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.
4
Two-dimensional binomial distributions
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(
5
Two-dimensional binomial distributions
06/07/2012 M. Martini
6
Two-dimensional binomial distributions
06/07/2012 M. Martini
7
Two-dimensional binomial distributions
06/07/2012 M. Martini
8
Two-dimensional binomial distributions
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
10
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
12
Laslett space charge self-field tune shift
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
13
Laslett space charge self-field tune shift
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
Total particle number = 160 1010
Full bunch length = 380 nsQx0 = 4.10 (tr=4)Qy0 = 4.21Ek = 160 MeVx
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
Application to the PSB
1806/07/2012 M. Martini
PSB MD: 6 June 2012
Total particle number = 160 1010
Full bunch length = 540 nsQx0 = 4.10 (tr=4)Qy0 = 4.21Ek = 160 MeVx
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
Application to the PSB
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