music: a multi-bunch/particle simulation code with an ... · with an alternative approach in...

M. Migliorati

MuSiC: a Multi-bunch/particle Simulation Code with an alternative approach in simulating wakefield effects

Thanks to: N. Biancacci, H. Damerau, E. Métral, B. Salvant, G. Sterbini

Outline

•  Common approach to wakefield simulations. •  Issues in using wakefield simulations. •  An alternative approach: impedance fit with resonant

modes. •  MuSiC: a Multi-bunch/particle Simulation Code. •  Comparisons with theory/other codes:

–  multi-bunch, single particle with LCBC (fb on and off) –  multi-bunch/particle: some preliminary results –  potential well distortion, synch phase shift and microwave inst. –  wakefield: PS Kickers and pure inductive

•  Conclusions.

Pag. 2 11/06/14 MuSiC: a Multi-bunch/particle Simulation Code with an alternative approach in simulating wakefield effects

Outline




•  Conclusions.


Common approach to wakefield simulations

•  Consider, e.g., the longitudinal beam dynamics and write the single particle equations of motion (z>0 at the tail of the bunch)

•  The total voltage V(z) has the contributions of RF voltage(s) and wake potential

where λ(z) is normalized to unity and w//(z) is the longitudinal wake with w//(z)=0 for z>0


Δz = z(t)− z(t −T0 ) = L0ηε

Δε = eV z( )−U0 +…( ) / E0

V z( ) =VRF z( )+Qtot w// z '− z( )λ(z ')dz '−∞

z

∫


•  In a simulation code the equations of motion are written for each macroparticle i at each turn n

•  By using Nm macroparticles instead of a continuous distribution for λ(z), the convolution intergal of the wake potential is transformed into a sum

At each turn, the evaluation of the effect of the wake potential is proportional to (Nm-1)*Nm/2


zin = zi

n−1 + L0ηεin−1

εin = εi

n−1 + eV zin( )−U0 +…( ) / E0

V zin( ) =VRF zi

n( )+Qtot

Nm

w// z jn − zi

n( )z jn<zi

n

j=1,Nm

∑ Vw(zin)


•  In order to reduce the computing time, in general the bunch is divided into Ns slices (bins) of width Δ and with centre ziΔ, each one containing ni(Δ) macroparticles, and we evaluate the wake potential acting on the centre of one slice due to all the other slices:


Vw ziΔ( ) = Qtot

Nm

nj (Δ)w// z jΔ − ziΔ( )z jΔ<ziΔ

j=1,Ns

∑zjΔ

nj(Δ) ziΔ

By linear interpolation, we can then obtain the wake potential acting on any charge inside the bunch

By repeating this operation for each slice, we get the wake potential at the centre of each slice.

+

+


•  A code based on this approach was developed for DAΦNE and used for the PS (see presentations at the 17th HDWG meeting and at the 51st ICE meeting in 2012)


Example of comparison of single bunch

simulation code with theory:

BBR impedance

-0.2 -0.1 0 0.1 0.2z [mm]

0

2

4

6

8

10

long

itudo

inal

dis

tribu

tion

[1/m

m]

tail head

-0.2 -0.1 0 0.1 0.2z [mm]

0

2

4

6

8

10

12

long

itudi

nal d

istri

butio

n [1

/mm

]

-0.2 -0.1 0 0.1 0.2z [mm]

0

2

4

6

8

long

itudi

nal d

istri

butio

n [1

/mm

]

-0.2 -0.1 0 0.1 0.2z [mm]

0

2

4

6

8

long

itudi

nal d

istri

butio

n [1

/mm

]

tail head

tail head tail head

Outline




•  Conclusions.


Issues in using wakefield simulations

•  SLICES: how many? The greater is the number of slices the better is the accuracy, but each slice must contain sufficiently high number of macroparticles è increase Nm.

•  Slices represent a non-physical artifice, which can introduce additional numerical noise making necessary, in some cases, a parametric study: e.g. what happens by changing number of slices and of macroparticles in the slices.


13

dove Vw zin( ) è calcolata tramite la (3.6), ed è stato cambiato il segno all'ultimo ter-

mine dal momento che non ha importanza se R viene preso positivo o negativo, es-sendo un numero aleatorio di valor medio nullo. L'algoritmo principale del codice di simulazione è dunque molto semplice. Con ogni macroparticella i , le equazioni (3.1) e (3.7) vengono fatte evolvere per un nu-mero di giri sufficientemente grande (pari a 8 ÷10 volte 1 D ) da permettere ai campi elettromagnetici scia a corto raggio di deformare la distribuzione iniziale ed even-tualmente dare luogo all'instabilità a microonde. I valori iniziali zi

o e !io non hanno

una eccessiva importanza, in quanto i processi di diffusione e di smorzamento pro-dotti dall'irraggiamento ed inclusi nell'equazione (3.7), unitamente alla funzione scia, determinano la forma del pacchetto indipendentemente dalle condizioni di partenza. Per accelerare comunque i tempi di calcolo si prendono dei valori aleatori derivanti da una funzione gaussiana. La distribuzione di particelle viene valutata al centro di ognuno dei B intervalli:

! zi"( ) = ni "( )Nm

(3.8)

e da essa si determina il valore della lunghezza !z . Naturalmente, la ! zi"( ) è tanto più precisa quanto più piccoli sono gli intervalli ! . Però questo può generare un ele-vato rumore, in quanto riducendo ! diminuiscono anche le macroparticelle che si trovano nell'intervallo considerato, e quindi ni !( ) può subire delle variazioni molto brusche passando da un intervallo ad un altro. In figura 3.1 sono mostrati due esempi di distribuzione ottenuti con B = 40 e B = 150 . Il secondo caso è leggermente più vi-cino al risultato analitico, ma il rumore è elevato. Oltre alla ! z( ) è anche possibile ri-cavare la funzione di distribuzione nel piano delle fasi in quanto, per ogni macroparti-cella, è noto il valore !i

n .

z

! z( ) ! z( )

z Figura 3.1: Distribuzione longitudinale con B = 40 e B = 150 .

low number of slices

high number of slices


•  If the bunch is long compared with the wavelength of the wakefield, a very high number of slices could be necessary, thus increasing enormously the computing time.

•  This problem can be very tricky because by changing the number of slices, say, by a factor of 2, one could get the same (wrong!) result.


-10 -5 5 10

5.0¥1010

1.0¥1011

1.5¥1011

2.0¥1011

2.5¥1011

3.0¥1011

Example: BBR with fr≈2.4 GHz, Q=1

(λw≈ 12.5 cm) Gaussian bunch

of RMS length=2m:

1000 slices/sigma

10000 slices/sigma

correct wake potential


•  Inverse dft of impedance: in order to obtain the impedance model of an accelerator, one usually uses theory and e.m. codes, such as CST Microwave Studio, GdfidL, ….

•  However these codes give the coupling impedance and the wake potential of a Gaussian distribution. In order to obtain the wakefield to be used in a simulation code, one has to do numerically the inverse Fourier transform of the impedance, thus adding other numerical noise, or use other tricks to find out the ‘Green function’.

•  In order to simulate the coupled bunch instabilities, long range wakefield is necessary. But how long? The em field in this case remains trapped over many turns, depending on the Q factor of the HOM … many machine lenghts!


Outline




•  Conclusions.


An alternative approach: impedance fit with resonant modes

•  To study coupled bunch instabilities a time domain simulation code, named LCBC (Longitudinal Coupled Bunch simulation Code), was developed and first used for DAΦNE.

•  Only dipolar oscillation modes are simulated. •  It includes the effect of a bunch-by-bunch feedback

system to damp coupled bunch instabilities. •  The code is used also to study the PS coupled bunch

instabilities. A frequency domain feedback system has been implemented in the code for this purpose.



•  See, e.g., presentations at the 84th ICE meeting (2013) and at the LIU Day 2014 by L. Ventura


22

2) Measurement 1) Simulation

Mode amplitude

2013 measurements vs. simulations in h=21 (10.1MHz)

Number of turns

Mode number

Pattern of the oscillation modes at 10 MHz compared with measurements done in 2013 which show the evolution of the mode spectra for LHC50ns beam during acceleration for a full machine (21 bunches). The ‘mode pattern’ depends on the initial conditions and the oscillation amplitude depends on the bunch number in the train.


•  In LCBC code, the impedance of resonant modes, responsible of coupled bunch instability, is used directly instead of the wakefield.

•  The wakefield of a resonant mode is known analytically

•  A point charge q1, due to interaction with a resonant

mode, produces this wakefield and, due to the beam loading theorem, loses an energy equal to:


w// (z) =ωrRsQ

eωr2Q

z/ccos(ωnz / c)+ ωr

2Qωn

sin(ωnz / c)!

"#

$

%&H −z( )

ε1 = e q1ωrRs2Q

!

"#

$

%& / E0


11/06/14 MuSiC: a Multi-bunch/particle Simulation Code with an alternative approach in simulating wakefield effects

Pag. 16

… q1 q2 q3 q4

z2z3

ε2 = e q1w// (z2 )+ q2ωrRs2Q

!

"#

$

%& / E0

ε1 = e q1ωrRs2Q

!

"#

$

%& / E0

ε3 = e q1w// (z3)+ q2w// (z3 − z2 )+ q3ωrRs2Q

"

#$

%

&' / E0

this sum is the convolution but …

ε3w

ε3bl

bunches


•  Instead of using this convolution, it is possible to demonstrate that, from the mathematical point of view, the previous result can also be obtained by using a matrix formalism:


M (z) = eωr2Q

z/ccos(ωnz / c)+

ωr

2Qωn

sin(ωnz / c)ωrRsQωn

sin(ωnz / c)

−ωrQRsωn

sin(ωnz / c) cos(ωnz / c)−ωr

2Qωn

sin(ωnz / c)

"

#

$$$$$

%

&

'''''

q = q0

!

"##

$

%&& ‘charge’

matrix ‘wakefield’ matrix



Pag. 18

… q1 q2 q3 q4

z2z3

ε2 = eωrRsQ

M z2( )•q1{ }1 +ε2bl!

"#

$

%& / E0

ε1 = eε1bl / E0

ε3 = eωrRsQ

M (z3 − z2 )• M (z2 )•q1 + q2( ){ }1 +ε3bl"

#$

%

&'

… new ‘charge’

matrix


•  Given the ‘wakefield’ matrix, we can see that it is possible to obtain the energy variation due to wakefield for any charge by transporting the matrix M� from one charge to another, adding another charge and continuing to transport …

•  This formalism has been used in LCBC, where every q is the total charge of each bunch.

•  It is possible to use this approach with any beam configuration (e.g. with gaps to simulate transients) and for any resonant mode.


qq


•  Can we use the same approach for any generic wakefield, also short range, to simulate simultaneusly single bunch and multi-bunch effects?

•  The advantage of using this approach in a simulation code is that we avoid the convolution integral (and slices) saving computing time, and the problem of the length of wakefield disappears.

•  We just need to order the particles inside the same bunch from the head to the tail and propagate the matrix between two particles: instead of (Nm-1)*Nm/2 operations at each turn, only about Nm-1 are needed!

•  If we consider a generic impedance, we can take a sum of some resonant modes to fit it and use them in a simulation code with this matrix transport formalism.


Outline




•  Conclusions.


MuSiC: a Multi-bunch/particle Simulation Code

•  The transport matrix of the wake is basically the idea behind the multibunch-multiparticle simulation code MuSiC.

•  No slices, no wakefield (total length and minimum step), no need of inverse Fourier transform from impedance.

•  Just need to order the particles before accounting for the wakefield effect.

•  It is possible to take into account at the same time short and long range wakefields.

•  Since MuSiC is based on the previous LCBC, the PS longitudinal frequency domain feedback system is already included (and the time domain feedback used in DAΦNE can be easily added).

•  Of course the disadvantage is that the impedance must be fitted with resonant modes (either narrow and broad band).


MuSiC: a Multi-bunch/particle Simulation Code

•  The impedance obtained as sum of resonant modes naturally respects all the properties of a coupling impedance:

–  Its inverse Fourier transform is real –  It respects the causality principle for β=1

equivalent to the Hilbert transforms The above expressions indicate that, in principle, knowing either the real or the imaginary part of a coupling impedance, one can construct the whole impedance.


ZRE ω( )cos ωτ( )dω−∞

∞

∫ = ZIM ω( )sin ωτ( )dω−∞

∞

∫

ZRE ω( ) = 1πP.V.

ZIM ω '( )ω '−ω

dω '−∞

∞

∫ ZIM ω( ) = 1πP.V.

ZRE ω '( )ω '−ω

dω '−∞

∞

∫

Outline




•  Conclusions.


Comparisons with theory/other codes multi-bunch, single particle with LCBC (fb off and on)


Pag. 25

7 bunches, 2 HOMs exciting oscillation modes 1 and 3, 1 macroparticle/bunch

fb off fb on

Comparisons with theory/other codes multi-bunch/particle: some preliminary results


Pag. 26

What happens if we use a distribution instead of a single macroparticle?

total bunch length=0.1 of RF wavelength total bunch length=0.16 of RF wavelength



Pag. 27

What happens if we use a distribution instead of a single macroparticle?

total bunch length=0.42 of RF wavelength linear RF



Pag. 28

Quadrupolar instability: strongly couple a HOM to a 2fs beam spectrum line (Q=2e6!)



Pag. 29

Quadrupolar instability: try to strongly couple a HOM to a 2ωs beam spectrum line (Q=2e6!)

Comparisons with theory/other codes potential well distortion and synch phase shift


Pag. 30

Single bunch, BBR model

synchtrotron phase shift with MuSiC: 0.024 rad analytical phase shift with BBR: 0.0238 rad

Comparisons with theory/other codes potential well distortion and synch phase shift


Pag. 31

Single bunch, BBR model, microwave instability regime

Comparisons with theory/other codes microwave instability


Pag. 32

Single bunch, BBR model, microwave instability threshold



Pag. 33



Pag. 34



Pag. 35



Pag. 36



Pag. 37



Pag. 38



Pag. 39



Pag. 40



Pag. 41



Pag. 42

clear evidence of instability



Pag. 43



Pag. 44



Pag. 45



Pag. 46



Pag. 47


threshold

Can we compare this threshold with theory?



Pag. 48


Linear theory of microwave instability in electron storage rings

Yunhai CaiSLAC National Accelerator Laboratory, Stanford University, Menlo Park, California 94025, USA

(Received 24 January 2011; published 14 June 2011)

The well-known Haissinski distribution provides a stable equilibrium of longitudinal beam distribu-

tion in electron storage rings below a threshold current. Yet, how to accurately determine this threshold,

above which the Haissinski distribution becomes unstable, is not firmly established in theory. In this

paper, we will show how to apply the Laguerre polynomials in an analysis of this stability that are

associated with the potential-well distortion. Our approach provides an alternative to the discretization

method proposed by Oide and Yokoya. Moreover, it reestablishes an essential connection to the theory

of mode coupling originated by Sacherer. Our new and self-consistent method is applied to study the

microwave instability driven by commonly known impedances, including coherent synchrotron radiation

in free space.

DOI: 10.1103/PhysRevSTAB.14.061002 PACS numbers: 29.27.Bd, 29.20.db

I. INTRODUCTION

The idea of using mode couplings to explain the insta-bility of a bunch beam was first proposed and then studiedover many years by Sacherer [1]. The theory essentiallyconsists of an integral equation, known as the Sachererintegral equation, derived as a perturbed Vlasov equation,which is solved as an eigenvalue problem of a matrix. Itwas extended by Besnier [2] who introduced orthogonalpolynomials to the theory. It was further developed byZotter [3] to include the radial modes and later bySuzuki, Chin, and Satoh [4] to make an expansion inboth azimuthal and radial modes.

Initially, the choice of the unperturbed distributions, forinstance Gaussian, was made on the basis of making theequation easier to solve. It is therefore not a self-consistentapproach for an electron bunch because it has its ownequilibrium distribution. To improve the theory, Oideand Yokoya [5] studied the perturbation near Haissinskidistribution [6]. They derived a generalized Sacherer in-tegral equation that includes the incoherent tune shift dueto the distorted potential. Moreover, they introduced theaction-angle variables so that a Fourier expansion can bemade in the angular variable to take the advantage of aperiodicity in the system of bunched beam. For the radialdirection, they numerically solved the integral equation bydiscretizing the action variable. Their method was suc-cessfully applied [7] to study the instability of the Stanfordlinear collider (SLC) damping ring, where many precisionmeasurements were made [8]. However, it was found outlater [9] that the method of discretization was not as robustas one might have expected largely due to the presence of

the incoherent spectrum in the system. In particular, theconvergence of the procedure is poor in many cases as onerefines the mesh. To improve the theory, the integralequation was ‘‘regularized’’ by Warnock, Stupakov,Venturini, and Ellison [10] by replacing the linear equa-tion with a highly nonlinear one, similar to the dispersionequation for a coasting beam. As demonstrated for theinstability in the SLC damping ring, it also worked ex-tremely well. However, the nonlinear equation itself in-troduced another layer of complication. Instead of solvingan eigenvalue problem, one has to perform a search of avery nonlinear equation. In general, it is not clear if themethod of search is indeed better than the one of Oide andYokoya.In this paper, we will continue the investigation of the

stability of the Haissinski distribution by analyzing thelinearized Vlasov (LV) equation. In particular, we willapply the orthogonal polynomials to solve the integralequation in the presence of the potential-well distortion.We will start with the Vlasov-Fokker-Planck (VFP)

equation in Sec. II; and then, in Sec. III, we will introducethe Haissinski distribution as a static solution of the VFPequation. In Sec. IV, we will derive the LV equation as asmall perturbation near the Haissinski distribution. Thenwe will reduce the linear equation to an integral equationusing a Fourier expansion in Sec. V. Finally, in Sec. VI, wewill solve the integral equation in terms of various matrixrepresentations.In the latter part of the paper, we will apply the linear

theory to commonly known impedances. First, the SLCdamping ring will be studied as a benchmark in Sec. VII;second, we will continue, in Sec. VIII, to investigate theinstability caused by the coherent synchrotron radiation(CSR) in free space; and third, we will study a broadbandresonance model in Sec. VIII. At the end of the paper, wewill conclude our study and discuss the microwaveinstability.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW SPECIAL TOPICS - ACCELERATORS AND BEAMS 14, 061002 (2011)

1098-4402=11=14(6)=061002(12) 061002-1 ! 2011 American Physical Society



Pag. 49


For Q ¼ 1, the broadband model was first systemati-cally studied [5] by Oide and Yokoya. Here, we repeat thecalculation with the formulation developed in this paper. Itis worth noting that the same difficulty of the double well isalso encountered in the original method proposed by Oideand Yokoya. As we discussed in the previous section, thereare some degrees of uncertainty in determining the exactthreshold due to numerical noise. To minimize the effect ofthe noise, we determined the threshold by extrapolatingfrom higher currents at which the noise is much lessimportant.

The results are summarized and shown in Fig. 11. Forthe VFP simulations, we simply plotted the thresholds inour previously published paper [18]. As one can see fromthe figure, the results from the four methods (including

Gaussian) are not too much different from each other. Onceagain, the coasting beam theory, !th ¼

ffiffiffiffiffiffiffi4"

p#2r , provides a

conservative estimate of the threshold.

X. DISCUSSION

We have rederived the Sacherer integral equation bylinearizing the Vlasov equation and expanding it in aFourier series in the presence of a potential-well distortion.To solve the integral equation, we employed the Laguerrepolynomials and reduced it to an eigenequation of aninfinite dimensional matrix. The matrix elements are writ-ten in terms of integrals that contain the incoherent spec-trum and a kernel driven by impedance. This linear theory

−8 −6 −4 −2 0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

q

λ(q)

−8 −6 −4 −2 0 2 4 6 80

5

10

15

20

25

q

V(q

)

K2K1

FIG. 10. Double-peak distribution on the left and its corresponding double-well potential on the right for the Q ¼ 1 broadbandresonance model with #r ¼ !r$z=c ¼ 0:5 and ! ¼ InR!r ¼ 18.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

20

25

30

νr = σ

zω

r/c

ξth (

I nRω r)

Valsov−Fokker−Planck simulationOide and Yokoya methodLaguerre polynomialGaussian modelCoasting beam theory

FIG. 11. Threshold as a function of #r ¼ !r$z=c in theQ ¼ 1broadband impedance model with various methods.

−6 −4 −2 0 2 4 6−2

0

2

4

6

8

10

12

14

16

18

q

V(q

)

qmin qmax

K

FIG. 12. The potential well at the threshold current Ithn ¼0:042 pC=V in the SLC damping ring. Note that most distortionoccurs near the bottom of the well.

YUNHAI CAI Phys. Rev. ST Accel. Beams 14, 061002 (2011)

061002-10

This point, exaclty superimposed to ‘Oide Yokoya method’, corresponds to 12e9 part

courtesy of Y. Cai, Phys Rev ST – AB 14 061002 (2011)

Comparisons with theory/other codes wakefield: PS Kickers and pure inductive


Pag. 50

PS kickers longitudinal impedance obtained with Tsutsui model



Pag. 51

PS kickers impedance and fit with resonant modes



Pag. 52

Comparison between MuSiC and the single bunch simulation code in a stable regime (capacitive impedance è shortening regime)



Pag. 53

Comparison between MuSiC and the single bunch simulation code in a stable regime (capacitive impedance è shortening regime)


•  Also a pure inductive impedance Z(ω)=iωL can be simulated with MuSiC (wake ∝ δ’)


Pag. 54

bunch spectrum (a.u.) Im(Zres)

Re(Zres)

pure inductance

Outline




•  Conclusions.


Conclusions

•  MuSiC is a new multibunch/multiparticle tracking code which simulates the longitudinal beam dynamics under the simultaneous effect of short range and long range wakefields.

•  The proposed alternative approach to wakefield simulations, which fits the impedance with multiple resonators, avoids the use of slices, the problem of the ‘Green’ function, and issues related to the length of the wakefield for coupled bunch instabilities.

•  Comparisons with a previous single bunch code, LCBC (coupled-bunch) and theory show very good agreements.

•  A frequency domain feedback system for controlling coupled bunch instabilities, similar to that used for the PS, is included and a time domain feedback system, similar to that used for DAΦNE, can easily be included.


Pag. 56

Pagina 57 11/06/14 MuSiC: a Multi-bunch/particle Simulation Code with an alternative approach in simulating wakefield effects

Thank you for your attention

music: a multi-bunch/particle simulation code with an ... · with an alternative approach in...

Documents