```Laslett self-field tune spread calculation
with momentum dependence
(Application to the PSB at 160 MeV)
M. Martini
Contents
• 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
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Two-dimensional binomial distributions
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.
 m

 a a
 2BD (m, a x , a y , x, y )   x y



with a x , y  2m  2  x , y
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 x2 y2 
1   
 a2 a2 
x
y 

m 1
0
 x2 y2 
for 1  2  2   1
 a
a y 
x

 x2 y2 
for 1  2  2   1
 a
a y 
x

and  u  u 2  u
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2
u  x,y
3
Two-dimensional binomial distributions
Kapchinsky-Vladimirsky beam distributions (m  0)
• Define the Kapchinsky-Vladimirsky distribution (K-V) as

KV
2D
 x2 y2 
1
(0, a x , a y , x, y ) 
 1  2  2  with a x , y  2 x , y
 ax a y  ax a y 
• Since the projections of B2D(m,ax,ay,x,y) for m  0 and KV2D(m,ax,ay,x,y) yield the same
1BD (0, a x , a y , x)  lim 
2
a y 1 x 2
ax
2
m 0  a y 1 x2
ax
1KV
D (a x , a y , x)  
2
a y 1 x 2
ax
2
 a y 1 x2
ax
1
 2BD (m, a x , a y , x, y ) dy 
 ax
1
 2KVD (ax , a y , x, y ) dy 
 ax
 x2 
1  2 
 ax 
 x2 
1  2 
 ax 
1/ 2
for x  a x
1 / 2
for x  a x
• The 2-dimensional distribution KV2D(m,ax,ay,x,y) can be identified to a binomial limiting
case m  0
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Two-dimensional binomial distributions
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Two-dimensional binomial distributions
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Two-dimensional binomial distributions
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Two-dimensional binomial distributions
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
 x2
y2 
 ( x ,  y , x)  lim  (m,a x ,a y , x, y ) 
Exp- 2 - 2 
m 
2 x y
 2 x 2 y 
G
2D
1
B
2D
with  u  u 2  u , u  x,y and
2
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a x , y  2m  2  x , y
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Projected binomial distributions
m 1 / 2
2
 m


 ( m)
x
1  2 

for x  a x
B
1D (m, a x , x)    a x (m  12 )  a x 

0
for x  a x

 1  x 2  1/ 2
 x2 
1
1  2 

for x  a x
G
KV
1D (ax , x) 
Exp- 2 
1D (a x , x)   a x  a x 
2


 2 x 
x

0
for x  a x

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Projected binomial distributions
m
ax  x  2m  2
x



2 x
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1/2
1
3/2
2
6

√2
√3
2
√5
√6
√14

1BD (m, ax , x) x 2 dx
1/2
1BD (m, ax , x) x 2dx
-
x
2 x
0
0.577 0.608 0.626 0.637 0.664 0.683
-
1
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0.984 0.975 0.960 0.955
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Laslett space charge self-field tune shift
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).
Q
spch
0, x , y
spch
a y2
ay
Nr0 R  x , y (ax , y )
spch
spch
 2 3

(
a
)


(
a
)

x
x, y
y
x, y
  Q0, x, y
a y2
a x (a x  a y )
a x a y
• 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
Bf 
I average
I peak
Q
spch
0, x , y
spch
Nr0 R  x , y (ax , y )
 2 3
  Q0, x , y a y2 Bf
• Considering binomial transverse beam distributions and using the rms beam sizes x,y
Q0spch
, x, y
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1

2

Nr0 R xspch
, y ( x , y )  ( 2m  2) y


1
 2 3Q0, x , y Bf 
2 y2


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for 0  m  
for m  
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Laslett space charge self-field tune shift
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

Q0spch
, x, y
n
x, y
 x2, y
 
 x, y
Nr0

  2 Bf
 xn, y

 x, y 
R
(smoothapproximation)
Qx , y
1
n
x, y
  yn, x  y , x  x , y

1


 (2m  2)

 12
for 0  m  
for m  
• Nonetheless this expression is not really useful due to contributions of the dispersion Dx,y
and relative momentum spread  to the rms beam sizes
 x, y
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 x , y xn, y
 
 Dx2, y 2

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Laslett space charge self-field tune shift
• For bunched beam with binomial or Gaussian longitudinal distribution the bunching
factor Bf can be analytically expressed as (assuming the buckets are filled)

 Gammam  12 
Binomial(bunch length  8m  8 z )

 2 Gammam  1
Bf  
  Erf 2  0.598 Gaussian beam (full bunch length  4 )
z

 8
 
m
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Laslett space charge tune spread with momentum
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).
ay 



Qxspch ( x, y, z )  Q0spch
1

,x 
a
x 


(1) n n n  j1 (2 j1 )!(2 j2 )!(2 j3 )!
  2n  
J ( j1 , j2 , j3 )
j1! j2 ! j3!
n 0 2
j1  0 j2  0
j1
j3  l
(2( j1  i  j2  k ))!
1
k  0 l  0 m  0 ( 2 j1  2i )!( 2 j2  2k )! i! k!l! m!
j2
j3
  
i 0

im
1
(i  m)!( j3  k  l  m)!( j1  j2  i  k )! ( j1  j2  i  k  l )!
 x
 
 ax



2 ( i  m 1)
 Dy Qy a y
 
 Raz
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


 y

a
 y




2 ( j3  k  m  l )
2 ( j3  m  l )
 z

 az
 Dx Qz a z

 Rax






2 ( j1  j2  k  m l )
2 ( j1 i )
 Dx Qx a x

 Raz
 D y Qz a z

 Ra
y

M. Martini







2m
2 ( j2  k )
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Laslett space charge tune spread with momentum
• 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
2
n
 4 a z 
  2 1
J (0,0, n)  ln
a a 
i 1 2i  1
y 
 x
2
J (1,0, j3 ) 
wit h   a y / a x
 1
  (n  1) J ( j1  1,0, j3 )
J ( j1 ,0, j3 ) 
(n  1 / 2)( 2  1)
   2 ( j1  1 / 2) J ( j1  1, j2  1, j3 )
J ( j1 , j2 , j3 ) 
j2  1 / 2
• It holds for bunched beams of ellipsoidal shape with radii defined as ax,y,z = 2x,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.
Q 
2
z
h eVrf
2 2 E0
• R is the machine radius, the other parameters Dx,y, , e, h, E0... are the usual ones.
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Application to the PSB
• All the space-charge tune spread have been computed to the 12th order but higher the
expansion order better is the tune footprint (15th order is really fine but time consuming)
PSB MD: 22 May 2012
Total particle number = 950 1010
Full bunch length = 627 ns
Qx0 = 4.10 (tr=4)
Qy0 = 4.21
Ek = 160 MeV
xn (rms) = 15 m
yn (rms) = 7.5 m
p/p = 1.44 10-3
Bunching factor (meas) = 0.473
RF voltage= 8 kV h = 1
RF voltage= 8 kV h = 2 in anti-phase
Qx0 = -0.247
Qy0 = -0.365
12th order run-time  11 h
Tune diagram on a PSB 160 MeV plateau for the CNGS-type long bunch
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.
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Application to the PSB
PSB MD: 4 June 2012
Total particle number = 160 1010
Full bunch length = 380 ns
Qx0 = 4.10 (tr=4)
Qy0 = 4.21
Ek = 160 MeV
xn (rms) = 3.3 m
yn (rms) = 1.8 m
p/p = 2 10-3
Bunching factor (meas) = 0.241
RF voltage= 8 kV h = 1
RF 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
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Application to the PSB
PSB MD: 6 June 2012
Total particle number = 160 1010
Full bunch length = 540 ns
Qx0 = 4.10 (tr=4)
Qy0 = 4.21
Ek = 160 MeV
xn (rms) = 3.4 m
yn (rms) = 1.8 m
p/p = 1.33 10-3
Bunching factor (meas) = 0.394
RF voltage= 8 kV h = 1
RF voltage= 4 kV h = 2 in antiphase
Qx0 = -0.176
Qy0 = -0.288
Tune diagram on a PSB 160 MeV plateau for the LHC-type long bunch
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