Proceedings of the 12th Annual Meeting of Particle Accelerator Society of Japan
August 5-7, 2015, Tsuruga, Japan

PASJ2015 WEOL06

２粒子モデルを使ったビーム不安定性に対する空間電荷力の効果の研究
A TWO PARTICLE MODEL FOR STUDY OF EFFECTS OF SPACE-CHARGE FORCEON BEAM INSTABILITIES
陳 栄浩#, A)，Alexander Wu Chao B)，Michael M. BlaskiwiczC)
Yong Ho Chin #, A), Alexander Wu Chao B), Michael M. BlaskiwiczC)
A)
High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki, JapanB)
Stanford Linear Accelerator Center, Menlo Park, CA, USA
C)
Brookhaven National Laboratory, Upton, NY, USA

Abstract

In this report, we present a new two particle model for study of beam instabilities in the presence of the space-charge
force. It is a simple expansion of the well-known two particle model for a strong head-tail instability and is still
analytically solvable. It leads to a formula for the growth rate as a function of the two dimensionless parameters: thespace charge tune shift (normalized by the synchrotron tune) and the wake field strength, Υ. The 3-dimensional contour
plot of the growth rate as a function of those two dimensionless parameters reveals stopband structures. Many
simulation results generally indicate that beam instability can be damped by a weak space-charge force, but the beam
becomes unstable again when the space charge force is further increased. The new two particle model indicates asimilar behavior. The damping of beam instabilities in weak space-charge regions and its loss in strong space-charge
regions are explained by a combination of linearized coherent kicks between the space-charge force and the wake fields,
not by an incoherent tune spread due to the non-linearity of the space-charge force.

1. INTRODUCTION
In low energy high-intensity hadron machines, thespace-charge tune shift is an important parameter in thedesign and operation of the machines. The space-charge
force is also believed to affect the behavior of beam
instabilities. Many theoretical and simulation studies havebeen made for a better understanding of their interplay [17]. They generally indicate that beam instability can bedamped when the space-charge force is weak, but thebeam becomes unstable again when it becomes too strong.
The mechanism of this loss of the damping effect due to astrong space-charge force has not been well understood sofar. If the damping of beam instabilities is caused by the
betatron tune spread (Landau damping) due to the nonlinearity of the space-charge force, one may naively thinkthat a stronger space-charge force will be more effectivein damping of beam instabilities. However, many
simulation results show the contrary. This inversion may
suggest that the damping phenomenon of beam
instabilities in a weak space-charge region may comefrom a different mechanism.
The two particle model has been applied to illustrate
the mechanism of the strong head-tail instability (or thetransverse mode-coupling instability [8]) in a very simple
but insightful way [Reference [9]: Chao, page 179]. This
two particle model can provide a superb framework forstudy of the space-charge force on beam instabilities justby adding new space-charge terms on top of the existing
wake potential ones. The crucial points in this new model
are that the resulting equations of motion need to beanalytically solvable and the final form of solutions
should be a continuous expansion from the one
dimensional (the wake strength only) case to twodimensional (the wake strength and the space-charge
strength) case.
We briefly summarize Chao’s no space-charge modelin Section II to review the premise and the solution
techniques of the original two particle model, and derivesome useful formulae for later use. In Section III and IV,
we show solutions and stability diagrams for the weak
and strong space-charge cases, respectively. The
procedure to identify unstable regions and to compute thegrowth rate is summarized in Section V. Contour plotsof the growth rate are presented in flat and 3-dimensional
ways. The paper is concluded with its findings in Section
VI.

2. NO SPACE-CHARGE CASE
Let us first review the premise and treatment of Chao’s
original two particle model by closely following his text
book. We assume that a beam is made of two macroparticles, each with charge of Ne/2 and each executing
synchrotron and betatron oscillations. We assume that
their synchrotron oscillations have equal amplitude, but
opposite phases. As for the betatron oscillations, we makeno such assumption. In what follows, we use s , thedistance along the circumference, as an independent
variable of motion. During the first half of thesynchrotron oscillation period, T𝑠𝑠 = 2π/ω𝑠𝑠 , the particle
2 leads the particle 1 on the synchrotron phase space, andonly the trailing particle (the particle 1 in the present# yongho.chin@kek.jp- 43 -

Proceedings of the 12th Annual Meeting of Particle Accelerator Society of Japan
August 5-7, 2015, Tsuruga, Japan

PASJ2015 WEOL06

case) receives transverse kicks from wake fields createdby the leading particle (the particle 2 in the present case),that is a function of the transverse displacement of theleading particle. For simplicity, we assume that the wakepotential is a constant, 𝑊𝑊0 , independent of the distance
between the two particles. The property of the wakepotential requires that 𝑊𝑊0 ≥ 0. The equations of motion
for the two particles arey1′′ + �y2′′ + �𝜔𝜔𝛽𝛽 2

� 𝑦𝑦1 =

𝑐𝑐𝜔𝜔𝛽𝛽 2𝑐𝑐

𝑁𝑁𝑟𝑟0 𝑊𝑊02𝛾𝛾𝛾𝛾

� 𝑦𝑦2 = 0,

𝑦𝑦2 ,

(1)
(2)

where y ′ = 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑, ωβ is the betatron angular frequency,c is the speed of light, γ is the Lorentz factor, C is thecircumference of the machine, and r0 is the classicalradius of the particle. Similarly during the second half
period of the synchrotron oscillation, we have the same
equations with indices 1 and 2 exchanged.
The solution for 𝑦𝑦2 is simply a free betatronoscillation:y� 2 (𝑠𝑠) = y� 2 (0)𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑠𝑠/𝑐𝑐 ,

(3)

𝑐𝑐

𝜔𝜔𝛽𝛽

𝑦𝑦2′ .

(4)

The solution for 𝑦𝑦1 is simplified when the betatronfrequency is much larger than the synchrotron one,𝜔𝜔𝛽𝛽 ≫ 𝜔𝜔𝑠𝑠 , which is mostly the case. It is thenapproximately given byy� 1 (𝑠𝑠) = y� 1 (0)𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑠𝑠/𝑐𝑐 + 𝑖𝑖Υy� 2 (0)𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑠𝑠/𝑐𝑐 ,(5)

where we have defined a positive dimensionlessparameter for the wake potential strengthΥ=

𝜋𝜋𝜋𝜋𝑟𝑟0 𝑊𝑊0 𝑐𝑐 24𝛾𝛾𝛾𝛾𝜔𝜔𝛽𝛽 𝜔𝜔𝑠𝑠.

(6)

We can write the solutions for the equations of motionsduring the period 0 < < 𝑇𝑇𝑠𝑠 /2 in a matrix form ascy�
1= 𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑇𝑇𝑠𝑠/2 �� 1�
y� 2 𝑠𝑠=𝑐𝑐𝑐𝑐 /2
0
𝑠𝑠

as

𝑖𝑖Υ y� 1�� � .
1 y� 2 𝑠𝑠=0

(7)

The transfer matrix during the second half of thessynchrotron oscillation period, 𝑇𝑇𝑠𝑠 /2 < < 𝑇𝑇𝑠𝑠 , is obtainedcby exchanging the indices 1 and 2 in the above treatment.
The total transfer matrix for one full synchrotron
oscillation period is then given by(8)

Let us find eigenvalues λ of this matrix by equating it
y�
y�
𝑖𝑖Υ
� � 1�
= 𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑇𝑇𝑠𝑠 ∙ 𝜆𝜆𝐈𝐈 � 1 � , (9)y� 2 𝑠𝑠=0
1 − Υ 2 y� 2 𝑠𝑠=01
𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑇𝑇𝑠𝑠 �𝑖𝑖Υwhere I is the unit matrix.
The two eigenvalues are found to be

λ=

2

2

2

⎧ 1 − Υ ± �Υ ∙ �Υ − 2�22𝑖𝑖𝑖𝑖 Υ 2 ≥ 4

2

Υ2
Υ2
Υ2
⎨
1 − ± 𝑖𝑖� ∙ �2 − �
2
2
2
⎩

.

(10)

𝑖𝑖𝑖𝑖 Υ 2 ≤ 4

Let us check the stability of the system for the twocases. If Υ 2 ≤ 4, the square of the absolute value of λbecomes|𝜆𝜆|2 = �1 −wherey� 2 = 𝑦𝑦2 + 𝑖𝑖y�
1 0 1 𝑖𝑖Υ y� 1
= 𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑇𝑇𝑠𝑠 �
��
�� �
� 1�
y� 2 𝑠𝑠=𝑐𝑐𝑐𝑐𝑖𝑖Υ
1 0 1 y� 2 𝑠𝑠=0
𝑠𝑠
y� 1
1
𝑖𝑖Υ
= 𝑒𝑒 −𝑖𝑖𝜔𝜔𝛽𝛽 𝑇𝑇𝑠𝑠 �� .
2 � �y𝑖𝑖Υ 1 − Υ � 2 𝑠𝑠=0

Υ2
2

2

� +

Υ2
2

∙ �2 −

Υ22

� = 1.

(11)

Namely, the system is stable. If Υ 2 ≥ 4, one of thesolutions,
𝜆𝜆 = 1 −

Υ2
2

−�

Υ2
2

∙�

Υ2
2

− 2� ≤ −1(12)with its amplitude larger than one, is unstable. At thethreshold value of Υ 2 = 4, the eigenvalue 𝜆𝜆 becomesexactly minus one (𝜆𝜆 = −1). It can be expressed in thecomplex phase space as
𝜆𝜆 = 𝑒𝑒 ±𝑖𝑖𝑖𝑖 .

(13)�𝜔𝜔𝛽𝛽 + ∆𝜔𝜔𝛽𝛽 � ∙ 𝑇𝑇𝑠𝑠 = 𝜔𝜔𝛽𝛽 ∙ 𝑇𝑇𝑠𝑠 ± 𝜋𝜋.

(14)

It indicates that during the full synchrotron oscillation
period, the coherent betatron frequency shift,∆𝜔𝜔𝛽𝛽 , generates a betatron phase advance by ±π. In otherwords,

It implies that the strong head-tail instability occurs by themode coupling between the two solutions when the
difference of their phase advances over one synchrotron
period becomes exactly 2π. The growth rate g, when
Υ 2 ≥ 4, is obtained by equating
Υ2|𝜆𝜆| = 𝑒𝑒 𝑔𝑔𝑇𝑇𝑠𝑠 = �

2

∙�

Υ2
2

− 2� +

Υ2
2

The formula for the growth rate is- 44 -− 1.

(15)

Proceedings of the 12th Annual Meeting of Particle Accelerator Society of Japan
August 5-7, 2015, Tsuruga, Japan

PASJ2015 WEOL06

𝑔𝑔 =

1

𝑇𝑇𝑠𝑠

log ��

Υ2
2

∙�

Υ2
2

− 2� +

Υ2
2

− 1�.

(16)

3. WEAK SPACE-CHARGE CASE
Now, let us introduce the space-charge force into thetwo particle model. Here, we make the following two
approximations:
1. The space-charge force is linear in the relative
distance between the two particles (the linear model).
2. The two particles interact continuously andcoherently with a space charge force in thetransverse plane (the continuous interaction model).
At a low energy where the space-charge force issignificant, the space-charge fields spread out angularly
with a large spread on the order of 1/γ, unlike a highenergy electron beam where the space-charge fields are
Lorentz contracted into a thin disk. In this sense, twoparticles behave like rods, not point charges, each with ahalf bunch length of the total beam.
Under this linearized continuous interaction model, we
have additional space charge terms in the equations ofsmotion for the period 0 < < 𝑇𝑇𝑠𝑠 /2 asy1′′ + �y2′′ + �

2

2

2

Γ2
Γ2
Γ2
⎨
1 − ± 𝑖𝑖� ∙ �2 − �222⎩Here we define Γ as
Γ2

2

2

1−𝑦𝑦 2= 2 ∙ 𝑦𝑦 2 ∙2Υ2Υ1−tanh2 � y�2

tanh2 � y�,𝑖𝑖𝑖𝑖 Γ 2 ≥ 4

.

(22)

2

𝑖𝑖𝑖𝑖 Γ ≤ 4(23)

where the wake potential strength parameter Υ is definedby Eq. (6) and y is given by
y = 2�𝑟𝑟(1 − 𝑟𝑟) .

(24)

The threshold value of Υ as a function of the ratio r
up to r=1 for the weak space-charge case (r=K/W≤1) isplotted in Fig. 1. Unstable regions are shown shaded.

c𝜔𝜔𝛽𝛽 2

� 𝑦𝑦1 = 𝐾𝐾(𝑦𝑦1 − 𝑦𝑦2 ) + 𝑊𝑊𝑦𝑦2 ,𝑐𝑐
𝜔𝜔𝛽𝛽 2

� 𝑦𝑦2 = 𝐾𝐾(𝑦𝑦2 − 𝑦𝑦1 ).

𝑐𝑐

λ=

2

⎧ 1 − Γ ± �Γ ∙ �Γ − 2�(17)Here, we define W as
𝑊𝑊 =𝑁𝑁𝑟𝑟0 𝑊𝑊0

𝐾𝐾 =

𝑁𝑁𝑟𝑟0

(18)

2𝛾𝛾𝛾𝛾

and K denotes the space-charge kick strength:
𝑎𝑎2 𝛽𝛽 2 𝛾𝛾3 𝐶𝐶

,

(19)

where a is the transverse beam size and β is the Lorenz β.
The properties of the wake potential and the space charge
force require that the both W and K are always positive(20)

W, K ≥ 0.

The ratio r=K/W can be expressed in a more familiar way,using Υ (defined by Eq. (6)) and the space-charge tune
shift parameter ∆ν𝑠𝑠𝑠𝑠 (normalized by the synchrotron tune𝜈𝜈𝑠𝑠 ), as
r=

𝜋𝜋

2Υ

�

∆ν𝑠𝑠𝑠𝑠𝜈𝜈𝑠𝑠

�.

Figure 1: The stability diagram for the weak space-chargecase (r=K/W≤1). Unstable regions are shown shaded.
The two eigenvalues for Γ 2 ≤ 4 can be expressed by
the left schematic of Fig. 2 in the complex phase plane.
The angles between these vectors and the positive real
axis are the phase advances of head-tail modes after one
synchrotron oscillation period. They line up on thenegative real axis (the right schematic of Fig. 2) when Γis two or when the difference of their phase advances over
one synchrotron period becomes exactly 2π.

(21)

The transfer matrix during the second half of thessynchrotron oscillation period, 𝑇𝑇𝑠𝑠 /2 < < 𝑇𝑇𝑠𝑠 , is obtainedcby exchanging the indices 1 and 2 in Eq. (17).
Hereafter, we assume that W ≥ K, namely we deal
with the weak space-charge case. Using the
eigenvalue/eigenvector technique, the equations of
motion can be readily solved. The eigenvalues are

Figure 2: Schematics of the eigenvalues in the complex
phase plane. Left for Γ 2 ≤ 4 and right for Γ = 2.

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Proceedings of the 12th Annual Meeting of Particle Accelerator Society of Japan
August 5-7, 2015, Tsuruga, Japan

PASJ2015 WEOL06

4. STRONG SPACE-CHARGE CASE

2.2 If

The eigenvalues for the strong space-charge case
(r=K/W≥1) can be expressed by Eq. (22) as well, though
the parameter Γ is now defined bytanh2 � y� ≥ y 2 ,

Γ2
2

=2∙

1+𝑦𝑦 2
𝑦𝑦 2

∙

Υ
2
Υ
1 +tan2 � y�2tan2 � y�.

(25)

Here the parameter y is defined by
y = 2�𝑟𝑟(𝑟𝑟 − 1).

Υthe beam is unstable and the growth rate is given by
𝑔𝑔 =

1

𝑇𝑇𝑠𝑠

where
(26)

The threshold value of Υ as a function of the ratio r forthe strong space-charge case (r=K/W≥1) is plotted in Fig.
3, together with the previous weak space-charge case(r≤1). Unstable regions are shown shaded.

(30)

2

Γ2
2

3.

log ��

=2∙1−𝑦𝑦 2
𝑦𝑦 2

Γ2

∙

2

∙�

Γ2
2

− 2� +
Υ

tanh2 � 2 y�Υ
21−tanh2 � y�

.

Γ2
2

− 1�,

(31)

(32)

If r≥1 (the strong space-charge case), calculate y as(33)

y = 2�𝑟𝑟(𝑟𝑟 − 1).
3.1 IfΥ

tan2 � y� ≤ y 2 ,

(34)

2

the beam is stable.
3.2 IfΥ

tan2 � y� ≥ y 2 ,

(35)

2

Figure 3: The stability diagram for the strong spacecharge case (r=K/W≥1). The stability diagram for the
weak space-charge case (r=K/W≤1) is also plotted forcompletion. Unstable regions are shown shaded.

5. PROCEDURETOIDENTIFYUNSTABLEREGIONS AND GROWTH RATE
We now have the total solution for both the weak andstrong space-charge regions. We can calculate thestability diagram using the following steps for given Υand:1.

Calculate the ratio r

r=

𝑊𝑊

𝐾𝐾

𝜋𝜋

∆ν𝑠𝑠𝑠𝑠�.

(27)

y = 2�𝑟𝑟(1 − 𝑟𝑟).

(28)

Υ

(29)

2.

=

2Υ

�

𝜈𝜈𝑠𝑠

If r≤1(the weak space-charge case), calculate y as

2.1 Iftanh2 � y� ≤ y 2 ,2the beam is stable.

the beam is unstable. The growth rate is given by Eq. (31),though the parameter Γ is now defined by
Γ2
2

=2∙

1+𝑦𝑦 2𝑦𝑦 2∙

Υtan2 � 2 y�Υ
2

1 +tan2 � y�

.

(36)

We calculate the growth factor g × T𝑠𝑠 as a function ofΥ and ∆ν𝑠𝑠𝑠𝑠 /𝜈𝜈𝑠𝑠 and plot it in a flat contour plot in Fig. 4and in a 3-dimension contour plot in Fig. 5. The threeparametersg × T𝑠𝑠 , Υ and ∆ν𝑠𝑠𝑠𝑠 /𝜈𝜈𝑠𝑠 are all dimensionlessparameters and these contour plots are universal.
One can see stopband structures in the stability
diagrams Figs. 4 and 5. The lowest two stopbands arestrongest (namely, very unstable), while other higherorder stopbands are considerably weaker.
The appearance of the stopbands can be explained asfollows. A mode-coupling instability takes place when thetwo eigenvalues line up on the negative real axis in thecomplex phase plane (see Fig. 2), in other words, when
the difference of their phase advances over one
synchrotron period is an odd integer times 2π. In thestrong space-charge region (∆ν𝑠𝑠𝑠𝑠 ≫ 𝜈𝜈𝑠𝑠 ), the tune shift ofone solution is close to −∆ν𝑠𝑠𝑠𝑠 , while other solution has
almost no tune shift. The mode-coupling condition
mentioned in the above corresponds to the case when∆ν𝑠𝑠𝑠𝑠 takes values around an odd integer times 𝜈𝜈𝑠𝑠 . That is
why the stopbands always start with ∆ν𝑠𝑠𝑠𝑠 /𝜈𝜈𝑠𝑠 equal to
odd integers at small Υ in Fig. 4. Pure space-charge
oscillations are stable, but even slight inclusion of wake
filed effects to them can make such oscillations unstable.

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Proceedings of the 12th Annual Meeting of Particle Accelerator Society of Japan
August 5-7, 2015, Tsuruga, Japan

PASJ2015 WEOL06

is not clear if we can achieve this state in reality or
computer simulations since such strong space-charge
force may expand the beam size, with a result of
reduction of the space-charge tune shift by itself.

Figure 4: Flat contour plot for the growth factor g × T𝑠𝑠∆ν
as a function of Υ and 𝑠𝑠𝑠𝑠.
𝜈𝜈𝑠𝑠

Figure 5: 3-dimensional contour plot for the growth factor∆νg × T𝑠𝑠 as a function of Υ and 𝑠𝑠𝑠𝑠.
𝜈𝜈𝑠𝑠

6. FINDINGS AND CONCLUSIONSLet us investigate how the space-charge force affects
the strong head-tail instability. Take a case of Υ = 4
where the beam is unstable without the space-charge∆νforce ( 𝑠𝑠𝑠𝑠 = 0). Figure 6 shows the growth rate (× T𝑠𝑠 ) as𝜈𝜈𝑠𝑠a function of the space-charge tune shift (normalized by
the synchrotron tune) at Υ = 4. If we gradually increase
the space-charge force, the beam moves from the unstableregion (the lowest stopband) to the stable region (thelowest passband). However, if we further increase thespace-charge force, the beam enters another unstableregion (the second lowest stopband). The maximum
growth rate in this unstable region is comparable to thatfor no space-charge case. One may conclude that the
space-charge force loses its damping effect when it is toostrong. In fact, many theoretical and simulation studies
show similar behaviors. If we further increase the spacecharge force, the beam would be stable again. However, it

Figure 6: The growth rate (× T𝑠𝑠 ) as a function of thespace-charge tune shift (normalized by the synchrotrontune) at Υ = 4.

The present two particle model has no tune spread
effect, since the space-charge force is linearized in thetransverse position. The damping of beam instabilities
with a weak space-charge force is caused by pure
coherent kicks of the space-charge force in a way topartially neutralize the coherent wake field kicks. The lossof the damping effect with a strong space-charge force isdue to an unfavorable combination between the coherent
space-charge kicks and the coherent wake field kicks. As
stated in the Introduction, if the damping of beam
instability with a weak space-charge force is caused by
the tune spread (Landau damping) due to the non-linearityof the space-charge force, its loss with a strong spacecharge force is hard to explain. The present model
suggests that the main damping mechanism of beam
instabilities with a weak space-charge force (as observedin many simulations) is linear coherent space-charge
kicks, not the Landau damping due to the non-linearity ofthe space-charge force.

7.

REFERENCES[1] M. Blaskiewicz, Phys. Rev. ST Accel. Beams 1, 044201(1998).
[2] M. Blaskiewicz, V. Ranjbar, in Proceedings of NAPAC13,
Pasadena, USA (JACoW, 2013), FROAA2.
[3] M. Blaskiewicz, A. Chao and Y. H. Chin, in Proceedings of
IPAC15, Richmond, USA (JACoW, 2015), MOPMN019.
[4] A. Burov, Phys. Rev. ST Accel. Beams 12, 044202 (2009)
and Phys. Rev. ST Accel. Beams 12, 109901 (2009).
[5] V. Balbekov, Phys. Rev. ST Accel. Beams 12, 124402(2009).
[6] V. Kornilov, O. Boine-Frankenheim, I. Hofmann, Phys. Rev.
ST Accel. Beams 11, 014201 (2008).
[7] V. Kornilov and O. Boine-Frankenheim, Phys. Rev. ST
Accel. Beams 15, 114201 (2012).
[8] Y. H. Chin, and K. Satoh, “Transverse Mode-Coupling in a
Bunched Beam", IEEE Trans. Nucl. Sci. NS-30, No. 4,
2566 (1983), and Nucl. Instrum. Methods, 207, 309 (1983).
[9] A. Chao, “Physics of Collective Beam Instabilities in HighEnergy Accelerators,” (John Wiley & Sons, Inc., New York,
1993).

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