A New Two Particle Model for Studyof Effects of Space-Charge Forceon Beam Instabilities*.
Yong Ho Chin (KEK), Alex Chao (SLAC)
and Mike Blaskiewicz (BNL)The 12th Annual Meeting of PASJ
Tsuruga, Japan, on 5 - 7August, 2015
*Paper Submitted to PRST-AB for Publication.
2015/8/26

Chin, Chao and Blaskiewicz

1Outline

Motivations of This Work

Chao’s Original Two Particle Model

New Two Particle Model with Space-Charge

Procedure to Identify Unstable Regions and toCompute Growth Rate
Findings and Conclusions

2015/8/26

Chin, Chao and Blaskiewicz

2

No Beam Instability Observed at RCS
No beam instability has been observed at RCS.
 It is generally believed that a beam at RCS isstabilized by a large incoherent tune spread (Landaudamping) due to non-linearity of the space-chargeforce.
 It is even proposed to shorten a bunch at RCS toincrease the space-charge force to achieve astronger damping of a beam, though it sounds
contrary to common belief (a longer bunch is morestable).
 Is the space-charge force really a “magic cure”?
2015/8/26

Chin, Chao and Blaskiewicz3

Mysterious Simulation/Analytic Results

During HB2014 Workshop, Kornilov and Blaskiewicz
reported mysterious simulation and analytical results
for beam instabilities with space-charge force.
Airbag model

V. Kornilov and O. Boine-Frankenheim
M. Blaskiewicz
PRST-AB, 1, 044201 (1998)
PRST-AB, 13, 114201 (2010)
Chin, Chao and Blaskiewicz
2015/8/26

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Beam Instabilities with Space-Charge
Many simulation results generally indicate that beam
instability can be damped by a weak space-charge
force, but the beam becomes unstable again when thespace charge force is further increased.
 If the damping of beam instabilities is caused by thebetatron tune spread (Landau damping) due to thenon-linearity of the space-charge force, A stronger space-charge force should be more
effective in damping of beam instabilities.
 Why do many simulation results show the contrary?
 This mystery has not been solved for ~20 years.
2015/8/26

Chin, Chao and Blaskiewicz

5Invitation by Chao

After the working session at HB2014, Chao has
invited use to collaborate on study for effects ofspace-charge force on beam instabilities by modifying
his famous two particle model for a strong head-tailinstability.
That was a fascinating idea.
 We may be able to solve the mystery by using a simple
model and mathematics for this complicated phenomenon.
 We found later though that his proposed new two particle
model did not work (a pity).
 So, it turned out that the crux of the problem is to find asuitable new two particle model which is A simple expansion of the original two particle model Still analytically solvable.
2015/8/26

Chin, Chao and Blaskiewicz

6Chao’s Original Two Particle Model

Let us first review the premise and treatment of
Chao’s original two particle model.
Two macro-particles executing synchrotron and betatronoscillations.
 Their synchrotron oscillations have equal amplitude, but
opposite phases.
𝛿

𝛿
21
𝜏

𝜏
1

2y1 ⇄ y22015/8/26

Chin, Chao and Blaskiewicz

7

Total Matrix for Full Synchrotron Periody1s−𝑖𝜔𝛽 𝑇𝑠 /2 1 𝑖Υ y1=𝑒for 0 < < 𝑇𝑠 /2cy2 𝑠=𝑐𝑇 /2y2 𝑠=001𝑠
Here𝜋𝑁𝑟0 𝑊0 𝑐 2
Υ=4𝛾𝐶𝜔𝛽 𝜔𝑠Dimensionless Wake Field Strength Parameter

Total Matrix
y11 0 1= 𝑒 −𝑖𝜔𝛽 𝑇𝑠
y2 𝑠=𝑐𝑇𝑖Υ 1 0𝑠y11𝑖Υ𝑒 −𝑖𝜔𝛽 𝑇𝑠.
2 y𝑖Υ 1 − Υ2 𝑠=0

2015/8/26

𝑖Υ y1=
1 y2 𝑠=0

Chin, Chao and Blaskiewicz

8Eigenvalues and Growth RateThe two eigenvalues areλ=Υ21− ±2

Υ2
Υ2
∙
−222

𝑖𝑓 Υ 2 ≥ 4

Υ21− ±𝑖2Υ2∙2𝑖𝑓 Υ 2 ≤ 4If Υ 2 ≥ 4, one of the solutions is unstable.
Υ2 𝜆 = 1−−2

Υ22−2Υ2Υ2
∙
−222

≤ −1

At the threshold value of Υ 2 = 4, the eigenvalue
𝜆 becomes exactly minus one (𝜆 = −1) or 𝜆 = 𝑒

2015/8/26±𝑖𝜋 .
Chin, Chao and Blaskiewicz

9Instability Mechanism
Phase advance of head-tail modes
after one synchrotron periodΥ2Υ2∙ 2−22

StableThresholdUnstable

U=2U>2

1

U=0

U=1

Υ21−2

Υ2
Υ2
Υ2
λ=1−±𝑖∙ 2−2222015/8/26

𝜆 = 𝑒 ±𝑖𝜋

Chin, Chao and Blaskiewicz

Υ2Υ2 Υ2
λ=1−±∙−222210

Transverse Mode-Coupling Instability

It implies that the strong head-tail instability occurs
by the mode coupling between the two solutions when
the difference of their phase advances over one
synchrotron period becomes exactly 2.
The growth rate g, when Υ 2 ≥ 4, is obtained byequating

𝜆 =𝑒𝑔𝑇𝑠

=

Υ2
Υ2
∙
−222

Υ2+ − 1.
2The formula for the growth rate:1
 𝑔 =log𝑇𝑠2015/8/26

Υ2
Υ2
∙
−222

Υ2+ −12.

Chin, Chao and Blaskiewicz

11New Two Particle Model with Space Charge
Two approximations:Linear Model The space-charge force is linear in the relative distance
between the two particles.
 Continuous Interaction Model The two particles interact continuously and coherently
with a space charge force in the transverse plane.
𝛿
2
𝜏
11/
2015/8/26s
For 0 < < 𝑇𝑠 /2c
𝜔𝛽 2′′y1 +𝑦1 = 𝐾 𝑦1 − 𝑦2 + 𝑊𝑦2𝑐
𝜔𝛽 2′′y2 +𝑦2 = 𝐾 𝑦2 − 𝑦1𝑐𝑊=Chin, Chao and Blaskiewicz

𝑁𝑟0 𝑊02𝛾𝐶𝐾=

𝑁𝑟0𝑎2 𝛽2 𝛾 3 𝐶

Weak Space-Charge Case (W≥K)
The two coupled equations of motion can be solved
using the eigenvalue/eigenvector technique.
The stability
diagram for theweak space-chargecase (r=K/W≤1).
Unstable regions
are shown shaded.
r=1/2y1′′ +y2′′ +𝜔𝛽 2𝑐
𝜔𝛽 2𝑐−

𝑊2

𝑦1 =𝑊
𝑦2 2−

𝑊2𝑦2 = −Absolutely stable
regardless of W
2015/8/26

Chin, Chao and Blaskiewicz

13𝑊
𝑦2 1

Strong Space-Charge Case (K≥W)
The stability diagram
for the strong spacecharge case
(r=K/W≥1).
The stability diagram
for the weak spacecharge case
(r=K/W≤1) is alsoplotted for completion.
Unstable regions areshown shaded.

2015/8/26

Chin, Chao and Blaskiewicz

14Procedure to Calculate Growth rate
For given Υ (the dimensionless wake field parameter)and ∆ν𝑠𝑐 /𝜈𝑠 (the dimensionless space-charge parameter),𝐾
𝜋 ∆ν𝑠𝑐r==𝑊 2Υ 𝜈𝑠r1

y = 2 𝑟(𝑟 − 1).

y = 2 𝑟(1 − 𝑟)
2

tanh
Yes
Stabler1Υy ≤ y2
2

tan2
Yes

No
Unstable

StableΥtanh2 yΓ2
1−𝑦 2
2=2∙ 2 ∙Υ .
22𝑦1−tanhyΥy2

≤ y2

No
UnstableΥtan2 yΓ21+𝑦 2
2=2∙ 2 ∙Υ .
2𝑦1 +tan2 y22

1Growth rate: 𝑔 = log
𝑇𝑠

2015/8/26

Γ2Γ2
∙
−22
2

Chin, Chao and Blaskiewicz

Γ2+ −1
2

15

Contour Plots for Growth Rate
These figures are all universal !

Flat contour plot for the growth factor g × T𝑠
∆νas a function of Υ and 𝑠𝑐 .
𝜈𝑠2015/8/26

3-dimensional contour plot for the growth∆νfactor g × T𝑠 as a function of Υ and 𝑠𝑐 .

Chin, Chao and Blaskiewicz

𝜈𝑠16

Instability Mechanism
In the strong space-charge regime:
Phase advance ofhead-tail mode
after one synchrotron
periodΓ2Γ2∙ 2−22

∆ν
- 𝑠𝑐2

∆ν
+ 𝑠𝑐2-∆ν𝑠𝑐2

Mode-couplingwhen equal toodd integers
times 𝜈𝑠Threshold

Unstable

G=2G>2StableΓ21−22

λ=1−

Γ
±𝑖2

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Γ22

∙ 2−

Γ22

Chin, Chao and Blaskiewicz

17Growth Rate as a Function ofSpace-Charge Tune ShiftΥ =4 case.

It shows that the space-charge force loses its damping effectwhen it is too strong.
It qualitatively reproduces typical behaviors shown in many
theoretical and simulation results.

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Chin, Chao and Blaskiewicz

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Two Cases of Absolutely Stable Coupled Motions
As the space-charge force
increases, Eqs. of motion
approach to those for twopendulums connected witha spring.

𝜔𝛽 2𝑦1 = 𝐾 𝑦1 − 𝑦2𝑐𝜔𝛽 2′′y2 +𝑦2 = 𝐾 𝑦2 − 𝑦1𝑐y1′′ +Another absolutely stablemotions.
y1′′ +y2′′ +2015/8/26

Chin, Chao and Blaskiewicz

𝜔𝛽 2𝑐
𝜔𝛽 2𝑐−

𝑊2𝑦1 =𝑊𝑦2 2−

𝑊2𝑦2 = −19𝑊𝑦2 1

Findings and Conclusions
The present two particle model has no tune spreadeffect, since the space-charge force is linearized in
the transverse position.
 The damping of beam instabilities with a weak spacecharge force is caused by pure coherent kicks of thespace-charge force in a way to partially neutralize
the coherent wake field kicks.
 The damping by linear coherent kicks is unusual ?


No. To damp beam instabilities externally, we often use
 Non-linear magnets such as octupoles for Landau damping
by an incoherent tune spread.
 Feedback system for linear (in the transverse
displacement of a beam from the center orbit) coherent
kicks to a beam.

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Chin, Chao and Blaskiewicz

20Biased Perception
The present model suggests that the main damping
mechanism of beam instabilities with a weak spacecharge force is linear coherent space-charge kicks,
not the Landau damping due to the non-linearity ofthe space-charge force.
 However, when we study on damping of beam
instabilities by a beam itself, we tend to think only
Landau damping as a damping mechanism of spacecharge force (because of a large tune spread).
 Further investigation of the present model and/or
inclusion of more effects will help us to have a better
understanding of effects of space-charge force on
beam instabilities.
2015/8/26

Chin, Chao and Blaskiewicz

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“The Geography of Thought” by R. Nisbett
How Asians and Westerners Think Differently… and WhyAccording to this book,
Westerners think that the World is simple and steady.
 It is ruled by simple laws of nature and can be describedby simple models.
 They value principles.
 Asians think that the World is complicated and rapidlychanging.
 It is too complicated even to describe.
 There is no law of nature, since such a law is also changing
all the time.
 They value practicality.
 That is why westerners succeeded in creating and developing
science called physics, while We Asians failed.


Let’s do it in Westerners’ way to see how it works.

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Chin, Chao and Blaskiewicz

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