# The 4th OVERSEAS CHINESE PHYSICS ASSOCIATION ACCELERATOR SCHOOL

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The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR
SCHOOL
Longitudinal dynamics

mhwang@nsrrc.org.tw
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.2
2006/7/27
•Longitudinal dynamics
–EM field in RF cavity
–Transit time factor
–Phase stability
–Equation of longitudinal motion
–Longitudinal phase space
–The synchrotron mapping equation
Outline
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.3
2006/7/27
R
L
EM field in RF cavity
R
c
R
c
R
c
J
tr
c
J
c
E
trB
tr
c
JEtrE
xJxJ
r
c
J
c
iE
rBerBtrB
r
c
JErEerEtrE
z
ti
z
ti
zz
405.2,405.20)(
mode TM010
sin)(),(
cos)(),(
)()(
)()(,)(),(
)()(,)(),(
0
1
0
00
10
1
0
00
==⇒=
−=
=
−=

==
==
ω
ωω
ω
ω
ω
ω
ω
ω
φ
φ
ω
φφ
ω
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.4
2006/7/27
Let the cavity gap be L, electric field amplitude be E, and speed of
the particle bev, the energy gain in a cavity with sinusoidal
varying electric field is reduced by a transit time factor.
vLu
u
u
TLTedz
z
eE
L
L
2/,
sin
,cos
trtr
2/
2/
ω
υ
ω
==Ε=Ε=Δ

Transit time factor
An efficient design of an RF cavity will have a large Ttr. For
example, one might want T
tr.= 0.9. This means u = 0.8. For a
relativistic beam with v ~ c, this means
3
2
8.0
2
405.2
2
≈⇒≈=
R
L
R
L
c
L
ω
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.5
2006/7/27
V=V0sin(ωrft+φ s)
ωrf=hω 0
We assume the longitudinal voltage across an RF cavity is
0rfsrf0
), t sin( V V
ω
ω
φ
ω
h=+=
What happen if = 200.000001?
0
rf
ω
ω
Phase stability
electron ofper turn loseenergy the:U

V
U
sinor
V
U
sin ,sin VU
0
0
0
1
s
0
0
1
ss00
ee
e
−−
−===
πφφφ
rf
ω
:RF frequency
:harmonic number
:RF phase angle of the synchronous particle
s
φ
h
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.6
2006/7/27
•Once ω rf
is chosen, the
beam —at least its
synchronous particle —will
in such a way that it becomes
exactly equal to ω rf
/h even
though its initial ω 0
is slightly
off
•A particle with slight
deviations in z,
δ
from the
synchronous particle will
oscillate around the
synchronous particle, and
these deviations will not grow
indefinitely with time
.
0n transitioabove>
η
s
φ
Phase stability
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.7
2006/7/27
Longitudinal (synchrotron) equations of motion
particle. theof timearrival at the phase RF
v
z
,
1
0
2
00
==
±=
Δ
=
Δ
=
t
E
E
p
p
rf
ωφ
τ
β
δ
vv
CC
T
v
v
C
C
T
TT
T
E
eV
eV
T
E
rfrf
s
s
Δ+
Δ+
=
Δ

Δ

=
Δ
=
−=
−==Δ
0
0
00
0
0
2
0
0
0
0
0
0
),(
)sin(sin
2
)sin(sin
2
turnonein deviation energy
ωω
φ
φ
φφ
βπ
ω
δ
φφ
π
ω
&
&
&
0
0
000
,0particle ssynchronou
v
C
TTEE,p, pz=====→
circumference
velocity
z of definitionby
z: longitudinal displacement relative to synchronous particle
)',(),,( yyxx

),(
δ
z
Transverse coordinate Longitudinal coordinate

0
τ
+=nTt
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.8
2006/7/27
noscillatio harmonic simple ,0))(cos
2
()(
)(cos
2
motion ofequation thelinearizecan we,1 i.e. small is if
0
2
0
2
0
2
2
0
2
0
0
0
0
≈−−−
−≈
<<−−
sss
ss
ss
E
heV
dt
d
E
eV
φφφ
βπ
ηω
φφ
φφφ
βπ
ω
δ
φφφφ
&
)(
ss
φ
φ
φ
φ
−+=
c
transc
α
γ
γ
αη
1
,
1
factor slippage phase
2
0
=−≡
c
α
η
γ
≈>>,1
0
δ
γ
δα
δ
ρ
θρ
θρθρ
2
0000
1
,
)(
=
Δ
==
−+
=
Δ

∫∫
v
v
C
ds
D
d
ddx
C
C
c
factor compaction
momentum

0
C
ds
D
c

=
ρ
α
δ
Dxx
D
==
ηδωφ
rf
=
&
)sin(sin
2
)(
0
2
0
2
0
2
2
0
srfs
E
h
eV
dt
d
φφ
βπ
ω
ηδηωφφφ
−==−=
&&&
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.9
2006/7/27
n tunesynchrotro is , cos
2
0
s
s
0
2
0
0
0s
ω
ω
νφ
βπ
η
ωω
=−=
s
E
heV
0cosor ,0cos
2
0
2
0
2
0
0
<<
ss
E
heV
φηφ
βπ
ηω
The stability of the motion require that
If above condition is true, the simple harmonic oscillation
angular frequency is
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.10
2006/7/27
Longitudinal phase space
]sin)(cos[cos
22
1
),(
)sin(sin
2
0
2
0
0
0
2
0
2
0
0
0
sssrf
rf
s
E
eVH
H
T
H
E
eV
φφφφφ
βπ
ω
ηδωδφ
δ
ηδω
φ
φ
φ
φφ
βπ
ω
δ
−+−+=

==
Δ
=

−=−=
&
&
The first term of the Hamiltonian resembles the kinetic energy part,
and the second term in the Hamiltonian can be visualized as the
potential. Stable particle motion is bounded by the potential well.
The area of stable motion is called rf bucket.
-2
0
2
4
6
8
10
12
0510152025
-2
-1
0
1
2
3
4
5
6
7
0510152025
-2
0
2
4
6
8
10
12
14
16
0510152025
φ s=45o
φ s=30o
φ s=20o
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.11
2006/7/27
The Hamiltonian torus that passes through the unstable fixed point
is called the separatrix.
]sin)2(cos2[
2
)0,(),(
sss
0
2
0
00
ssx
φφπφ
πβ
ω
φπδφ
−+−=−==
E
eV
HHH
0]sin)(cos[cos
sss
0
2
0
0
2
sx
=−−−++
φφφπφφ
ηπβ
δ
hE
eV
The phase space area enclosed by the
separatrixis called the rf bucket, where
particle motion around the stable fixed
point is elliptical. The motion around the
unstable fixed point is hyperbolical. The
bucket area is defined as
)0( UFP
)0( SFP
,0,0point fix
,
,
=−=
==
==
δφπφ
δφφ
δφ
s
s
&&
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.12
2006/7/27
The bucket area is defined as
2/1
s
s
s
2/1
0
2
0
0
s
2/1
ssssb
sbsb
s
s
sb
0
2
0
0
sxB
sin
2
2
cos)
||
2
( isheight bucket
,)-( islength bucket
]sin)(cos[cos
24
1
)(
sin1
sin1
)()(
|cos|||
16
)(
||2
16)(
~
φ
φπ
φ
ηπβ
δ
φφπ
φφφπφφ
η
η
φα
φ
φ
φαφα
φη
ν
φα
ηπβ
φφδ
φπ
φ

−=

−−−+−=
+

≈=
==

hE
eV
h
hE
eV
dA
B
u
s
s
u
u
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.13
2006/7/27
π
ν
φ
β
ηπ
1
1cos
2
0
2)(
0
0
2
0
<
<−<

s
s
eV
E
h
Mtrace
1
2
1
01
2
),sin(sin
++
+
Δ+=
−+Δ=Δ
nnn
snnn
E
E
h
eVEE
β
ηπ
φφ
φ
φ
V=V0sin(ωrft+φ s)
ωrf=hω 0
0)(
2
)(
ringtheamong
0
2
0
=ΔΔ
Δ=ΔΔ
E
E
E
h
β
ηπ
φ
φφ
φφ
φ
Δ≈
−=ΔΔ
=ΔΔ
s
s
eV
eVE
cos
)sin(sin)(
0)(
RFcavity
p
assing
0
0
Synchrotron mapping equation
in
s
out
E
eV
E

Δ
Δ

=

Δ
Δ
φ
φ
φ
1cos
01
0
inout
E
E
h
E

Δ
Δ

=

Δ
Δ
φ
β
ηπ
φ
10
2
1
0
2
0

+
=

Δ
Δ
=

Δ
Δ

=

Δ
Δ
+
1cos
2
cos
2
1
1cos
01
10
2
1
0
0
2
0
0
0
2
0
0
0
2
0
1
s
s
nn
s
n
eV
E
h
eV
E
h
M
E
M
E
eV
E
h
E
φ
β
ηπ
φ
β
ηπ
φφ
φ
β
ηπ
φ
One turn map
The 4th OVERSEAS CHINESE PHYSICS
ASSOCIATION ACCELERATOR SCHOOLp.14
2006/7/27
Reference
•S. Y Lee, “Accelerator Physics”, 2nd ed. (World
Scientific 2004).
•Alex Chao, “Lecture Notes on Accelerator physics”.
•D.A. Edward and M.J. Syphers,”AnIntrouctionto the
physics of high energy accelerator”, (Wiley, 1993).