Laslett self-field tune spread calculation with momentum dependence (Application to the PSB at 160...
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Transcript of Laslett self-field tune spread calculation with momentum dependence (Application to the PSB at 160...
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
a
y
a
x
a
y
a
x
a
y
a
x
aa
m
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
ayx
BD
myx
BD ax
a
x
adyyxaamxaa xa
xy
xa
xy
for11
),,,,(lim),,,0(2/1
2
21
12
01
2
2
2
2
xxx
a
ayx
KVDyx
KVD ax
a
x
adyyxaaxaa xa
xy
xa
xy
for11
),,,(),,(2/1
2
21
121
2
2
2
2
yxyxyxyx
yxKVD a
a
y
a
x
aayxaa ,,2
2
2
2
2 2with11
),,,,0(
8
Two-dimensional binomial distributions
06/07/2012 M. Martini
Gaussian transverse beam distributions (m )
• The 2-dimensional Gaussian distribution G2D(x,y,x,y) can be identified to a binomial
limiting case m since
2y
2
2x
2
22 2
y-
2
x-Exp
2
1),,,,(lim),,(
yxyx
BD
myx
GD yxaamx
yxyxu max,yuuu ,,
22 22and,with
9
Projected binomial distributions
06/07/2012 M. Martini
x
x
m
xxxBD
ax
axa
x
m
m
a
mxam
for0
for1)(
)(),,(
2/1
2
2
21
1
x
xxxx
KVD
ax
axa
x
axa
for0
for11
),(
2/1
2
2
1
2x
2
1 2
x-Exp
2
1),(
x
xGD 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 )(
Ba
a
Q
RNrQ
I
IB
y
yxyx
yxyxf
yx
yyxy
yxx
yyxx
y
yxyx
yxyx aa
aa
aaa
aa
a
a
Q
RNrQ
)(
)()(
)(,
spch2
,spch
2
,spch,
,,0320spch
,,0
m
mm
BQ
RNrQ
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
B
NrQ
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
Gamma
2
z
z21
m
m
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
zzy
ij
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
Ra
aQD
Ra
aQD
Ra
aQD
Ra
aQD
a
z
a
y
a
x
lkijjkijjmlkjmi
mi
mlkikjij
kjij
jjjJjjj
jjj
a
aQzyxQ
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(
12
12
4ln),0,0(
2
32112
321
231
31
3
1
2
j
jjjJjjjjJ
n
jjJnjjJ
aajJ
iaa
anJ
xy
n
iyx
z
02
2
2 E
eVhQ rf
z
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 mD Qx0 = -0.247D 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 phaseD Qx0 = -0.221D 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-phaseD Qx0 = -0.176D Qy0 = -0.288
Tune diagram on a PSB 160 MeV plateau for the LHC-type long bunch
Application to the PSB