LONGITUDINAL DYNAMICS
Boris Vodopivec
7XWRULDOH[HUFLVHV
)0)
Joël Le DuFF
summary
•RadioFrequency Acceleration and Synchronism
•Properties of RadioFrequency cavities
•Principle of Phase Stability and Consequences
•Synchronous linear accelerator
•The Synchrotron
•RF cavities for Synchrotron
•EnergyPhase Equations in a Synchrotron
•Phase space motions
Bibliography:Old Books
M. Stanley LivingstonHighEnergyAccelerators
(IntersciencePublishers, 1954)
J.J. LivingoodPrinciplesof cyclicParticleAccelerators
(D. Van NostrandCoLtd, 1961)
M. Stanley LivingstonandJ. B. BlewettParticleAccelerators
(Mc GrawHill Book Company, Inc1962)
K.G. SteffenHighEnergyoptics
(IntersciencePublisher, J. Wiley& sons, 1965)
H. BruckAccelerateurscirculaires de particules
(PUF, Paris 1966)
M. Stanley Livingston(editor) Thedevelopmentof HighEnergyAccelerators
(DoverPublications, Inc, N. Y. 1966)
A.A. Kolomensky& A.W. LebedevTheoryof cyclicAccelerators
(NorthHollandPublihersCompany, Amst. 1966)
E. Persico, E. Ferrari, S.E. SegrePrinciplesof ParticlesAccelerators
(W.A. Benjamin, Inc. 1968)
P.M. Lapostolle& A.L. SeptierLinearAccelerators
(NorthHollandPublihersCompany, Amst. 1970)
A.D. VlasovTheoryof LinearAccelerators
(Programmfor scientifictranslations,Jerusalem1968)
Bibliography:New Books
M. Conte, W.W. Mac Kay AnIntroductionto the Physicsof particleAccelerators
(World Scientific, 1991)
P. J. Bryant andK. JohnsenThe Principlesof CircularAcceleratorsandStorageRings
(Cambridge UniversityPress, 1993)
D. A. Edwards, M. J. SyphersAnIntroductionto the Physicsof HighEnergyAccelerators
(J. Wiley& sons,Inc,1993)
H. WiedemannParticleAcceleratorPhysics
(SpringerVerlag, Berlin, 1993)
M. ReiserTheoryandDesignof ChargedParticlesBeams
(J. Wiley& sons, 1994)
A. Chao, M. Tigner
Handbookof AcceleratorPhysicsandEngineering
(World Scientific1998)
K. WilleThe Physics of Particle Accelerators: An Introduction
(Oxford University Press, 2000)
E.J.N. WilsonAn introduction to Particle Accelerators
(Oxford University Press, 2001)
Methods of Acceleration
1_ Electrostatic Field
Energy gain: W=n.e(V
2V1)
limitation : Vgenerator
=Σ
Vi
2_ Radiofrequency Field
Synchronism:
L=vT/2
v=particle velocity
T= RFperiod
Wideroestructure
Electrostatic accelerator
The advantage of Resonant Cavities
Considering RF acceleration, it is obvious that when particles get high
velocitiesthe drift spaces get longer and one loses on the efficiency. The
solution consists of using a higher operating frequency.
The power lost by radiation, due to circulating currents on the electrodes,
is proportional to theRFfrequency. Thesolution consistsofenclosingthe
systemin a cavitywhich resonant frequency matches the RF generator
frequency.
Each such cavity can be independently
powered from the RF generator.
The electromagnetic power is now
constrained in the resonant volume.
Note however that joule losses will
occur in the cavity walls(unlessmade
of superconductingmaterials)
ωRF
E
z
r
JouH
rr
The Pill Box Cavity
0
2
2
0
0
2
=
∂
∂
−
∇
tA
A
μ
ε
)(HouEA
=
(
)
kr
J
E
z
0
=
()
kr
J
Z
j
H
1
0
−=
θ
Ω====37762,2
2
0
Z
a
c
k
λ
ω
λ
π
e
tj
ω
}
E
z
Hθ
From Maxwell’s equations one can derive
the wave equations:
Solutions for E and H are oscillating modes,
at discrete frequencies, of types TM orTE.
For l<2a the most simple mode, TM010, has
the lowest frequency ,and has only two field
components:
The Pill Box Cavity (2)
The design of a pillboxcavity can
be sophisticated in order to
improve its performances:
A nose cone can be introduced in
order to concentrate the electric
fieldaround the axis,
Round shaping of the corners
allows a betterdistribution of the
magnetic field on the surface and a
reduction of the Joule losses.
A good cavity is a cavity which
efficiently transforms the RF
power into accelerating voltage.
Transit Time Factor
t
g
V
t
EE
z
ωω
coscos
0
==
v
t
z
=
Oscillating field at frequencyωand which amplitude
is assumed to be constant all along thegap:
Consider a particle passing through the middle of
the gap at time t=0 :
The total energy gain is:
dz
v
z
g
eV
W
g
g
∫
=Δ
−
2/
2/
cos
ω
eVTeVW==Δ
2/
2/sin
θ
θ
angletransit
v
g
ω
θ
=
factortimetransitT
( 0 < T < 1 )
Transit Time Factor (2)
(
)
dzezEeW
tj
g
ze
ω
∫
ℜ=Δ
0
()
⎥
⎦
⎤
⎢
⎣
⎡
∫
ℜ=Δ
−
dzezEeeW
v
z
j
g
z
j
e
p
ω
ψ
0
()
⎥
⎦
⎤
⎢
⎣
⎡
∫
ℜ=Δ
−
g
v
z
j
z
j
j
e
dzezEeeeW
i
p
0
ω
ψ
ψ
()
φ
ω
cos
0
∫
=Δ
g
v
z
j
z
dzezEeW
(
)
()
∫
∫
=
g
z
g
tj
z
dzzE
dzezE
T
0
0
ω
p
v
z
t
ψωω
−=
ip
ψ
ψ
φ
−
=
Ψp
is the phase of the particle entering the gap with respect to the RF.
and considering the phase which yields the
maximum energy gain:
Consider the most general case and make use of complex notations:
Introducing:
Let’s consider a succession of accelerating gaps, operating in the 2πmode,
for which the synchronism condition is fulfilled for a phase Φ
s .
For a 2πmode,
the electric field
is the same in all
gaps at any given
time.
s
Ve
s
eVΦ=sin
ˆ
is the energy gain in one gap for the particle to reach the next
gap with the same RF phase: P
1 ,P2, ……are fixed points.
Principle of Phase Stability
If an increase in energy is transferred into an increase in velocity, M1 & N1
will move towards P1(stable), while M
2
& N2
will go away from P
2
(unstable).
Transverse Instability
00<
∂
∂
⇒>
∂
∂
z
z
E
t
V
Longitudinal phase stability means:
The divergence of the field is
zero according to Maxwell :
000.>
∂
∂
⇒=
∂
∂
+
∂
∂
⇒=∇
x
E
z
E
x
E
E
x
z
x
defocusing
RF force
External focusing (solenoid,quadrupole) is then necessary
A Consequence of Phase Stability
The Traveling Wave Case
(
)
()
0
0
cos
ttvz
v
k
kztEE
RF
RFz
−=
=
−
=
ϕ
ω
ω
velocityparticlev
velocityphasev
=
=
ϕ
The particle travels along with the wave, and
k represents the wave propagation factor.
⎟
⎠
⎞
⎜
⎝
⎛
−−=
00
cos
φωω
ϕ
t
v
v
tEE
RFRFz
00
cos
If synchronism satisfied:
φ
ϕ
EEandvv
z
=
=
where φ0
is the RF phase seen by the particle.
Multigaps Accelerating Structures:
ALow Kinetic Energy Linac(protons,ions)
Mode πL= vT/2
Mode 2πL= vT=
βλ
In«WIDEROE»structureradiated power
∝ωCV
In order to reduce the
radiated power the gap is
enclosed in a resonant
volume at the operating
frequency.A common wall
can be suppressed if no
circulating current in it
for the chosen mode.
ALVAREZstructure
CERN Proton Linac
The Synchrotron
The synchrotron is a synchronous accelerator since there is a synchronous RF
phase for which the energy gain fits the increase of the magnetic field at each
turn. That implies the following operating conditions:
B
e
P
B
h
Ve
rRF
s
⇒=
=
Φ=Φ
Φ
ρ
ρ, R
ωω
sin
^
Energy gain per turn
Synchronousparticle
RF synchronism
Constant orbit
Variable magnetic field
Ifv =c,
ωr
hence
ωRF
remain constant (ultrarelativistic e

)
Energy ramping is simply obtained by varying the B field:
v
BRe
r
TBe
turn
pBe
dt
dp
eBp
′
=
′
=Δ⇒
′
=⇒=
ρπ
ρρρ
2
)(
Since:
pvEcpEEΔ=Δ⇒+=
222
0
2
•The number of stable synchronous particles is equal to the harmonic
number h. They are equally spaced along the circumference.
•Each synchronous particle satifiesthe relation p=eBρ. They have the
nominal energy and follow the nominal trajectory.
The Synchrotron (2)
()
(
)
φ
ρπ
s
sturn
VeRBe
WE
sin
ˆ
'2==
Δ
=
Δ
The Synchrotron (3)
During the energy ramping, the RF frequency
increases to follow the increase of the
revolution frequency :
hence :
),(
s
RF
r
RB
h
ω
ω
ω
==
)(
)(
2
2
1)(
)(
2
1
2
)()(
tB
s
R
r
t
s
E
ec
h
t
RF
f
tB
m
e
s
R
tv
h
t
RF
f
πππ
=⇒><==
Since , the RF frequency must follow the variation of the
Bfield with the law : which asymptotically tends
towards when B becomes large compare to (m
0c2
/ 2πr) which corresponds to
v c (pc >> m0c2
). In practice the
B field can follow the law:
222
0
2
cpcmE+=
2
1
2
)(
2
)/
2
0
(
2
)(
2
)(
⎭
⎬
⎫
⎩
⎨
⎧
+
=
tBecrcm
tB
s
R
c
h
t
RF
f
π
R
c
f
r
π
2
→
tBt
B
tB
2
2
sin)cos1(
2
)(
ω
ω
=−=
Longitudinal Dynamics
It is also often called “synchrotron motion”.
The RF acceleration process clearly emphasizes two coupled
variables, the energy gained by the particle and the RF
phase experienced by the same particle. Since there is a
well defined synchronous particle which has always the same
phase φs, and the nominal energy Es, it is sufficient to follow
other particles with respect to that particle. So let’s
introduce the following reduced variables:
revolution frequency : Δfr
= fr
–f
rs
particle RF phase : Δφ= φφs
particle momentum : Δp = p p
s
particle energy : ΔE = E –E
s
azimuth angle : Δθ= θθs
First EnergyPhase Equation
θ
R
∫
=
Δ
−
=
Δ
⇒
=
dtwithhhff
rrRF
ω
θ
θ
φ
For a given particle with respect to the reference one:
()
()
dt
d
hdt
d
hdt
d
r
φ
φθω
11
−=Δ−=Δ=Δ
Since:
s
r
rs
s
dp
d
p
⎟
⎠
⎞
⎜
⎝
⎛
=
ω
ω
η
one gets:
(
)
φ
ηω
φ
ηωω
&
rs
ss
rs
ss
rs
h
Rp
dt
d
h
Rp
E
−=
Δ
−=
Δ
and
c
p
EE
2
2
2
0
2
+=
pRpvE
srss
Δ
=
Δ
=
Δ
ω
Second EnergyPhase Equation
The rate of energy gained by a particle is:
π
ω
φ
2
sin
ˆ
r
Ve
dt
dE
=
The rate of relative energy gain with respect to the reference
particle is then:
)sin(sin
ˆ
2
s
r
Ve
E
φφ
ω
π
−=
⎟
⎠
⎞
⎜
⎝
⎛
Δ
&
leads to the second energyphase equation:
()
s
rs
Ve
E
dt
d
φφ
ω
π
sinsin
ˆ
2−=
⎟
⎠
⎞
⎜
⎝
⎛
Δ
()
()
ET
dt
d
ETTEETTETE
rsrsrrsrr
Δ=Δ+Δ=Δ+Δ≅Δ
&&&&&
Expanding the left hand side to first order:
Equations of Longitudinal Motion
()
s
rs
Ve
E
dt
d
φφ
ω
π
sinsin
ˆ
2−=
⎟
⎠
⎞
⎜
⎝
⎛
Δ
()
φ
ηω
φ
ηωω
&
rs
ss
rs
ss
rs
h
Rp
dt
d
h
Rp
E
−=
Δ
−=
Δ
deriving and combining
()
0sinsin
2
ˆ
=−+
⎥
⎦
⎤
⎢
⎣
⎡
s
rs
ss
Ve
dt
d
h
pR
dt
d
φφ
π
φ
ηω
This second order equation is non linear. Moreover the parameters
within the bracket are in general slowly varying with time…………………
Small Amplitude Oscillations
()
0sinsin
cos
2
=−
Ω
+
s
s
s
φφ
φ
φ
&&
(for small Δφ)
0
2
=ΔΩ+
φφ
s
&&
ss
srs
s
pR
Veh
π
φηω
2
cos
ˆ
2
=
Ω
γ< γtr
η> 0 0 < φs
< π/2 sinφs
> 0
γ> γtr
η< 0 π/2 < φs
< πsinφs
> 0
with
Let’s assume constant parameters Rs,p
s,ω
s
and η:
(
)
φ
φ
φ
φ
φ
φ
φ
Δ
≅
−
Δ
+
=−
ssss
cossinsinsinsin
Consider now small phase deviations from the reference particle:
and the corresponding linearizedmotion reduces to a harmonic oscillation:
stable for and Ωs
real
0
2
>
Ωs
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the
second order differential equation is nonlinear:
()
0sinsin
cos2
=−
Ω
+
s
s
s
φφ
φ
φ
&&
(Ωs
as previously defined)
Multiplying by and integrating gives an invariant of the motion:
φ
&
()
I
s
s
s
=+
Ω
−
φφφ
φ
φ
sincos
cos2
2
2
&
which for small amplitudes reduces to:
()
I
s
=
Δ
Ω+
22
2
2
2
φφ
&
(the variable is Δφand φ
s
is constant)
Similar equations exist for the second variable : ΔE∝dφ/dt
Large Amplitude Oscillations (2)
()
()()()
sss
s
s
s
s
s
φφπφπ
φ
φφφ
φ
φ
sincos
cos
sincos
cos2
22
2
−+−
Ω
−=+
Ω
−
&
(
)()
ssssmm
φ
φ
π
φ
π
φ
φ
φ
sincossincos
−
+
−
=
+
Second value φm
where the separatrixcrosses the horizontal axis:
Equation of the separatrix:
When φreaches πφs
the force goes
to zero and beyond it becomes non
restoring. Hence πφs is an extreme
amplitude for a stable motion which
in the phase space( ) is shown
as closed trajectories.
φ
φ
Δ
Ω
,
s
&
Energy Acceptance
From the equation of motion it is seen that reaches an extremum
when , hence corresponding to .
Introducing this value into the equation of the separatrixgives:
φ
&
0=
φ
&&
s
φ
φ
=
(
)
{
}
sss
φπφφ
tan222
22
max
−+Ω=
&
That translates into an acceptance in energy:
This “RF acceptance”depends strongly on φs
and plays an important role
for the electron capture at injection, and the stored beam lifetime.
()
⎭
⎬
⎫
⎩
⎨
⎧
−=
⎟
⎠
⎞
⎜
⎝
⎛
Δ
φ
ηπ
β
s
ss
G
E
h
Ve
E
E
ˆ
2
1
max
m
(
)
(
)
[
]
φ
π
φφφ
ssss
Gsin2cos2
−
+
=
RF Acceptance versus Synchronous Phase
As the synchronous phase
gets closer to 90ºthe
area of stable motion
(closed trajectories) gets
smaller. These areas are
often called “BUCKET”.
The number of circulating
buckets is equal to “h”.
The phase extension of
the bucket is maximum
for φs =180º(or 0°) which
correspond to no
acceleration . The RF
acceptance increases with
the RF voltage.
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