Introduction to particle
accelerators
Walter Scandale
CERN - AT department
Roma, marzo 2006
Lecture II - single particle dynamics
topics

Guiding fields and transverse motion

Weak versus strong focusing

Equation of motion





Unperturbed case
Orbit errors
Quadrupole errors
Chromaticity
Resonances and dynamic aperture

Low-ß insertion

Longitudinal stability
Synchrotron ring:




Synchrotron: guiding field
particle trajectories at fixed radius r
to keep r constant B should increase as p increases during acceleration,
RF frequency synchronized to the particle revolution
bending and focusing fields
Dipole: the Lorenz force provides the centripetal acceleration
mv 2
L
r
v
B
 evB0  r 
pGeV /c 
Br Tm 
0.2998
L
L
L  r    arc 
F
r
By
f
L
f
X0
ºº
º
L
p
eB0
r
bending
radius
magnetic
rigidity
bending
angle
x
Quadrupole: the Lorenz force focuses the trajectories
p 

s
f
By 
p
dBy
dx
x  Bx
L
 eBy L
v
p eBy L eBxL x




p
p
p
f
1 eBL B


f
p
Br
p  Ft  evBy
Weak focusing
Weak focusing of the transverse particle motion:


to get vertical stability, the bending field should decrease with r, as in cyclotrons,
to get horizontal stability the the decrease of B with r should be moderate, so that, for
r > r0, the Lorenz force exceeds the centripetal force.
y
y
N
F
y
N
n<0
no vertical stability
Br
Br
Fy
r
Br
Fy
F
r
y
S
r 0 
By  B0  
guiding field
 r 
r By 
n     field index
By r r  r 0

n>0
vertical stability
Br
S
horizontal stability:
n
centripetal force ≤
Lorenz force
 r0 n

mv 2
mv 2  x 
x 


1  evBy  evB0 
  evB01 n 
r
r0  x r0  r0 
r0 
r0  x 

mv 2
horizontal stability  n  1
weak focusing
vertical stability
n0
Strong focusing
d
Horizontal and vertical
focusing for a large
range of f1 f2 and d
1 1 1
d
  
f
f1 f 2 f1 f 2
separated functions: the alternate gradient is made
with quadrupoles of opposite focusing strength
 combined functions: the alternate gradient is made
with dipoles with radial shape of opposite sign

f1
f2

normalized quadrupole gradient
quadrupole strength
Examples
Ring
p
[GeV/c]
B0
[T]
CERS
5.2
Tevatron
1000

1

B
Tm


eB B
K

 K m2  0.2998
p Br
pGeV /c 
1
 KL
f
Br
[Tm]
1/r2 [m-2]
weak focus
Lquad
[m]
B’
[T/m]
K
[m-2]
0.18 96.4
17.3
10-4
0.5
5
0.298
4.4
3335
1.7•10-6
1.7
76
0.0228
r
[m]

758
Particle equation of motion
F  ev  B
B-field expansion
Maxwell equations and quadrupolar gradients
Bx
y  Bx
y

By
By  B0 
x  By
x

1 By
1 Bx
g
k

k

k



x
y

  B  0 
B0 r x x 0 B0 r y y 0
 


1 By
k  k x  k y  1 Bx
  B  0
gskew


B0 r x y 0 B0 r y x 0

Bx 
Bs  Bs
Equation of motion
d 2z


K
s
z
   F s
2
ds
transport matrix approach

zs
z0

 M  s,s0 


z
s
z
  
0




1
x  kx (s)  2 x  

r s rs

y  k y  0

y
p

momentum error
p
Weak versus strong focusing
d 2z
 K sz  F s
2
ds
Equation of motion
Strong focusing



x  kx (s)  21 x  

r s rs

y  k y  0

y
p

momentum error
p



zs
z0

 M s,s0 
z
zs
0
 
Weak focusing
 r 
By x   B0 

r  x 
n
1/ 2
zs  z0 cossK 1/ 2  z
sin sK 1/ 2 
0K
if K  0
zs  z0  z
0L
if K = 0

zs  z0 cosh sK
1/ 2

 z
0K
1/ 2

sinh sK
1/ 2

if K  0
 cosK 1/ 2 L
K 1/ 2 sin K 1/ 2 L

M F  
1/ 2
K 1/ 2 sin K 1/ 2 L

cos
K
L






1 L
M o  
FODO transfer matrices

0 1 
1/ 2
1/ 2
 cosh K 1/ 2 L
 
K  sinh K  L

MD 
1/ 2
K 1/ 2 sinh K 1/ 2 L
cosh
K
L








1
K

1 n 
x


n
r2
kx  k y   2  
r
K y   n2

r


d 2 x 1
d 2 x
2

1 nx  0 

  0 1 n x  0 
2
ds2 r 2 
  1 n 0
 dt2
  x
 2

d y  1 ny  0  d y  n 2 y  0
 y  n 0
2

dt 2
r2
ds


Strong focusing



Smaller pipe
Smaller magnet
Reduced cost



cosmotron
AGS
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Unperturbed equation of motion
Hill equation
periodicity condition
d 2z
 K sz  0
2
ds
K s  C   K s
envelop
solution : Floquet theorem
zs    s  cos s  0 
  and  depend on the lattice arrangement
  and 0 depend on the initial conditions of
 s  S   s
s

s0
1

cos-like orbit
Here ß is NOT the
-function (periodic) relative speed v/c
s /s0    s   s0  
 
the trajectory
ds
 s
phase advance
between s0 and s
sin-like orbit
phase advance variation
2  2 
 2K     1
 0
  4 
envelop equation
several orbits
Courant & Snyder invariant and more
Courant-Snyder parameters
 s   
 s 
 s
Poincaré section
1  s 
 s  
 s 
2
z’
2
zs     s  cos s   
0


zs     scos s    sin  s   
0
0
 s

C
s
S0
Slope=-/
Courant-Snyder invariant

  z  2    z  z   z  
2
2

emittance


phase plane (z,z’)
 beam is the (1·rms) beam emittance if the area beam
encloses 39 % of the circulating particles N



 is the particle emittance
 is the area of the ellipse mapped turn by turn in the

z’




z




Liouville theorem


z



Aellips  

beam/N is a constant of the motion (Liouville theorem)
The Liouville theorem holds in absence of acceleration, losses, scattering effects and
radiation emission

Adiabatic invariant
The Courant-Snyder invariant emittance ε decreases if we the accelerate the particle.
This is called “adiabatic damping”
(a pure cinematic effect, since there is no damping process involved).
The slope of the trajectory is z’ = pz/ps.
Accelerate the particle: ps increases to ps+∆ps, but pz doesn’t change => slope changes.
p
ps
pz
z’
p+p
ps+ps
pz
z’+z’

pz

z



ps

pz
z z
 p1  

p

p

s
s
  z2  z 2    2zz 2zz (assuming  = 0)
z  z    2zz 2z  2 sin    0 
2
d
dp
   
   

p
p
 p  p0   0
p
2
 -> Courant Snyder parameter

Invariant of the motion
 In a stationary Poincaré section
-> 
 In an accelerating Poincaré section -> 
normalized emittance
n  

v



 = the relative speed
c

 = the relativistic factor  2  1
1  2

Stability of the motion
Transfer matrix from s0 to s



 cos    0  sin  
0

M s / s 0   
  0    cos   1   0  sin 
 
 0
0





0
 cos     sin  



 0  sin 
One turn transfer matrix
cos    0  sin 
 0  sin  
M s0  C /s0   

cos    0  sin 
  0  sin 

cos 

 
0


 0


 0




Slope=-/

z’



A
1
ellips  
  M 11  M 22  Condition for the stability of the motion


2
M
tr M  M1,1  M 2.2  2
 12

sin 

Condition for the invariance of 
Int(Q y)+1

M
  21
det M  1    costant of motion
sin 

Q
y
M 11  M 22
 
2  sin 
tune Q 

 
z



 


2
4D resonance condition --> order n x  n y
n x  Qx  n y  Q y  n
Int(Q y)
Int(Q x)
Qx
Int(Q x)+1
Exact & approximate solution
Exact solution of the Hill equation


 cos    0  sin   zs0    0  sin   zs0 
zs 
0


  

1  0
0

0

z
s



cos



sin


z
s

 cos     sin   zs0 






0
 



 0


0

Exact solution in compact form

zs   s coss  0  with




s 
1

1
Q

 s
2 2

Ten cell lattice

S

Approximate solution (smooth approximation)
1
C
  sds Q  2   s  2 
1
C
ds
C
S
L2
R TrM L,0  2  2  0.77  2 stable motion
4f


zs    cos  t  0  with    Q 0

ds
This is a pseudo-harmonic oscillation modulated both in amplitude and in frequency
Q is the total number of oscillation per turn
The phase advances faster in the sections with a smaller 
 

  s
Cell length
 L = 1 m
Ring length C = 10 m
Focal length f = 0.45 m

  L
sin  
 0.56    1.178 (67.5o )
2  4 f

 
L1 sin  
2 

 0 
 1.68 m
 0  0
sin 
S  2Q  10  11.78  Q  1.8748
R 
S
 1.59 m
2
 
R
 0.85 m
Q
Perturbed equation of motion
Solution with dipoles, quadrupoles sextupoles and octupoles
d 2z
 K sz  F s
ds2

Fx x, y,s 

 kx  12 mx 2  y 2  16 rx 3  3xy 2  ...
r
dispersion
chromaticity sextupole
octupole
Fy x, y,s  kx  12 mxy  16 r3x 2 y  y 2  ...
dispersive orbit
x  x   x co  x  ...

y  y   y co  ...
betatron motion
nonlinear terms
closed orbit
Uncoupled motion (x-plane)
orbit distortion
dispersive orbit
betatron oscillation



 x co  K sx co   r1   12 mxco2  ...
with
x K sx  r1   r1  2  Kx  ...
with
2
1

x 

K
s
x

K

mD

x

mx







  ...
2

natural
chromaticity
geometric
chromaticity
correction by
aberration
sextupoles
B
B
x  Ds

aberration

Dipolar and quadrupolar field errors
Dipole error
Bs
d 2z
 K sz 
2
ds
Br

B localized in sk over the length L (kick approximation)
 BL 
Br
Periodicity of the closed orbit
zsk  zsk  C
 zsk  



M
s

C,s


 


k
k  
zsk  zsk  C
zsk    
At every turn the perturbation is compensated
zco s 
  s sk 
coss  sk   Q
2 sin Q
At every turn the perturbation is enhanced

Q = 1/2 integer
Q = integer

kick
Avoid tune close to integer
Quadrupole error
kick
B’ localized in sk over the length L (thin lens approximation)
Bs
d 2 z 
 K s 
z  0
2
ds 
Br 
1  BL
1

  kL
Q 
0 sk  kL first order
4
f
Br
 s
 s 
  kL 0 k cos2o s  o sk   Q second order
 0 s
2sin 2Q0
Avoid tune close to 1/2 integer - the range of 
forbidden tunes is called stop-band

Momentum dispersion and chromaticity
d 2z

 K sz 
2
ds
rs
Design orbit
Design orbit
First order solution
xs   s coss  0  Ds
Chromatic
close orbit
On-momentum
particle trajectory
Off-momentum
particle trajectory
Dispersion function
Ds 
 s

2sin Q
 u
 du ru cosu  s  Q
Divergent for Q=integer
C

Qnatural
 Q

4 Qnatural
m1L1  
Ds1  x s1 
Gradient error induced by momentum dispersion
K  K  Q  

4
 dssks  Q
C
Chromaticity correction with sextupoles


x  k 1  x  12 mx 2  y 2 

r

y  k 1  y  mxy  0
  
Qx,y
1
4
 ds smsDs  k sks
x,y
C
Sextupole strength

BT/m 2 
B
3
m
with mm  0.2998
Br
pGeV/c 
x,y
m 2 L2  

sext H

4 Qnatural
sext V
Ds2  y s2 
Why chromaticity should be corrected




The beam rigidity increases with p
K decreases with p
the tune decreases with p
Q’ is negative



Q’ non zero produces a tune shift with p
In a beam Q’ produces a tune spread
Be aware of resonance crossing
Int(Qy)+1
Qy
Low energy particles
High energy particles
Int(Qy)
Int(Qx)
Qx
Int(Qx)+1
Dynamic aperture
Nonlinear fields imply multiple traversal of resonances





Emittance distortion and growth
Tune shift and spread with the amplitude
Coupling of the degrees of freedom
Chaotic motion
Particle loss -> dynamic aperture
Phase space with only linear fields
Distortion
due to
sextupoles
Distortion
due to
octupoles
Low ß insertion
A low-ß insertion is used to focalize the
beams at the collision point
 This is achieved with triplets or doublets
of quadrupoles
 In the drift space where the
experimental devices sit ß growth with
the square of the length

  s 2 

 s   * 
*  
1 
   
 * 
ds
1 L
  * 
 2tg  *   since L*>>*
 L*  s
 
1
L*
*
* 2
ˆ
   L 
The chromaticity induced by the triplet can be large (local correction scheme may be needed)
Acceleration mechanism
Longitudinal stability
Momentum compaction

1
 tr
df
2
  dp f 
Slip factor


C
dp

dR
dT
T
dp
p

R
dp
p
p

dC
p
1

2

1  measures how closely packed
  2 orbits with different momenta are
Q
1
 tr
2
 tr  Q
 measures how how much offmomentum particles slip in time
relative to on-momentum ones

Phase stability
principle
< tr
 B is late
respect to A
 B will receive a larger voltage and
will increase its speed
 B will be closer to A one turn later
> tr
 B is late respect to A
 B will receive a smaller voltage and
will see a shorter circumferential
path
 one turn later B will be closer to A

small
Ý
Ý s2  0 oscillations

c 2 heVˆ cos  s
2
s   
R 
2E s
LHC luminosity
Performances limitations
protons
in a bunch
Luminosity:
L=
event rate
cross section •
=
1
•
no. of bunches
revolution frequency
N1 N2 k f
for equal, round, bi-Gaussian beams: N
•
S
beam cross section
2
1 N2 = N
S --> 4š  2
* =



*
L=
invariant emittance

L=
*
Transverse beam density:
• head-on beam-beam
• space-charge in the injectors
• transfers dilution
N
*
N kf 
 

N
²t
Beam current:
• long range beam-beam
• collective instability
• synchrotron radiation
• stored beam energy
Head-on beam-beam:
detuning
 rp N

 nb. of interactions Š 0.02
Lecture II - single particle dynamics
reminder

In a circular accelerator the guiding fields provide the required forces
to keep the particles in a closed orbit along the magnet axis

Strong focusing allows building much smaller magnets and is a
fundamental progress respect to weak focusing

The particle trajectory is a pseudo-harmonic function modulated both
in amplitude and phase rather well approximated by a sinusoidal
function oscillating at the betatron frequency (tune•revolution
frequency) with an amplitude proportional to the square root of the
emittance

The imperfections of the guiding field and of the momentum particles
produce resonances and eventually chaotic motion

The low ß insertions are basic devices to focus the beam size at the
collision point of a collider ring

The phase stability principle guarantees the stability of the longitudinal
motion