m . martini

Laslett self-field tune spread calculation with momentum dependence

(Application to the PSB at 160 MeV)

M. Martini

2

Contents

06/07/2012 M. Martini

• Two-dimensional binomial distributions

• Projected binomial distributions

• Laslett space charge self-field tune shift

• Laslett space charge tune spread with momentum

• Application to the PSB

3

Two-dimensional binomial distributions

06/07/2012 M. Martini

11for0

11for1),,,,(

2

2

2

2

2

2

2

21

2

2

2

2

2

yx

yx

m

yxyxyx

BD

ay

ax

ay

ax

ay

ax

aam

yxaam

x,yuuuma uyxyx 22,, and22with

Binomial transverse beam distributions• The general case is characterized by a single parameter m > 0 and includes the waterbag

distribution (uniform density inside a given ellipse), the parabolic distribution... (c.f. W. Joho, Representation of beam ellipses for transport calculations, SIN-Report, Tm-11-14, 1980.

• The Kapchinsky-Vladimirsky distribution (K-V) and the Gaussian distribution are the limiting cases m 0 and m .

• For 0 < m < there are no particle outside a given limiting ellipse characterized by the mean beam cross-sectional radii ax and ay.

• Unlike a truncated Gaussian the binomial distribution beam profile have continuous derivatives for m 2.

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06/07/2012 M. Martini

Kapchinsky-Vladimirsky beam distributions (m 0)• Define the Kapchinsky-Vladimirsky distribution (K-V) as

• Since the projections of B2D(m,ax,ay,x,y) for m 0 and KV

2D(m,ax,ay,x,y) yield the same Kapchinsky-Vladimirsky beam profile

• The 2-dimensional distribution KV2D(m,ax,ay,x,y) can be identified to a binomial limiting

case m 0

xxx

a

a yxBDmyx

BD ax

ax

adyyxaamxaa xa

xy

xax

y

for11),,,,(lim),,,0(2/1

2

21

1 20122

22

xxx

a

a yxKVDyx

KVD ax

ax

adyyxaaxaa xa

xy

xax

y

for11),,,(),,(2/1

2

21

1 2122

22

yxyxyxyx

yxKVD a

ay

ax

aayxaa ,,2

2

2

2

2 2with11),,,,0(

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06/07/2012 M. Martini

6


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06/07/2012 M. Martini

Gaussian transverse beam distributions (m )• The 2-dimensional Gaussian distribution G

2D(x,y,x,y) can be identified to a binomial limiting case m since

2y

2

2x

2

22 2y-

2x-Exp

21),,,,(lim),,(

yxyx

BDmyx

GD yxaamx

yxyxu max,yuuu ,,22 22and,with

9

Projected binomial distributions

06/07/2012 M. Martini

x

x

m

xxxBD

ax

axax

mm

am

xamfor0

for1)(

)(),,(

2/1

2

2

211

x

xxxx

KVD

ax

axax

axafor0

for11),(

2/1

2

2

1

2

x

2

1 2x-Exp

21),(

xx

GD xa

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Projected binomial distributions

06/07/2012 M. Martini

m 0 1/2 1 3/2 2 6

√2 √3 2 √5 √6 √14

1/2 0.577 0.608 0.626 0.637 0.664 0.683

- - 1 0.984 0.975 0.960 0.955

x

x

dxxxam xBD

2

1 ),,(

22 ma xx

x

x

dxxxam xBD

2

2

21 ),,(

11

Laslett space charge self-field tune shift

06/07/2012 M. Martini

Space charge self-field tune shift (without image field)• For a uniform beam transverse distribution with elliptical cross section (i.e. binomial

waterbag m=1) the Laslett space charge tune shift is (c.f. K.Y. Ng, Physics of intensity dependent beam instabilities, World Scientific Publishing, 2006; M. Reiser, Theory and design of charged particle beams,Wiley-VCH, 2008).

• For bunched beam a bunching factor Bf is introduced as the ratio of the averaged beam current to the peak current the tune shift becomes

• Considering binomial transverse beam distributions and using the rms beam sizes x,y instead of the beam radii ax,y yields

f2

,spch,

,,0320spch

,,0peak

average )(Baa

QRNrQ

II

By

yxyx

yxyxf

yx

yyxy

yxx

yyxx

y

yxyx

yxyx aa

aa

aaaa

aa

aQRNrQ

)(

)()(

)(,

spch2

,spch

2,

spch,

,,0320spch

,,0

m

mm

BQRNr

Q

y

y

yx

yxyxyx

for 2

1

0for)22(

1)(

2

2

f,,032

,spch,0spch

,,0

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06/07/2012 M. Martini

Space charge self-field tune shift (without image field)• The self-field tune shift can also be expressed in terms of the normalized rms beam

emittances defined as

• Nonetheless this expression is not really useful due to contributions of the dispersion Dx,y and relative momentum spread to the rms beam sizes

ion)approximat(smooth,

,,

2,n

,yx

yxyx

yxyx Q

R

m

mm

BNrQ

yxxyxyyxyx

yx

for 21

0for)22(

11

,,n,

n,

n,f

20spch

,,0

22,

n,,

,

yxyxyx

yx D

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06/07/2012 M. Martini

)4lengthbunch (fullbeamGaussian598.02Erf

8

)88mlength(bunch Binomial1Gamma

Gamma2

z

z21

mm

B f

• For bunched beam with binomial or Gaussian longitudinal distribution the bunching factor Bf can be analytically expressed as (assuming the buckets are filled)

m

14

Laslett space charge tune spread with momentum

06/07/2012 M. Martini

Space charge self-field tune spread (without image field)• Tune spread is computed based on the Keil formula (E. Keil, Non-linear space charge

effects I, CERN ISR-TH/72-7), extended to a tri-Gaussian beam in the transverse and longitudinal planes to consider the synchrotron motion (M. Martini, An Exact Expression for the Momentum Dependence of the Space Charge Tune Shift in a Gaussian Bunch, PAC, Washington, DC, 1993).

)(2)(2)(2

2)(2)(2)1(2

21213

0 0 0 0 21

21

0 0321

321

321

02

spch,0

spch

213

213

1 2 3 3

1

1

2

)!(1

)!()!()!(

!!!!1

)!22()!22())!(2(

),,(!!!

)!2()!2()!2(2

)1(

1),,(

kj

y

zzyij

x

zzx

lmj

z

yyy

m

z

xxx

lmkjj

z

lmkj

y

mi

x

j

i

j

k

j

l

lj

m

n

j

jn

jnn

n

x

yxx

RaaQD

RaaQD

RaaQD

RaaQD

az

ay

ax

lkijjkijjmlkjmimi

mlkikjijkjij

jjjJjjj

jjj

aa

QzyxQ

15

Laslett space charge tune spread with momentum

06/07/2012 M. Martini

Tune spread formula

• In the above formula j1+j2+j3=n where n is the order of the series expansion. The function J(j1+j2+j3) is computed recursively as

• It holds for bunched beams of ellipsoidal shape with radii defined as ax,y,z = 2x,y,z with Gaussian charge density in the 3-dimensional ellipsoid. It remains valid for non Gaussian beams like Binomial distributions with ax,y,z = (2m+2)x,y,z (0 m < ).

• x,y are the rms transverse beam sizes and z the rms longitudinal one, x, y, z are the synchro-betatron amplitudes. Qx,y,z are the nominal betatron and synchrotron tunes.

• R is the machine radius, the other parameters Dx,y, , e, h, E0... are the usual ones.

2/1),1,1()2/1(),,(

)1)(2/1(),0,1()1(),0,(

/with1

2),0,1(

12124ln),0,0(

2

32112

321

231

31

3

1

2

jjjjJjjjjJ

njjJnjjJ

aajJ

iaaanJ

xy

n

iyx

z

02

2

2 EeVh

Q rfz

16

Application to the PSB

06/07/2012 M. Martini

Tune diagram on a PSB 160 MeV plateau for the CNGS-type long bunch

PSB MD: 22 May 2012

Total particle number = 950 1010

Full bunch length = 627 nsQx0 = 4.10 (tr=4)Qy0 = 4.21Ek = 160 MeVx

n (rms) = 15 my

n (rms) = 7.5 mp/p = 1.44 10-3

Bunching factor (meas) = 0.473RF voltage= 8 kV h = 1RF voltage= 8 kV h = 2 in anti-phasePSB radius = 25 m Qx0 = -0.247 Qy0 = -0.36512th order run-time 11 h

The smaller (blue points) tune spread footprint is computed using the Keil formula using a bi-Gaussian in the transverse planes while the larger footprint (orange points) considers a tri-Gaussian in the transverse and longitudinal planes.

• All the space-charge tune spread have been computed to the 12 th order but higher the expansion order better is the tune footprint (15th order is really fine but time consuming)

1706/07/2012 M. Martini

PSB MD: 4 June 2012



n (rms) = 3.3 my

n (rms) = 1.8 mp/p = 2 10-3

Bunching factor (meas) = 0.241RF voltage= 8 kV h = 1RF voltage= 8 kV h = 2 in phase Qx0 = -0.221 Qy0 = -0.425

Tune diagram on a PSB 160 MeV plateau for the LHC-type short bunch


1806/07/2012 M. Martini

PSB MD: 6 June 2012



n (rms) = 3.4 my

n (rms) = 1.8 mp/p = 1.33 10-3

Bunching factor (meas) = 0.394RF voltage= 8 kV h = 1RF voltage= 4 kV h = 2 in anti-phase Qx0 = -0.176 Qy0 = -0.288

Tune diagram on a PSB 160 MeV plateau for the LHC-type long bunch


m . martini

Documents

bunched beam

binomial waterbag

laslett selffield tune

beam radii ax

binomial limiting case

rms beam sizes x

averaged beam current

image field