BNL-107540-2015-IR

C-A/AP/536 February 2015

Ion Emittance Growth Due to Focusing Modulation

from Slipping Electron Bunch G. Wang

Collider-Accelerator Department Brookhaven National Laboratory

Upton, NY 11973

U.S. Department of Energy Office of Science, Office of Nuclear Physics

Notice: This document has been authorized by employees of Brookhaven Science Associates, LLC under Contract No. DE-SC0012704 with the U.S. Department of Energy. The United States Government retains a non- exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this document, or allow others to do so, for United States Government purposes.

Ion Emittance Growth Due to Focusing Modulation from Slipping Electron Bunch

Low energy RHIC operation has to be operated at an energy ranging from 4.1γ = to 10γ = . The energy variation causes the change of revolution frequency. While the rf system for the circulating ion will operate at an exact harmonic of the revolution frequency ( h = 60 for 4.5 MHz rf and h = 360 for 28 MHz rf.), the superconducting rf system for the cooling electron beam does not have a frequency tuning range that is wide enough to cover the required changes of revolution frequency. As a result, electron bunches will sit at different locations along the ion bunch from turn to turn, i.e. the slipping of the electron bunch with respect to the circulating ion bunch. At cooling section, ions see a coherent focusing force due to the electrons’ space charge, which differs from turn to turn due to the slipping. We will try to estimate how this irregular focusing affects the transverse emittance of the ion bunch. Time Domain Analysis Assuming the line charge distribution of the electrons and the ions are Gaussian, their currents can be written as ( ) ( )2

2exp22

eee

n ee

t n TQI t

σπσ

∞

=−∞

− ⋅Δ= −

, (1) and1 Ii t( ) = Qe

2πσ e

exp −t − m ⋅Trev( )2

2σ e2

m=−∞

∞

. (2) The electron current as observed by an ion bunch at its M th revolution is given by ( ) ( )

( )( )

2

2

2

2

, exp22

exp22

rev eee

n ee

e e e ee

n ee

M T n TQI M

n T M h Round M h TQ

ττ

σπσ

τσπσ

∞

=−∞

∞

=−∞

⋅ + − ⋅Δ= −

− ⋅Δ + ⋅ − ⋅ Δ = −

, (3)

where ,e repe

rev

fh

f= (4) with ,e repf being the repetition frequency of the electron beam and revf is RHIC revolution frequency, i.e. ~78 KHz. The ions sitting at the center of the bunch, i.e.

0τ = , get a kick at its M th revolution of 1 We consider one circulating ion bunch.

Figurobsefromiσ =

Figusectiordincente

re 1: mountrved by an m eq. (3) ( )5.8 / im cβ

ure 2: focusion. The absnate is the er as calcula

tain range pion bunch for 4.iγ =

19.95ns= a

ng strengthscissa is thefocusing stated from eq

plot of electrat the cooli1 , 1e nsσ =and 0.7iN =

h seen by ione independerength of thq. (5) and (8

ron current ing section. s , 10eTΔ =

975 10× .

ns at the cenent time vahe electron8) (they ove

evolution fThe electro0ns , 0eQ =

nter of the iariable in nuns at the locerlap).

for 5 succeson current 0.8nC , ,i rff

ion bunch inumber of tucation of th

sive turns ais calculate4.56MHz=

n the coolinurns and thhe ion bunc

as ed z ,

ng he ch

( )( )( ) 2

2

,0

exp22

M e

e e e ee

n ee

I M

n T M h Round M h TQ

ε α

ασπσ

∞

=−∞

= ⋅

− ⋅Δ + ⋅ − ⋅ Δ = ⋅ −

. (5) Assuming e eT σΔ >> , only the closest electron bunch, with 0n = , has contribution and hence eq. (5) simplifies to ( ) 2 2

2exp22

e e eeM

ee

M h Round M h TQε ασπσ

⋅ − ⋅ Δ = ⋅ −

. (6) Splitting eh into the integer part, ,e Ih , and the fractional part, eν , i.e. ,e e I eh h ν= + , (7) eq. (6) becomes ( ) 2 2

2exp22

e e eeM

ee

M Round M TQ ν νε α

σπσ

⋅ − ⋅ Δ = ⋅ −

. (8) The coefficient α can be derived as (See APPENDIX), ( )

20

2 3 2 23 20

2

2avgi cool cool

ei x e

rZ e L L

a cem a c

βα

γ β βε γ ω π β= = , (9) where avg

x

cββω

= , (10) and 2

0 204i

i

Z er

m cπε= . (11) Inserting eq. (9) into eq. (8) leads to ( ) 2 2

03 2 2 2

2exp

22e e eavg cool e

Me ee

M Round M Tr L Q

a ce

ν νβε

γ β β σπσ

⋅ − ⋅ Δ = ⋅ −

. (12) Applying eq. (77) to eq. (12), we obtain ( )

,

2 2

2

4

4 exp2

M e M

e e epeak

e

M Round M T

ε π ν

ν νπ ν

σ

= ⋅Δ

⋅ − ⋅ Δ = ⋅Δ ⋅ −

(13) with eν is the fractional part of , /e rep revf f and 0

3 3 24 2 2 2avge e cool

peakee e

rQ N L

a c

βν α

γ βπ πσ π πσΔ = ⋅ = . (14) Using eq. (9) from [1], we have

( )( ) ( )

ˆ,ˆexp ,

0J N

u NJ

ψψ = , (15) with ( ) ( ) ( )

2

0 0

1ˆ ˆ ˆ, sin 4 2 cos 4 22

N N

k kk k

u N kQ kQψ π ψ δε π ψ δε= =

= − + + +

, (16)

1

1 N

avg kkN

ε ε=

= , (17)

4avgQ Q

επ

= + , (18) and k k avgδε ε ε= − . (19) The evolution of the phase is given by 2

0

0

ˆ2 sin

1ˆ ˆ2 cos 4 22

N

N i ii

N

ii

QN

QN Qi

ψ π ψ ψ δε

π ψ π ψ δε

=

=

= + +

≈ + − +

(20)

Fourier Analysis In order to find out the resonant condition for emittance growth, we Fourier decompose eq. (1) as

( ) ( )2

2

0

1

exp22

2 2cos sin22

eee

n ee

ek k

k e ee

t n TQI t

Q a k t k ta b

T T

σπσ

π ππσ

∞

=−∞

∞

=

− ⋅Δ= −

= + + Δ Δ

, (21)

with 2 2 2

2

2 2 2expe ek

e e

ka

T T

πσ π σ ≈ − Δ Δ

, (22) and 0kb ≈ . (23) for e eTσ << Δ . Inserting eq. (22) and (23) into eq. (21) leads to ( ) ( )2

2

2 2 2

21

exp22

2 21 2exp cos2

eee

n ee

e e

ke e e

t n TQI t

Q k k t

T T T

σπσ

π σ π

∞

=−∞

∞

=

− ⋅Δ= −

= + − Δ Δ Δ

. (24)

The average electron current seen by ions is thus ( )2 2 2

21

2 21 , exp2

e ee eavg

ke e

Q kI N

T T

π σν∞

=

= + Θ − Δ Δ , (25) with ( ) ( )

( )( )

sin 2 11, 12 1 sin

ee

e

N kN

N k

πνν

πν + Θ ≡ + +

. (26) Inserting eq. (24) and (25) into eq. (5) and (19) yields ( )2 2 2

21

2 2 2exp cos ,e e revl e

ke e e

Q k k lTN

T T T

α π σ πδε ν∞

=

= − ⋅ − Θ Δ Δ Δ

. (27) Applying eq. (27) to the first term of eq. (16) produces ( ) ( ) ( ) ( )

( ) ( ) ( )

2 2 2

2

2 2 2

2

2 2 2

2

32

2

10 1

3 22

1 0

2

1

4 2 2ˆ ˆ, sin 4 2 e cos ,

2 2ˆ2 e , sin 4 2

e sin 2 2

e

e

e

e

e

e

kN

e peak T reve

l ke e

kN

e peak Te

k le

k

T rev

k

k lTu N lQ N

T T

N lQT

kTl Q

T

π σ

π σ

π σ

π σ ν πψ π ψ ν

π σ νν π ψ

π

∞ −Δ

= =

∞ −Δ

= =

∞ −Δ

=

Δ = − + ⋅ − Θ Δ Δ

Δ = − − Θ + +Δ

+Δ

2 2 2

22

0 0 1

ˆ ˆ2 e sin 2 2 2e

e

kN N

T rev

l l ke e

kTl Q

T

π σ

ψ π ψ∞ −

Δ

= = =

+ + − + Δ

. (28) The second term of eq. (16) reads

( ) ( )( )

( )

( ) ( ) ( )

2 2 2 2

2

2

20

2 '23

, 1 , 0

1ˆ ˆ, cos 4 22

264 e cos ,

2 ˆ ˆcos , cos 4 2 cos 4 2

e

e

N

mm

k lN

e peak T reve

l k m ne e

reve

e

u N mQ

k mTN k

T T

l nTN l nQ mQ

T

π σ

ψ π ψ δε

σ ν ππ ν

π ν π ψ π ψ

=

+∞ −

Δ

= =

= +

Δ = − Θ Δ Δ

− Θ + + Δ

. (29) For 0eν ≠ , taking the leading term of 1/ N in eq. (29) leads to

( )

( ) ( )

( ) ( )

2 2 2

22 4

32

1 , 0

ˆ, 8 e

ˆ ˆcos 2 2 4 cos 2 2 4

cos 2 2 cos 2 2

e

e

kN

e peak T

k m ne

rev rev

e e

rev rev

e e

u NT

T Tn m Q k n m Q k

T T

T Tn m Q k n m Q k

T T

π σσ νψ π

π ψ π ψ

π π

∞ −Δ

= =

Δ ≈ Δ

+ ⋅ + + + + ⋅ − + Δ Δ

+ − ⋅ + + − ⋅ − Δ Δ

. (30) Hence the resonant condition is

2 rev

e

kTQ K

T± =

Δ , (31) with K being an integer. Letting 2 decQ be the fractional part of 2Q and eν be the fractional part of rep

rev

f

f, eq. (31) becomes 2 dec eQ K kν= ± . (32) If 2 0rev

e

TQ k

T− =

Δ , eq. (28) becomes

( ) ( ) ( )2 2 2

2

32

2

1

2 2ˆ ˆ, e sin 2

e

e

k

e peak T

e

u N NT

π σπ σ νψ ψ

−ΔΔ

= −Δ

(33) eq. (29) becomes ( ) ( )2 2 2

22 4

3 22 ˆ ˆ, 8 e 1 cos 4

e

e

k

e peak T

e

u N NT

π σσ νψ π ψ

−ΔΔ

≈ + Δ . (34) Inserting eq. (33) and (34) into eq. (15) yields ( )

( ) ( )ˆ,

ˆexp ,0 k

J Nu N

J

ψψ = , (35) where ( ) ( ) ( )2 2ˆ ˆ ˆ, 2 sin 2 1 cos 4k k ku N g N g Nψ ψ ψ = − + + , (36) and ( )

2 2 2

223

22 ee

e

k

e peak Tk

e

gT

π σσ νπ

−ΔΔ

≡Δ

. (37) Inserting eq. (36) into eq. (35) and averaging over the initial phase yields ( ) ( ) ( )

( ) ( ) ( ){ }

2

02

2 2

0

1 ˆ ˆ0 exp ,2

1 ˆ ˆ ˆ0 exp 2 sin 2 1 cos 42

avg k

k k

J N J u N d

J g N g N d

π

π

ψ ψπ

ψ ψ ψπ

=

= − + +

. (38)

For ( )ˆ, 1ku N ψ << , eq. (38) becomes

( ) ( ) ( ) ( ){ }( ) ( ){ }( )( )

22 2

02

2 2 2 2 2

0

2 2

1 ˆ ˆ ˆ0 exp 2 sin 2 1 cos 42

0ˆ ˆ1 2 sin 2

2

0 1 2

avg k k

k k

k

J N J g N g N d

Jg N g N d

J g N

π

π

ψ ψ ψπ

ψ ψπ

= − + +

≈ + +

= +

. (39)

Using and the o and The tgrowactioxQ =fract(39) comp

Figurwith

g the norma

one turn tran(nPtracking is wth of the an and even

28.15 and ional part ofully agreeparison betw

re 3: compadirect time

alized canon nsfer map c

( 1nX k +

) (1 cosk + =done in Maaverage actinly uniformthe electroof , /e rep revf fes with theween the in

arison of av domain tra

Simpnical variabx

pP

Q=

X x Q=can be writt) (1 cos 2 Qπ=

( ) (2 x nQ P kπathcad and tion growth mly distribuon bunch r is 0.3. As se tracking nitial and fin

erage actionacking accor

le Trackin les:

2 cox

x

pJ

Q=

2 sinxQ J=en as ) ( )x nQ X k +

) (sin 2k Qπ−the results of 300 mauted phaserepetition fhown in figresults. Fignal distribut

n growth asrding to eq.

ng

osψ nψ ,

( )sin 2 xQ Pπ

) ( )x nQ X k ε+are shown cro-particles. The betafrequency ig. 3, the squg. 8 showstion after 20

s calculated(40) and (4

( )nP k ( )1k nX kε + .in fig. 7 fores with the atron tune is taken suuare growths the phase000 turns.

d from eq. (343).

(40(41(42

. (43r the relativsame initiais taken auch that thh formula, eqe space plo

38) and (39

0) 1) 2) 3) ve al as he q. ot

9)

Figurspacedistr ReIn ortwo, whic

Eq. (4space If eq.

re 4: compae distributiibution afteequiremerder to storthe highesth can be re-45) suggeste charge tun. (46) is not

arison of phon in normer 2000 turnent on Elere the ions order reson

kg =-written as 2k >

ts that the gne shift is smsatisfied, eq

ase space dmalized varins. ctron Bunfor storeN tunances to av( )

322 e

T

σ νπ

Δ=

Δ

2

2 2 ln2

e

e

T

π σ

Δ

rowth will amaller than

peakνΔ <q. (45) requ

distribution.iables and nch Repet

urns with thvoid is given2 2 2

22

ee

e

k

peak T

eT

π σν −Δ

( )322 2 stN

T

πΔalways slow

( )322 2

e

st

T

Nπ

Δ

uires

. The blue cthe red cutition Frehe action grn by eq. (392

12

e

storeN<

tore e peak

eT

σ ν Δ wer than the

tore eσ .

curve is the urve is the equency Arowth below9), i.e. ,

. e requireme

initial phasphase spacAccuracyw a factor o

(44(45

ent if the (46

se ce

of 4) 5) 6)

FigurbeameQ =

Assu i.e. Figuras a fparamrepet

re 5: requirm store tim4nC , peaνΔ

ming eν is cmin

2repfδ π<

re 5 shows tfunction of Rmeters listetition frequ

red electronme. The plo43 10k

−= × ak >

close to zero(n 2 ,1 2dec

e

Q

ν−

merev

e

fT

σπ ⋅ ⋅ ⋅Δthe dependeRHIC ion beed in the capency is 3 KH

n repetition ot is generand 4.1γ = .ln

2e

e

T

πσ

Δ

o, applying e

)2

2decQ T

πΔ>

(min 2 ,1decQ

ence of the eam storing ption of fig. Hz.

frequency rated from. ( )

322 2 sN

T

πΔeq. (32) yiel2

lne

e

T

πσ

)2 lndecQ

− ⋅

required eletime as calc5, the requi

accuracy lem eq. (49)store e peak

eT

σ ν Δ lds

( )322 2 store

e

N

T

πΔ

( )322 2

nπ

ectron buncculated fromired accurac

evel as a funfor 10eTΔ =

. e e peakσ ν Δ

,e peak st

e

N

T

σ νΔΔch repetitiom eq. (49). Fcy of the ele

nction of io0ns , 1e nsσ =

(47, (48

12

tore

−

. (49n frequencyFor ectron bunc

on s ,7) 8) 9) y h

Estimation of Required Arriving Time Jitter Level In the presence of arriving time jitter, eq. (13) becomes ( ) 2

,24 exp

2e e e e t M

M peake

M T Round M Tτ ν ν δε π ν

σ

+ ⋅ Δ − ⋅ Δ + = ⋅Δ ⋅ −

, (50) with ,t Mδ being the arriving time error in the thM turn andτ is the location of the ion. If 0eν = , eq. (50) becomes ( )2

,24 exp

2t M

M peake

τ δε π ν

σ

+ = ⋅Δ ⋅ −

. (51) The maximal kick variation happens at

2 2

2 2 2

2 2

2 2 2

4 exp2

14 exp 12

0

M peake e

peake e e

d d

d d

τ τε π ντ τ σ σ

τ τπ νσ σ σ

= ⋅Δ ⋅ − −

= ⋅Δ ⋅ − −

=

, (52) i.e. 0 eτ σ= . (53) If the arriving time jitter is much smaller than the bunch length, eq. (51) becomes

( )

( )

2

,

,

,

,

1 /4 exp

2

1 2 /4 exp

2

4 exp

4 1

t M e

M peak

t M e

peak

t Mpeak

e

t Mpeak

e

e

e

δ σε π ν

δ σπ ν

δπ νσ

δπ νσ

+ = ⋅Δ ⋅ − +

≈ ⋅Δ ⋅ −

= ⋅Δ ⋅ −

≈ ⋅Δ ⋅ −

, (54)

which leads to the average kick of 1

1 4N

i peakiN e

πε ν=

≈ ⋅Δ , (55) with the rms variation of

( )

( )

22

1

2

,1

22

,1

2

2

1

1 4

4 1

4

N

ii

N

peak t Mi e

Npeak

t Mie

peakt

e

N

N e

Ne

e

δε ε ε

π ν δσ

π νδ

σ

π νδ

σ

=

=

=

= −

≈ ⋅Δ ⋅

Δ=

Δ= ⋅

(56)

Eq. (13) of [1] reads ( )

2 2

20

1ln2 1 2 cos 4

NJ eN

J e e Q

α

α α

δεπ

−

− −

−= + − , (57) with α being the characteristic parameter of the time correlation of the jitter, i.e. ( )2 expk m k mδε δε δε α+ = − . (58) Assuming α → ∞ , i.e. there is no correlation between arriving time jitters for two different turns, eq. (57) reduces to 2

0

ln2

NJN

J

δε =

. (59) Inserting eq. (56) into eq. (59) yields

2

20

0

4exp

2

exp

peakN t

e

grow

NJ J

e

tJ

π νδ

σ

τ

Δ = ⋅

≡

, (60) with the growing time of 2

2 22

18

egrow rev

peakt

eT

στπ νδ

=Δ

. (61) Requiring the growing time to be longer than the store time leads to 2 12 2

et

peak store

e

N

σδπ ν

<Δ

. (62) Fig. 6 shows the required arriving time jitter level as a function of the beam store time. For the action growth below a factor of e, the required arrival time jitter is 37 ps.

Figurthere The melect with Acco Inser whic

re 6: require is no correEsti

maximal foctron bunch pα defined

2δε =rding to eq.rting eq. (65h combines

red time jitelation betwimation o

cusing strenpeak currenin eq. (9). T

(1

1 N

iiN

ε ε=

−. (14), the co5) into (64) s with eq. (5

tter level asween time jitof Require

ngth is at thnt by Mε =The rms vari

)2 2 1N

ε α= ⋅oefficient α4α =produces

2 216δε π=57) leads to

s calculatedtters of twoed Peak C

he electron ( ),0eI Mα= ⋅iation of the

( )(1

,0N

ei

I iN =

α can be wri( )

40

peak

eI

νπ

Δ

2

peak

δνΔ

d from eq. (o different tuCurrent Jit

bunch cent) , e focusing st) ( ) )2

0eI−itten as .( )

( )

2

2

0

0

e

e

I

I

δ ,

(62). It is aurns. tter Leve

ter and is rtrength is th

(2 0eIα δ=

assumed thal

elated to th(63hus )20 . (64

(65(66

at

he 3) 4) 5) 6)

( )( ) ( )

2222

2 20

0 1ln 81 2 cos 40

eN

peak

e

IJ eN

J e e QI

α

α α

δπ ν

π

−

− −

−= Δ + − . (67) Assuming peak current jitters from two different turns are uncorrelated, i.e. α → ∞ , eq. (67) reduces to ( )

( )

222

20

0ln 8

0

eN

peak

e

IJN

J I

δπ ν

= Δ

, (68) which can be rewritten as ( )

( )

222

0 02

0exp 8 exp

0

e

N peakgrowe

I NJ J N J

NI

δπ ν

= Δ ≡

, (69) with

( )( )

222

2

10

80

grow

e

peak

e

NI

I

δπ ν

=

Δ ⋅

. (70) Hence requiring the action growth time longer than the beam store time (or cooling time) gives the requirement for the current jitter as ( )

( )

20 120 2

erev

storee peak

I T

tI

δ

π ν<

Δ . (71) Fig.7 plots the required peak current jitter level as calculated from eq. (71) with parameters given in the caption of fig. 5. As shown in fig. 7, in order to have the local growth time smaller than 1000 seconds, the required peak current jitter level is below 4.3%.

Figurjitterappli Now [1]. A with Inse

re 7: requirrs from twoied are give

we will finAccording to rting eq. (73

red relative o different n in the cap

d the coeffio eq. (1) of [ d

dθ 3) into eq. (

2

2

dx

dt=

=

=

peak curreturns are aption of fig. 5

APPicient α so [1], the equa2

22

dx Q x

θ= −

θ(72) leads t( )

( )

(

22

2

22

2

2

2

2rev

rev

x p

Q xT

Q xT

x t

π

π

ω δ

−

−

− −

nt jitter levassumed to 5.

PENDIX Athat Mδε haation of mot( )p xQδ θ ε−

2 / revt Tπ= . to ( ) (

( ) (

) ( )

2/

2

p

prev

x M

tx

d dt T

x tT

t t x

δθ

πδ

ω ε

−

−

vel as calcul be uncorr

A: as the sametion reads( )Mε θ ,

) ( )

) ( )

2

2

2M

rev

Mv

Q tT

Q t x

πε

πε

lated from eelated. The

e definition ) x

.

eq. (71). The parameter

of referenc (72 (73)

(74

he rs

ce )

4)

Taking ( )cos e revx A Q tν ω = + Δ (75) and inserting it into eq. (74) leads to ( )( ) ( ) ( )2 2

2rev

e p M

TQ t Q t Q tν δ ε

π+ Δ = + . (76) Integrating eq. (76) for the thM turn leads to

( )( )

( )

( )( )

( )

1

,1

2 02

1 1 04

rev

rev

rev

rev

MT

reve M

M T

MT

e M e Mrev M T

Tt dt

t dtT

ν επ

ν ν επ

−

−

Δ =

Δ = Δ =

. (77)

The space charge field from the electron beam in the co-moving frame is 0

'' '2

eenE r

ε⊥ ⊥= , (78) ' 0zE = , (79) and ' 0B = . (80) Explicitly, eq. (78) can be written as

0

'' '2

ex

enE x

ε−= (81) and

0

'' '2

ey

enE y

ε−= . (82) In lab frame, the field is Lorentz transformed to

02e

x

enE x

ε−= , (83)

02e

y

enE y

ε−= , (84)

0

'2

y ex

E enB y

c cγβ β

ε= − = , (85) and

0

'2

x ey

E enB x

c cγβ β

ε= = − . (86) Thus the space charge force that the ions experienced from the electrons is ( )

2

202

i ex i x y

Z e nF Z e E cB xβ

ε γ= − = − , (87) and

( )2

202

i ey i y x

Z e nF Z e E cB yβ

ε γ= + = − . (88) The equation of motion is thus ( )

222 2

2 30

12

i e coolx x x p

i i

Z e n Ldx x F x t x

dt m m cω ω δ

γ ε γ β= − + = − − . (89) Comparing eq. (89) with eq. (74) yields ( ) ( ) ( )

( )

2 2

23 3 20 0

2 2i e i cool e

Mi x i x e

Z e n t Z e L I tt

m m a c eε

ε γ ω ε γ ω π β= = . (90) Comparing eq. (90) with eq. (5) leads to

( )2

02 3 2 23 2

0

2

2avgi cool cool

ei x e

rZ e L L

a cem a c

βα

γ β βε γ ω π β= = , (91)

References [1] M. Blaskiewicz, in C-A/AP/#363, 2009.