2.3 Off-momentum closed orbit and dispersion function
We have discussed the closed orbit for a reference particle with
momentum p0, including dipole field errors and quadrupole
misalignment. By using closed-orbit correctors, we can achieve
an optimized closed orbit that essentially passes through the
center of all accelerator components. This closed orbit is called
the “golden orbit,” and a particle with momentum p0 is called a
synchronous particle. However, a beam is made of particles
with momenta distributed around a synchronous momentum
p0. What happens to particles with momenta different from p0?
Here we study the effect of off-momentum on the closed orbit.
For a particle with momentum p, the momentum deviation is
∆p=p-p0 and the fractional momentum deviation is δ=∆p/p0,
which is typically small of the order of 10–6 to 10–3. Since δ is
small, we can study the motion of off-momentum particles
perturbatively.

 1

x   2  K ( s )  x 



K ( s )  K1 ( s ) 
B
B1
, B1  z
x
B
The bending angle resulting from a dipole field is different for
particles with different momenta. i.e. nonzero δ. The resulting
betatron equation of motion is inhomogeneous. The solution of an
in-homogeneous linear equation of motion is a linear superposition
of the particular solution and the solution of the homogeneous
equation, i.e.
x  x  D
x  x  D
x  K x( s ) x  0,
D  K x( s ) D 
1
K x( s ) 
1
2
 K ( s)

The solution of the homogeneous equation is the betatron oscillation
we have discussed earlier. The solution of the inhomogeneous
equation is called the dispersion function, or the off-momentum
closed orbit.
x 
x
B p
x
  z 0 (1  ) 2 ,
2


B p
z   
Bx p0
x
(1  ) 2 .

B p
0
K ( s )  K1 ( s ) 
p
p  p0  p,  
p0
x 
 1
x  1
    Kx 
2

 1 
 
B1 
B1
,
B
Bz
x

x x2 
1  2  2 
  

 1 
K (s) 

 x 

x   2



(
1

)
1


(
1
 )


 1


x   2  K ( s )  K x ( s )  x 
,
 (1   )



K 
z  0
z    

 ) 
(
1

z   K  K z z  0,
 2

K x ( s )    2  K ( s )   O( 2 )
 

K z   K  O( 2 )
x  x  xco  x  D

 1
1
D   2  K ( s )  D  ,



For a pure dipole (K=0):
1
 cos 
 sin   (1  cos  ) 



1
sin 
M     sin  cos
  0



0
1
0

0
 2 


1 

0 1


1
2
Example: FODO cell
For pure quadrupoles:
0
0 0
 1


0     1/f 0 0 
1   0
0 1 

1
sinh | K | 0 
 cosh | K |
|K |
 1
0 0


 

0   1/f 0 0 
M ( s, s0 )  | K | sinh | K | cosh | K |

0 1 
0
1   0
0


1
sin K 
 cos K 
K

M ( s, s0 )    K sin K  cos K 

0
0

Closed orbit condition:
Using the Courant-Snyder parameterization for the transfer matrix, we
obtain
For combined function magnets:
The dispersion is proportional to the cell length L times the bending
angle θ, and inversely proportional to the square of the phase advance.
The dispersion at other locations can be obtained by using the 3×3
transfer matrix M(s2,s1).
The AGS (33 GeV proton synchrotron built in 1960) is made of 60
(5×12) FODO cells. The CPS (28 GeV) is made of 50 FODO cells.
The betatron amplitude
functions for one superperiod of
the AGS lattice, made of 20
combined-function magnets.
The upper plot shows βx (solid)
and βz (dashed). The middle
plot shows the dispersion
function Dx. The lower plot
shows schematically the
placement of combined-function
magnets. The superperiod can
be approximated by five FODO
cells. The phase advance of each
FODO cell is about 52.8o.
 max 
L1
) 2 L1 (1  sin  )
2f
2

sin 
sin 
2 L1 (1 
What is the effect of bending radius on dispersion function?
Ldip
DF / D
L [1  12 sin( / 2)]

sin 2 ( / 2)
L=10m, LQ=1m, P=18
θ=π/P
Damping
re-partition
number
Ldip = 2, 4, 6, 8 m
Example: APS lattice is made
of 40 Double-bend Achromat
(DBA) cells with a total length
of 1104m. The momentum
compaction factor for all DBA
lattice is αc=ρθ2/(6R).
Because of its simplicity and
flexibility, DBA lattice is
commonly used as basic cells
of synchrotron light source
design.
Question: can you
identify dispersion phase
space coordinates with
accelerator elements
The transfer matrix of a periodic cell: D=MD
The normalized betatron phasespace coordinates:
y2+Py2=y2+(αy+βy’)2=2βJ.
Define the normalized
dispersion function coordinates:
The H-function of the dispersion invariant is defined as:
In a straight section, Jd is invariant and Φd, aside from a constant,
is identical to the betatron phase. In a region with dipoles, Jd is not
constant. The change of dispersion function across a thin dipole is
D = 0 and D′ = θ, i.e.
x' ' K x ( s ) x 
xco ( s ) 
B
B
 (s)
2 sin 
s C
 ds
 ds G(s, s )
0
0
s
 ( s0 ) cos[  |  ( s )  ( s0 ) |]
s
s C

0
B ( s0 )
B
For a FODO cell, the dispersion H-function at the defocussing
quadrupole is larger than that at the focusing quadrupole, i.e. HF≤HD,
where
B ( s0 )
B
∆
We obtain the Integral representation of the dispersion function:
Left: Normalized dispersion phase-space coordinates Xd and Pd are plotted in one
superperiod of the AGS lattice. Right: the coordinates are shown in Xd vs Pd. The
scales for both Xd and Pd are m1/2. Note that Xd is indeed nearly constant, and
(Pd,Xd) propagate in a very small region of the dispersion phase-space
Path length, momentum compaction and phase-slip factors:
How to measure D(s)? xco(s)=D(s).
The path length of Frenet-Serret coordinate system is
d  (1  x ) 2 (ds ) 2  (dx) 2  (dz ) 2  ds[(1  x ) 2  x2  z 2 ]1/ 2  ds[1  x ]
x  D
x
C   d   ds   ds  C0   
ds
C  
D


ds ,
K=45 – 270 keV
I=0-4 A

1
dC 1 D
c 
  ds    Di  i
Cd C 
C i
Here αc is called the momentum compaction factor, which is a
measure of the compactness of the orbit length for particles with
different momenta. The orbiting time for particle is T=C/v. Thus
C=86.8 m
Here η is called the phase slip-factor. Thus
An accelerator with circumference 360 m is made of 18 FODO
cells. The betatron tunes of the synchrotron are νx=4.793 and
νz=4.783 respectively. If one of the focusing quadrupoles has
1% error, estimate the maximum change in ΔDx.
1
∆
∆
⋯
xF
xD
f
DxF
DxD
35.0
5.17
6.73
4.33
1.99
Effect of half-integer stopband on dispersion function
/
Floquet transformation: η=ΔD/, ϕ=(1/ν)ds/
/
/
1
2
1
2
35 ∆
1
4.33
≅
/
0.0061
cos
For the above problem, we find |η|~0.03 m,
and the maximum ΔD~0.18 m.
We know the effect of
half integer stopbands on
bettatron amplitude
functions!
What is the perturbation on
the dispersion function?
Qx=4.406
What happens if the −2% is changed to +2%?
Dispersion Suppression and Dispersion Matching
The first-order achromat theorem states that a lattice of n repetitive
cells is achromatic to first order if and only if Mn = I or each cell is
achromatic. Here M is the 2 × 2 transfer matrix of each cell, and I
is a 2 × 2 unit matrix.
Left: Normalized dispersion
phase-space coordinates Xd and
Pd of the IUCF Cooler lattice are
plotted. Right: The coordinates
are shown in Xd vs Pd at the end
of each lattice elements. The
accelerator is made of six 60◦bends forming a 3 double-bend
achromat modules. The scales for
both Xd and Pd are m1/2.
Achromat Transport Systems
The double-bend achromat
thus
The phase-slip factor and synchronization:
The (momentum) compaction factor αc is defined as
c 
1
dC 1 D
  ds    Di  i
Cd C 
C i
which is a measure of the compactness of the orbit length for particles
with different momenta. The importance of the orbit length is that the
particles in synchrotron must synchronize with the rf accelerating
voltage. Since the orbiting period is T=C/v, we find
Here η is called the phase slip-factor.
Phase stability and the synchrotron equation of motion:
-- stability of synchrotron motion
V=V0sin(ωrft+φ)
ωrf=hf0
V=V0sin(ωrft+φ)
ωrf=hf0

1 p
1 E
 2
 2 2
  p0   E0
E

E 0
 0

E  E 0
0
(
1
0

d  E 
1  E  E 0  (1 / 0 )

)E 
E
E  
dt  0 
0
E

Illustration of the Phase stability: A
beam bunch consists of particles with
slightly different momenta. A particle
with momentum p has its own offmomentum closed orbit Dδ. Since the
energy gain depends sensitively on the
synchronization of rf field and particle
arrival time, what happens to a particle
with a slightly different momentum when
the synchronous particle is accelerated?
Lower Energy
Synchronous Energy
Acceleration
V0
Higher Energy
0
φs
η>0
Deceleration
0
π/2
En 1  En  eV (sin n  sin s )
φs
π/2
η<0
−V0
Synchrotron equation of motion:
π
φ
3π/2
The key answer is the discovery of the phase stability of synchrotron motion by
McMillan and Veksler. If the revolution frequency f is higher for a higher
momentum particle, i.e. df/dδ>0, the higher energy particle will arrive at the rf gap
earlier, i.e. φ<φs. Therefore if the rf wave synchronous phase is chosen such that
0<φs<π/2, higher energy particles will receive less energy gain from the rf gap.
Similarly, lower energy particles will arrive at the same rf gap later and gain more
energy than the synchronous particle. This process provides the phase stability of
synchrotron motion. In the case of df/dδ<0, phase stability requires π/2<φs<π.
2π
The separatrix orbits for η > 0
(γ>γT) with ϕs=2π/3, 5π/6, π,
(top) and for η < 0 (γ<γT)
with ϕs= 0, π/6, π/3 (bottom).
The phase space area
enclosed by the separatrix is
called the bucket area. The
stationary buckets that have
largest phase space areas
correspond to ϕs=0 (bottom)
and π (top) respectively.
n 1  n 
2
En 1
 2E
Summary:
1. Particle motion in an accelerator can be described by 3D simple
harmonic motion. The transverse oscillations is called betatron
motion and the longitudinal motion is called the synchrotron
motion.
2. The betatron tunes are number of betatron oscillations per
revolution, and the synchrotron tune is the number of synchrotron
oscillations per revolution. The betatron tunes increase with the
size of the accelerator, while the synchrotron tune is about 10–4 to
10–2.
3. The momentum compaction factor plays an important role in the
accelerator. Typically, the momentum compaction factor for
FODO cell lattice is αc~1/νx2. Thus the transition energy is γT~νx.
However, the momentum compaction for accelerators can be
changed by changing the dispersion function in dipoles.
C  
D

ds ,
c 
1
dC 1 D
  ds    Di  i
Cd C 
C i
Dispersion function plays a very important role in the performance
of high energy and synchrotron light source accelerators. For the
synchrotron light source, the H-function plays a particular
important role in determining the natural emittance of electron
beams, i.e.
55
 x  FCq 2 3 , Cq 
 3.83 10 13 m
32 3mc
The factor F is lattice dependent factor, F~1 for FODO cell,
F~1/(4√15) for DBA lattice and F~1/(12√15) for the minimum
emittance lattice.
Pulse from
linac
Extracted beam
Lambertson septum
Kicker 1
Kicker2
1.
2.
Beam in and out in one
revolution satisfies the
CRANE requirement of
steady state experiment.
The accelerator can
accumulate 250 nC of
charge in 10 or more
turns and extracted in one
turn for transient mode
experiment (15J).
Note that a large compaction factor is necessary for achieving de-bunching
for the electron beams in a single path!