′=eR8π3βsγs2c∫02πdψx∫02πdψy∫02πdϕL ×Φc[(2βx(s)Jxps)1/2cosψx−D(s)R(2JLνs0cβs|η|psω0)1/2cosϕL,(2βy(s)Jyps)1/2cosψy, ch(2JL|η|ω0Esνs0)1/2sinϕL+D(s)ps(2Jxβx(s))1/2(αx(s)cosψx+sinψx) +dDds(2βx(s)Jxps)1/2cosψx] ,$
$Ux$ and $Y′$ are the effect of the horizontal wake and the space charge forces, respectively.

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

(B53)
$(2βx(s)Jxps)1/2≪|D|R(2JLνs0cβs|η|psω0)1/2,$

(B54)
$|Dps(2Jxβx(s))1/2(αx(s)+1)+dDds(2βx(s)Jxps)1/2|≪ch(2JL|η|ω0Esνs0)1/2.$
In this case, the function $Y′$ is approximated as
(B55)
$Y′=Ycoh,0′(JL)+Y′coh,2(JL)(βx(s)Jxps+βy(s)Jyps) +Y′coh,4(JL)(3βx2(s)Jx22+3βy2(s)Jy22+2βx(s)βy(s)JxJy) ,$
where
(B56)
$Ycoh,0′(JL)=eRZ0eNb2π2βsγsσx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλexp(−λ2γs2σz22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] ,$

(B57)
$Y′coh,2(JL)=eRZ0eNb2π2σx2βsγs∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ24+ρ′24σx4−12σx2) ×exp(−λ2γs2σz22)[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] +eRZ0eNb4σx4π2βsγs∫0adρ′ρ′2exp(−ρ′22σx2)∫0∞dλλexp(−λ2γs2σz22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] −eRZ0eNb8π2βsγs2σx2σz(π2)1/2exp(−c2JL|η|2σz2h2ω0Esνs0)I0(c2JL|η|2σz2h2ω0Esνs0),$

(B58)
$Y′coh,4(JL)=eRZ0eNb16π2σx2βsγsps2∫0adρ′ρ′exp(−ρ′22σx2)J0[γsλch(2JL|η|ω0Esνs0)1/2] ×∫0∞dλ(λ48+ρ′2λ22σx4+ρ′48σx8−λ2σx2−ρ′2σx6+1σx4)exp(−λ2γs2σz22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)] +eRZ0eNb8π2σx4βsγsps2∫0adρ′exp(−ρ′22σx2)J0[γsλch(2JL|η|ω0Esνs0)1/2] ×∫0∞dλ(ρ′4λ4σx4+ρ′2λ34−ρ′2λσx2)exp(−λ2γs2σz22)[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)] +eRZ0eNb64π2σx6βsγsps2∫0adρ′exp(−ρ′22σx2)ρ′3∫0∞dλλ2exp(−λ2γs2σz22) ×[K2(λρ′)−K2(λa)I2(λa)I2(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] +eRZ0eNbβsγs2π2σx2ps2(π2)1/2exp(−c2JL|η|2h2ω0Esνs0σz2) ×[(−1128γs2σz3+164σx2σz+c2JL|η|128γs2σz5h2ω0Esνs0) ×I0(c2JL|η|2h2ω0Esνs0σz2)−c2JL|η|I1(c2JL|η|2h2ω0Esνs0σz2)128γs2σz5h2ω0Esνs0].$
Finally, they are simplified to Eqs. (7), (8), and (9) as in the text, under a typical parameter region.

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1After 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.
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