cas chios, 18-30 september 20111 longitudinal dynamics frank tecker based on the course by joël le...
TRANSCRIPT
CAS Chios, 18-30 September 2011 1
LONGITUDINAL DYNAMICS
Frank Tecker
based on the course byJoël Le Duff
Many Thanks!
CAS on Intermediate Level Accelerator Physics Course
Chios, 18-30 September 2011
CAS Chios, 18-30 September 2011 2
Summary
•Radio-Frequency Acceleration and Synchronism Condition
•Principle of Phase Stability and Consequences•The Synchrotron•Dispersion Effects in Synchrotron•Energy-Phase Equations•Longitudinal Phase Space Motion•Stationary Bucket•Injection Matching•From Synchrotron to Linac•Adiabatic Damping•Dynamics in the vicinity of transition energy
CAS Chios, 18-30 September 2011 3
Bibliography
M. Conte, W.W. Mac Kay An Introduction to the Physics of particle Accelerators (World Scientific, 1991)P. J. Bryant and K. Johnsen The Principles of Circular Accelerators and Storage Rings (Cambridge University Press, 1993)D. A. Edwards, M. J. Syphers An Introduction to the Physics of High Energy Accelerators (J. Wiley & sons, Inc, 1993)H. Wiedemann Particle Accelerator Physics
(Springer-Verlag, Berlin, 1993)
M. Reiser Theory and Design of Charged Particles Beams (J. Wiley & sons, 1994)A. Chao, M. Tigner Handbook of Accelerator Physics and Engineering (World Scientific 1998)K. Wille The Physics of Particle Accelerators: An Introduction (Oxford University Press, 2000)E.J.N. Wilson An introduction to Particle Accelerators (Oxford University Press, 2001)
And CERN Accelerator Schools (CAS) Proceedings
CAS Chios, 18-30 September 2011 4
Main Characteristics of an Accelerator
ACCELERATION is the main job of an accelerator.• It provides kinetic energy to charged particles, hence increasing their
momentum. • In order to do so, it is necessary to have an electric field , preferably along the
direction of the initial momentum.dp
dt=eEz
BENDING is generated by a magnetic field perpendicular to the plane of the particle trajectory. The bending radius obeys to the relation :
Be
p
FOCUSING is a second way of using a magnetic field, in which the bending effect is used to bring the particles trajectory closer to the axis, hence to increase the beam density.
E
Newton-Lorentz Force on a charged particle:
rF
dpdt
eE +
v×
B( )
2nd term always perpendicular to motion => no acceleration
B [Tm] ≈p[GeV/c]
0.3in practical units:
CAS Chios, 18-30 September 2011 5
Radio-Frequency Acceleration
Cylindrical electrodes (drift tubes) separated by gaps and fed by a RF generator, as shown above, lead to an alternating electric field polarity
Synchronism condition L = v T/2 v = particle velocityT = RF period
D.Schulte
Similar for standing wave cavity as shown
CAS Chios, 18-30 September 2011 6
Radio-Frequency Acceleration (2)
L = vT/2 (π mode) L = vT (2π mode)
Single Gap Multi-Gap
CAS Chios, 18-30 September 2011 7
Energy Gain
RF Acceleration
(neglecting transit time factor)
The field will change during the passage of the particle through the cavity=> effective energy gain is lower
Newton-Lorentz Force Relativistics Dynamics
cpEE 222
02 + dpvdE
dE
dz=v
dpdz
=dpdt
=eEz
dE=dW=eEzdz W =e Ez dz∫
Ez =Ez sinω RFt= Ez sinφ t( )
Ez dz=V∫
sinVeW
rF =
drp
dt=e
rE +
rv×
rB( )
2nd term always perpendicular to motion => no acceleration
p =mv=Ec2 βc=β
Ec=βγm0c
2
11
γβ
c
v γ =E
E0
=m
m0
=1
1 − β 2
CAS Chios, 18-30 September 2011 8
Transit time factor
In the general case, the transit time factor is:
Ta =E1(s,r) cos ωRF
sv
⎛⎝⎜
⎞⎠⎟
ds−∞
+∞
∫
E1(s,r) ds−∞
+∞
∫
Defined as: Ta eneγy γain o paticle ωith vβc
m axim um eneγy γain (paticle ωith v→ ∞)
const.),(1 g
VrsE RF
• Ta < 1• Ta 1 for g 0, smaller ωRF
Important for low velocities (ions)
E(s,r, t) =E1(s,r)⋅E2 (t)
Examplefor g=βλ/3:
for
Simple modeluniform field:
Ta =sinωRFg2v
ωRFg2v
follows:
CAS Chios, 18-30 September 2011 9
Let’s consider a succession of accelerating gaps, operating in the 2π mode, for which the synchronism condition is fulfilled for a phase s .
For a 2π mode, the electric field is the same in all gaps at any given time.
sVeseV sinˆ is the energy gain in one gap for the particle to reach the next gap with the same RF phase: P1 ,P2, …… are fixed points.
Principle of Phase Stability (Linac)
If an energy increase is transferred into a velocity increase => M1 & N1 will move towards P1 => stable M2 & N2 will go away from P2 => unstable
(Highly relativistic particles have no significant velocity change)
CAS Chios, 18-30 September 2011 10
00
z
zE
t
VLongitudinal phase stability means :
The divergence of the field is zero according to Maxwell : 000.
+
x
E
z
E
x
EE xzx
defocusing RF force
External focusing (solenoid, quadrupole) is then necessary
A Consequence of Phase Stability
Transverse focusing fields at the entrance and defocusing at the exit of the cavity.Electrostatic case: Energy gain inside the cavity leads to focusingRF case: Field increases during passage => transverse defocusing!
CAS Chios, 18-30 September 2011 11
Circular accelerators: 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:
BePB
cteRcte
h
cte
Ve
rRF
s
ωω
sin^
Energy gain per turn
Synchronous particle
RF synchronism (h - harmonic number)
Constant orbit
Variable magnetic fieldIf v≈c, wr hence wRF remain constant (ultra-relativistic
e- )
B
injection extraction
r
R
E
Bending magnet
CAS Chios, 18-30 September 2011 12
Energy ramping is simply obtained by varying the B field (frequency follows v):
p =eB ⇒
dp
dt=e &B ⇒ (Δp)turn = eρ &BTr =
2π eρ R &B
v
Since: E2 =E0
2 + p2c2 ⇒ ΔE =vΔp
•The number of stable synchronous particles is equal to the harmonic number h. They are equally spaced along the circumference.•Each synchronous particle satisfies the relation p=eB. They
have the nominal energy and follow the nominal trajectory.
The Synchrotron (2)
turnΔE( ) =
sΔW( ) =2π eρ R &B=eV sin sϕ
Stable phase φs changes during energy ramping
CAS Chios, 18-30 September 2011 13
Dispersion Effects in a Synchrotron
If a particle is slightly shifted in momentum it will have a different orbit and the length is different.
The “momentum compaction factor” is defined as:
a =p
L
dL
dp
If the particle is shifted in momentum it will have also a different velocity. As a result of both effects the revolution frequency changes:
dpdf
fp r
rh
p=particle momentum
R=synchrotron physical radius
fr=revolution frequency
E+E
E
cavity
Circumference
2R a =dL
Ldp
p
⇒
h =
d frfr
d pp
⇒
CAS Chios, 18-30 September 2011 14
Dispersion Effects in a Synchrotron (2)
a =p
L
dL
dp
x
0ss
pdpp +
d
x
ds0 =dds= + x( )d
The elementary path difference from the two orbits is: dl
ds0
=ds−ds0
ds0=
x=
Dx
dpp
leading to the total change in the circumference:dL = dl
C∫ =
x∫ ds0 =
Dx
dpp
ds0∫
With ρ=∞ in straight sections we get:
RD
mxa
< >m means that the average is considered over the bending magnet only
definition of dispersion Dx
a =1
L
Dx (s)
ρ (s)ds0
C∫
CAS Chios, 18-30 September 2011 15
Dispersion Effects in a Synchrotron (3)
dfr
fr
=hdpp
fr =βc2pR
⇒dfrfr
=dββ
−dRR
=dββ
−adpp
p mvβγE0
c⇒
dpp
dββ
+d 1−β 2( )
−12
1−β 2( )−1
2 1−β 2( )
−1
γ 21 24 34
dββ
pdp
fdf
r
r
aγ2
1 aγh 2
1
=0 at the transition energy a
γ 1tr
definition of momentum
compaction factor
CAS Chios, 18-30 September 2011 16
Phase Stability in a Synchrotron
From the definition of it is clear that an increase in energy gives- below transition (η > 0) a higher revolution frequency (increase in velocity dominates) while - above transition (η < 0) a lower revolution frequency (v c and longer path) where the momentum compaction (generally > 0) dominates.
Stable synchr. Particle for < 0
> 0
Crossing transition during acceleration makes the previous stable synchronous phase unstable. RF system needs to make a ‘phase jump’.
CAS Chios, 18-30 September 2011 17
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 s, 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 : fr = fr – frs
particle RF phase : = - s
particle momentum : p = p - ps
particle energy : E = E – Es
azimuth angle : = - s
CAS Chios, 18-30 September 2011 18
First Energy-Phase Equation
fRF =hfr ⇒ Δφ=−hΔ with = ω r dt∫
For a given particle with respect to the reference one:
( ) ( )dtd
hdtd
hdtd
rω 11 ΔΔΔ
Since:
h =ps
ω rs
dω r
dp
⎛
⎝⎜⎞
⎠⎟s
one gets:
( ) hω
hωω
rs
ss
rs
ss
rs hRp
dtd
hRpE ΔΔ
and
2E = 02E + 2p 2c
ΔE = vsΔp = ω rsRsΔp
particle ahead arrives earlier=> smaller RF phase
qs
Dq
R
v
CAS Chios, 18-30 September 2011 19
Second Energy-Phase Equation
The rate of energy gained by a particle is: p
ω2
sinˆ rVedtdE
The rate of relative energy gain with respect to the reference particle is then:
2pΔ
&Eω r
⎛
⎝⎜
⎞
⎠⎟ eV(sinφ−sinφs)
leads to the second energy-phase equation:
2pddt
ΔEω rs
⎛
⎝⎜⎞
⎠⎟=eV sinφ−sinφs( )
Δ &ETr( ) ≅ &EΔTr + Trs Δ &E = ΔE &Tr + Trs Δ &E =
d
dtTrs ΔE( )
Expanding the left-hand side to first order:
CAS Chios, 18-30 September 2011 20
Equations of Longitudinal Motion
( )srs
VeEdtd ωp sinsinˆ2
( ) hω
hωω
rs
ss
rs
ss
rs hRp
dtd
hRpE ΔΔ
deriving and combining
( ) 0sinsin2
ˆ +
srs
ss Vedtd
hpR
dtd p
hω
This second order equation is non linear. Moreover the parameters within the bracket are in general slowly varying with time.
We will study some cases later…
CAS Chios, 18-30 September 2011 21
Hamiltonian of Longitudinal Motion
( )sVedt
dW sinsinˆ
WRp
hdtd
ss
rshωp
21
Introducing a new convenient variable, W, leads to the 1st order equations:
pREW srs
Δ
Δ pωp 22
The two variables ,W are canonical since these equations of motion can be derived from a Hamiltonian H(,W,t):
WH
dtd
Hdt
dW
( ) ( ) WpRhVetWH
ss
rssss
2
41sincoscosˆ,, hωp
CAS Chios, 18-30 September 2011 22
Small Amplitude Oscillations
( ) 0sinsincos
2
+ ss
s
(for small )
&&φ+s
2Δφ = 0
ss
srss pR
Vehp
hω2
cosˆ2with
Let’s assume constant parameters Rs, ps, s and :
( ) ΔΔ+ ssss cossinsinsinsin
Consider now small phase deviations from the reference particle:
and the corresponding linearized motion reduces to a harmonic oscillation:
where s is the synchrotron angular frequency
CAS Chios, 18-30 September 2011 23
Stability is obtained when s is real and so s2 positive:
s2 =
e VRFη hω s
2π Rs ps
cosφs ⇒ Ωs2 > 0 ⇔ η cosφs > 0
2
pp
2
3p
VRF
cos (fs)
acceleration deceleration
0h 0h0h 0hStable in the region if
Stability condition for ϕs
< tr < tr > tr > tr
CAS Chios, 18-30 September 2011 24
Large Amplitude Oscillations
For larger phase (or energy) deviations from the reference the second order differential equation is non-linear:
( ) 0sinsincos
2+ s
s
s (s as previously defined)
Multiplying by and integrating gives an invariant of the motion:
( ) Iss
s sincos
cos222
which for small amplitudes reduces to:
&φ2
2+ Ωs
2 Δφ( )2
2= ′I (the variable is , and s is constant)
Similar equations exist for the second variable : Ed/dt
CAS Chios, 18-30 September 2011 25
Large Amplitude Oscillations (2)
( ) ( ) ( )( )ssss
ss
s
s pp sincos
cossincos
cos2222
++
( ) ( ) ssssmm pp sincossincos ++
Second value m where the separatrix crosses the horizontal axis:
Equation of the separatrix:
When reaches -s the force goes to zero and beyond it becomes non restoring.Hence -s is an extreme amplitude for a stable motion which in the
phase space( ) is shown as
closed trajectories.
&φs
, Δφ
Area within this separatrix is called “RF bucket”.
CAS Chios, 18-30 September 2011 26
Energy Acceptance
From the equation of motion it is seen that reaches an extreme
when , hence corresponding to .
Introducing this value into the equation of the separatrix gives:
0 s
&φmax
2 = 2Ωs2 2 + 2φs − π( ) tanφs{ }
That translates into an acceptance in energy:
This “RF acceptance” depends strongly on s and plays an important role for the capture at injection, and the stored beam lifetime.
max
ΔE
sE⎛⎝⎜
⎞⎠⎟
mβ −eV
phh sEG sφ( )
G sφ( )= 2cos sφ + 2 sφ −p( )sin sφ⎡⎣ ⎤⎦
CAS Chios, 18-30 September 2011 27
RF Acceptance versus Synchronous Phase
The areas of stable motion (closed trajectories) are called “BUCKET”.
As the synchronous phase gets closer to 90º the buckets gets smaller.
The number of circulating buckets is equal to “h”.
The phase extension of the bucket is maximum for s =180º (or 0°) which correspond to no acceleration . The RF acceptance increases with the RF voltage.
CAS Chios, 18-30 September 2011 28
Potential Energy Function
( ) Fdtd 2
2
( ) UF
( ) ( ) FdFU ss
s00
2
sincoscos
∫ +
The longitudinal motion is produced by a force that can be derived from a scalar potential:
The sum of the potential energy and kinetic energy is constant and by analogy represents the total energy of a non-dissipative system.
CAS Chios, 18-30 September 2011 29
Stationnary Bucket - Separatrix
This is the case sins=0 (no acceleration) which means s=0 or . The equation of the separatrix for s= (above transition) becomes:
+ 222
cos2 ss
2sin22
222
s
Replacing the phase derivative by the canonical variable W:
ωh
pω
p rs
ss
rs hRpEW 22 Δ
and introducing the expression for s leads to the following equation for the separatrix:
W =±2Cc
−eV sE2phh
sinφ2
=±Wbk sinφ2with C=2Rs
W
0 2
Wbk
CAS Chios, 18-30 September 2011 30
Stationnary Bucket (2)
Setting = in the previous equation gives the height of the bucket:
The area of the bucket is:
bkA =8Wbk =16Cc
−eV sE2phh
∫ p 202 dWAbk
Since: p 20 4
2sin d
one gets:
hp hEVe
cCW s
bk 2
ˆ2
8AW bk
bk
ΔEmax =ωrs
2πWbk = β s 2
−eVRFEs
πηh
This results in the maximum energy acceptance:
CAS Chios, 18-30 September 2011 31
Bunch Matching into a Stationnary Bucket
A particle trajectory inside the separatrix is described by the equation:
W
0 2
Wbk
Wb
m 2-m
( ) Iss
s sincos
cos2
22 s= Is + cos2
22
mss coscos
222
2
+
( ) coscos2 ms
W Wbk2cos m2
− 2cos2
The points where the trajectory crosses the axis are symmetric with respect to s=
cos(φ) =2cos2φ2−1
φ
CAS Chios, 18-30 September 2011 32
Bunch Matching into a Stationnary Bucket (2)
Setting in the previous formula allows to calculate the bunch height:
2cos
8mbk
bAW bW = bkW cos mφ
2= bkW sin
φ2
or:
b
ΔE
sE⎛⎝⎜
⎞⎠⎟=
RF
ΔE
sE⎛⎝⎜
⎞⎠⎟
cos mφ2
=RF
ΔE
sE⎛⎝⎜
⎞⎠⎟
sinφ2
This formula shows that for a given bunch energy spread the proper matching of a shorter bunch (m close to , small)will require a bigger RF acceptance, hence a higher voltage
φ
W bkA16
φ2− 2Δφ( )2
16W
bkA φ⎛
⎝⎜⎞
⎠⎟+
2Δφφ
⎛
⎝⎜⎞
⎠⎟=1
bA =p16 bkA φ2
For small oscillation amplitudes the equation of the ellipse reduces to:
Ellipse area is called longitudinal emittance
CAS Chios, 18-30 September 2011 33
Effect of a MismatchInjected bunch: short length and large energy spreadafter 1/4 synchrotron period: longer bunch with a smaller energy spread. W W
For larger amplitudes, the angular phase space motion is slower (1/8 period shown below) => can lead to filamentation and emittance growth
stationary bucket accelerating bucket
CAS Chios, 18-30 September 2011 34
Capture of a Debunched Beam with Fast Turn-On
CAS Chios, 18-30 September 2011 35
Capture of a Debunched Beam with Adiabatic Turn-On
CAS Chios, 18-30 September 2011 36
From Synchrotron to Linac
In the linac there is no bending magnets, hence there is no dispersion effects on the orbit and =0 and =1/2.
sRC p2
sRC p2
cavity
Ez
s or z
lβ RFhC
lβ RF
CAS Chios, 18-30 September 2011 37
From Synchrotron to Linac (2)
Moreover one has:
Since in the linac =0 and =1/2, the longitudinal frequency becomes:
pRVeh
ss
srss p
ωγ2
cosˆ22
vmpERVh sssRFs 002ˆ γpωω
leading to:
vmEe
s
sRFs γ
ω3
0
02 cos
Since in a linac the independent variable is z rather than t one gets by using
in terms of ϕ(z):
vmEe
s s
sRF33
0
02
cos2γ
ωlp
0 sγ
dz =vsdt
The longitudinal distribution is fixed for higher energy!
CAS Chios, 18-30 September 2011 38
Adiabatic Damping
Though there are many physical processes that can damp the longitudinal oscillation amplitudes, one is directly generated by the acceleration process itself. It will happen in the synchrotron, even ultra-relativistic, when ramping the energy but not in the ultra-relativistic electron linac which does not show any oscillation.
As a matter of fact, when Es varies with time, one needs to be more careful in combining the two first order energy-phase equations in one second order equation:
( )
( ) 0
0
2
2
2
Δ++
Δ++
Δ
sss
s
ssss
sss
EEE
EEE
EEdtd
The damping coefficient is proportional to the rate of energy variation and from the definition of s one has:
s
s
s
s
EE
2
CAS Chios, 18-30 September 2011 39
Adiabatic Damping (2)
∫ .constdWI
( ) WpRhVetWH
ss
rss
22
41cos
2ˆ
),,( ωhp Δ
tWW s cosˆ
( ) ts sin
So far it was assumed that parameters related to the acceleration process were constant. Let’s consider now that they vary slowly with respect to the period of longitudinal oscillation (adiabaticity).
For small amplitude oscillations the hamiltonian reduces to:
with
Under adiabatic conditions the Boltzman-Ehrenfest theorem states that the action integral remains constant:
(W, are canonical variables)
WpR
hWH
dtd
ss
rsωhp
21
dtWpRhdt
dtdWI
ss
rs 2
21 ωhp
Since:
the action integral becomes:
CAS Chios, 18-30 September 2011 40
Adiabatic Damping (3)
leads to:
s
WdtWˆ 2
2 pPrevious integral over one period:
.ˆ
2
2
constWpR
hIsss
rs
ωh
From the quadratic form of the hamiltonian one gets the relation:
ωh
p ˆ2ˆ Δrs
sss
hRp
W
Finally under adiabatic conditions the long term evolution of the oscillation amplitudes is shown to be:
EVRE
s
sss
4/1
2
4/1
cosˆˆ
Δ
h EEorW s
4/1ˆˆ Δ
W ⋅Δφ=invariant
CAS Chios, 18-30 September 2011 41
Dynamics in the Vicinity of Transition Energy
one gets:
γγaγ
h 222
1 tIntroducing in the previous expressions:
Δ
γγγ
224/1
cosˆ1ˆ t
sV
Δ
γγγ
224/1
cosˆ1ˆ t
sVE
γγγ
22
2/1
cosˆ t
ss V
CAS Chios, 18-30 September 2011 42
Dynamics in the Vicinity of Transition Energy (2)
In fact close to transition, adiabatic solution are not valid since parameters change too fast. A proper treatment would show that:
will not go to zero
E will not go to infinityt
t t
s Δ
EEs
ˆ