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Lecture A3: Damping Rings
Linear beam dynamics
overview
Damping rings, Linear Collider School 2015
Yannis PAPAPHILIPPOU
Accelerator and Beam Physics group
Beams Department
CERN
Ninth International Accelerator School for Linear Colliders
26 October – 6 November 2015,
Whistler BC, Canada
1
Outline – Transverse transfer matrices
 Hill’s equations
 Derivation
 Harmonic
oscillator
 Transport Matrices
 Matrix
formalism
 Drift
Damping rings, Linear Collider School 2015
 Thin
lens
 Quadrupoles
 Dipoles
 Sector magnets
 Rectangular magnets
 Doublet
 FODO
2
Equations of motion – Linear fields
 Consider s-dependent fields from dipoles and normal
quadrupoles
 The total momentum can be written
 With magnetic rigidity
gradient
and normalized
Damping rings, Linear Collider School 2015
the equations of motion are
 Inhomogeneous equations with s-dependent coefficients
 Note that the term 1/ρ2 corresponds to the dipole week
focusing
 The term ΔP/(Pρ) represents off-momentum particles
3
Hill’s equations
 Solutions are combination of the ones from the
homogeneous and inhomogeneous equations
 Consider particles with the design momentum.
The equations of motion become
George Hill
Damping rings, Linear Collider School 2015
with
 Hill’s equations of linear transverse particle motion
 Linear equations with s-dependent coefficients (harmonic
oscillator with time dependent frequency)
 In a ring (or in transport line with symmetries), coefficients
are periodic
 Not straightforward to derive analytical solutions for whole
accelerator
4
Harmonic oscillator – spring
 Consider K(s) = k0 = constant
 Equations of harmonic oscillator
with solution
u
u
Damping rings, Linear Collider School 2015
with
for k0 > 0
for k0 < 0

Note that the solution can be written in matrix form
5
Matrix formalism
 General transfer matrix from s0 to s
 Note that
which is always true for conservative systems
Damping rings, Linear Collider School 2015
 Note also that
 The accelerator can be build by a series of matrix multiplications
S1
S0
S2
S3
… Sn-1
from s0 to s1
Sn
from s0 to s2
from s0 to s3
from s0 to sn
6
Symmetric lines
 System with normal symmetry
Damping rings, Linear Collider School 2015
S
 System with mirror symmetry
S
7
4x4 Matrices
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 Combine the matrices for each plane
to get a total 4x4 matrix
Uncoupled motion
8
Transfer matrix of a drift
 Consider a drift (no magnetic elements) of length L=s-s0
L
Damping rings, Linear Collider School 2015
 Position changes if particle has a slope which remains unchanged.
After
u’
Before
u’L
0
L
Real Space
u
s
Phase Space
9
(De)focusing thin lens
 Consider a lens with focal length ±f
 Slope diminishes (focusing) or increases
(defocusing) for positive position, which remains
unchanged.
u’
u
Damping rings, Linear Collider School 2015
Before
After
0
f
After
u
Before
0
u’
f
10
Quadrupole
 Consider a quadrupole magnet of length L = s-s0.
The field is
Damping rings, Linear Collider School 2015
 with normalized quadrupole gradient (in m-2)
The transport through a quadrupole is
u’
u
0
L
s
11
(De)focusing Quadrupoles
 For a focusing quadrupole (k>0)
Damping rings, Linear Collider School 2015
 For a defocusing quadrupole (k<0)
 By setting
 Note that the sign of k or f is now absorbed inside the symbol
 In the other plane, focusing becomes defocusing and vice
versa
12
Sector Dipole
 Consider a dipole of (arc) length L.
Damping rings, Linear Collider School 2015
 By setting in the focusing quadrupole matrix
transfer matrix for a sector dipole becomes
the
L
with a bending radius
 In the non-deflecting plane
and
θ
 This is a hard-edge model. In fact, there is some edge
focusing in the vertical plane
 Matrix generalized by adding gradient (synchrotron magnet)13
Rectangular Dipole
ΔL
θ
Damping rings, Linear Collider School 2015
 Consider a rectangular dipole with bending angle θ. At each edge of
length ΔL, the deflecting angle is changed by
i.e., it acts as a thin defocusing lens with focal length
 The transfer matrix is
with
 For θ<<1, δ=θ/2
 In deflecting plane (like drift),
in non-deflecting plane (like sector)
14
Quadrupole doublet
 Consider a quadrupole doublet,
i.e. two quadrupoles with focal
lengths f1 and f2 separated by a
distance L.
x
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L
 In thin lens approximation the
transport matrix is
with the total focal length
 Setting f1 = - f2 = f
 Alternating gradient focusing seems overall focusing
 This is only valid in thin lens approximation
15
FODO Cell
 Consider defocusing quad
“sandwiched” by two focusing
quads with focal lengths ± f.
 Symmetric transfer matrix from
center to center of focusing quads
L
L
Damping rings, Linear Collider School 2015
with the transfer matrices
 The total transfer matrix is
16
Outline – Betatron functions
General solutions of Hill’s equations
 Floquet
theory
Betatron functions
Transfer matrices revisited
 General
and periodic cell
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General transport of betatron functions
 Drift
 Beam
waist
17
Solution of Betatron equations
 Betatron equations are linear
with periodic coefficients
 Floquet theorem states that the solutions are
Damping rings, Linear Collider School 2015
where w(s), ψ(s) are periodic with the same period
 Note that solutions resemble the one of harmonic oscillator
 Substitute solution in Betatron equations
0
0
18
Betatron functions
 By multiplying with w the coefficient of sin
 Integrate to get
 Replace ψ’ in the coefficient of cos and obtain
Damping rings, Linear Collider School 2015
 Define the Betatron or twiss or lattice functions (CourantSnyder parameters)
19
Betatron motion
 The on-momentum linear betatron motion of a particle is
described by
Damping rings, Linear Collider School 2015
with
the twiss functions
the betatron
phase
and the beta function
is defined by the envelope equation
 By differentiation, we have that the angle is
20
Courant-Snyder invariant
 Eliminating the angles by the position and slope we define
the Courant-Snyder invariant
 This is an ellipse in phase space with area πε
 The twiss functions
have a geometric meaning
Damping rings, Linear Collider School 2015
 The beam envelope is
 The beam divergence
21
General transfer matrix
 From equation for position and angle we have
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 Expand the trigonometric formulas and set ψ(0)=0 to get
the transfer matrix from location 0 to s
with
and
the phase advance
22
Periodic transfer matrix
 Consider a periodic cell of length C
 The optics functions are
and the phase advance
Damping rings, Linear Collider School 2015
 The transfer matrix is
 The cell matrix can be also written as
with
and the Twiss matrix
23
Stability conditions
 From the periodic transport matrix
and the following stability criterion
Damping rings, Linear Collider School 2015
 From transfer matrix for a cell
we get
24
Tune and working point
 In a ring, the tune is defined from the 1-turn phase
advance
Damping rings, Linear Collider School 2015
i.e. number betatron oscillations per turn
 Taking the average of the betatron tune around the ring we
have in smooth approximation
 Extremely useful formula for deriving scaling laws
 The position of the tunes in a diagram of horizontal versus
vertical tune is called a working point
 The tunes are imposed by the choice of the quadrupole
strengths
 One should try to avoid resonance conditions
25
Transport of Betatron functions
 For a general matrix between position 1 and 2
and the inverse
Damping rings, Linear Collider School 2015
 Equating the invariant at the two locations
and eliminating the transverse positions and angles
26
Example I: Drift
 Consider a drift with length s
 The transfer matrix is
Damping rings, Linear Collider School 2015
 The betatron transport matrix is
from which
γ
β
α
s
27
Simplified method for betatron transport
 Consider the beta matrix
the matrix
and its transpose
Damping rings, Linear Collider School 2015
 It can be shown that
 Application in the case of the drift
and
28
Example II: Beam waist
 For beam waist α=0 and occurs
at s = α0/γ0
 Beta function grows quadratically
and is minimum in waist
γ
β
α
s
Damping rings, Linear Collider School 2015
waist
 The beta at the waste for having beta minimum
in the middle of a drift with length L is
 The phase advance of a drift is
which is π/2 when
. Thus, for a drift
29
Outline – Off-momentum dynamics
 Off-momentum particles
 Effect
from dipoles and quadrupoles
 Dispersion equation
 3x3 transfer matrices
 Periodic lattices in circular accelerators
Damping rings, Linear Collider School 2015
 Periodic
solutions for beta function and dispersion
 Symmetric solution
 3x3 FODO cell matrix
30
Effect of dipole on off-momentum particles
Damping rings, Linear Collider School 2015
 Up to now all particles had the same momentum P0
 What happens for off-momentum particles, i.e. particles
with momentum P0+ΔP?
 Consider a dipole with field B and
ρ+δρ
bending radius ρ
ρ
 Recall that the magnetic rigidity is
θ
and for off-momentum particles
P0+ΔP
P0
 Considering the effective length of the dipole unchanged
 Off-momentum particles get different deflection (different
orbit)
31
Off-momentum particles and quadrupoles
 Consider a quadrupole with gradient G
 Recall that the normalized gradient is
P0+ΔP
P0
Damping rings, Linear Collider School 2015
and for off-momentum particles
 Off-momentum particle gets different focusing
 This is equivalent to the effect of optical lenses
on light of different wavelengths
32
Dispersion equation
 Consider the equations of motion for off-momentum
particles
Damping rings, Linear Collider School 2015
 The solution is a sum of the homogeneous equation (onmomentum) and the inhomogeneous (off-momentum)
 In that way, the equations of motion are split in two parts
 The dispersion function can be defined as
 The dispersion equation is
33
Dispersion solution for a bend
 Simple solution by considering motion through a sector
dipole with constant bending radius ρ
Damping rings, Linear Collider School 2015
 The dispersion equation becomes
 The solution of the homogeneous is harmonic with
frequency 1/ρ
 A particular solution for the inhomogeneous is
and we get by replacing
 Setting D(0) = D0 and D’(0) = D0’, the solutions for
dispersion are
34
General dispersion solution
Damping rings, Linear Collider School 2015
 General solution possible with perturbation theory and use of Green
functions
 For a general matrix
the solution is
 One can verify that this solution indeed satisfies the differential
equation of the dispersion (and the sector bend)
 The general betatron solutions can
be obtained by 3X3 transfer
matrices including dispersion
 Recalling that
and
35
General solution for the dispersion
 Introduce Floquet variables
Damping rings, Linear Collider School 2015
 The Hill’s equations are written
 The solutions are the ones of an harmonic oscillator
 For the dispersion solution
, the
inhomogeneous equation in Floquet variables is written
 This is a forced harmonic oscillator with solution
 Note the resonance conditions for integer tunes!!!
36
3x3 transfer matrices - Drift, quad and sector bend
Damping rings, Linear Collider School 2015
 For drifts and quadrupoles which do not create
dispersion the 3x3 transfer matrices are just
 For the deflecting plane of a sector bend we have seen
that the matrix is
and in the non-deflecting plane is just a drift.
37
3x3 transfer matrices - Synchrotron magnet
 Synchrotron magnets have focusing and bending included
in their body.
 From the solution of the sector bend, by replacing 1/ρ with
Damping rings, Linear Collider School 2015
 For K>0
 For K<0
with
38
3x3 transfer matrices - Rectangular
magnet
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 The end field of a rectangular magnet is simply the one of
a quadrupole. The transfer matrix for the edges is
 The transfer matrix for the body of the magnet is like for
the sector bend
 The total transfer matrix is
39
Periodic solutions
 Consider two points s0 and s1 for which the magnetic
structure is repeated.
 The optical function follow periodicity conditions
Damping rings, Linear Collider School 2015
 The beta matrix at this point is
 Consider the transfer matrix from s0 to s1
 The solution for the optics functions is
with the condition
40
Periodic solutions for dispersion
 Consider the 3x3 matrix for propagating dispersion
between s0 and s1
Damping rings, Linear Collider School 2015
 Solve for the dispersion and its derivative to get
with the conditions
41
Symmetric solutions
 Consider two points s0 and s1 for which the lattice is mirror
symmetric
 The optical function follow periodicity conditions
Damping rings, Linear Collider School 2015
 The beta matrices at s0 and s1 are
 Considering the transfer matrix between s0 and s1
 The solution for the optics functions is
with the condition
42
Symmetric solutions for dispersion
 Consider the 3x3 matrix for propagating dispersion
between s0 and s1
Damping rings, Linear Collider School 2015
 Solve for the dispersion in the two locations
 Imposing certain values for beta and dispersion,
quadrupoles can be adjusted in order to get a
solution
43
Periodic lattices’ stability criterion revisited
 Consider a general periodic structure of length 2L
which contains N cells. The transfer matrix can be
written as
Damping rings, Linear Collider School 2015
 The periodic structure can be expressed as
with
 Note that because
 Note also that
 By using de Moivre’s formula
 We have the following general stability criterion
44
3X3 FODO cell matrix
Damping rings, Linear Collider School 2015
 Insert a sector dipole in between the quads and
consider θ=L/ρ<<1
 Now the transfer matrix is
which gives
and after multiplication
45
Longitudinal dynamics
Damping rings, Linear Collider School 2015
 RF acceleration
Energy gain and phase stability
Momentum compaction and transition
Equations of motion
Small amplitudes
Longitudinal invariant
Separatrix
Energy acceptance
Stationary bucket
Adiabatic damping
46
RF acceleration
Damping rings, Linear Collider School 2015
 The use of RF fields allows an arbitrary number of
accelerating steps in gaps and electrodes fed by RF
generator
 The electric field is not longer continuous but sinusoidal
alternating half periods of acceleration and deceleration
 The synchronism condition for RF period TRF and particle
velocity v
p
L = vTRF / 2 = bc
= bl / 2
wRF
47
Energy gain
Assuming a sinusoidal electric field Ez = E0 cos(wRF t + fs )
where the synchronous particle passes at the middle of
the gap g, at time t = 0, the energy is
g/ 2
z
W (r,t) = q ò Ez dz = q ò E0 cos(wRF + f s )dz
v
-g/ 2
Damping rings, Linear Collider School 2015
And the energy gain is
sin Q / 2
= qV T with the transit time
and finally DW = qV
Q/2
sin(wg/ 2v)
T=
factor defined as
g/ 2
wg/ 2v
It can be shown that in general
T=
ò E(0, z)coswt(z)dz
-g/ 2
g/ 2
ò E(0, z)dz
-g/ 2
48
Damping rings, Linear Collider School 2015
Phase stability
 Assume that a synchronicity condition is fulfilled at the phase
fs and that energy increase produces a velocity increase
 Around point P1, that arrives earlier (N1) experiences a
smaller accelerating field and slows down
 Particles arriving later (M1) will be accelerated more
 A restoring force that keeps particles oscillating around a
stable phase called the synchronous phase fs
 The opposite happens around point P2 at -fs, i.e. M2 and
N2 will further separate
49
RF de-focusing
In order to have stability, the time derivative of the Voltage
and the spatial derivative of the electric field should satisfy
¶V
¶E
>0Þ
<0
¶t
¶z charge
In the absence
of electric
Damping rings, Linear Collider School 2015
the divergence of the field is
given by Maxwell’s equations
where x represents the generic transverse direction.
External focusing is required by using quadrupoles or
solenoids
50
Momentum compaction
 Off-momentum particles on the dispersion orbit travel in a
different path length than on-momentum particles
 The change of the path length with respect to the
momentum spread is called momentum compaction
P+ΔP
Damping rings, Linear Collider School 2015
 The change of circumference is
D(s)ΔP/P
P
ρ
Δθ
 So the momentum compaction is
51
Transition energy
 The revolution frequency of a particle is
 The change in frequency is
 From the relativistic momentum
we have
Damping rings, Linear Collider School 2015
for which
and the revolution frequency
The slippage factor is given by
For vanishing slippage factor,
the transition energy is defined
52
Damping rings, Linear Collider School 2015
Synchrotron
 Frequency modulated but
also B-field increased
synchronously to match
energy and keep revolution
radius constant.
 The number of stable
synchronous particles is
equal to the harmonic
number h. They are equally
spaced along the
circumference.
ESRF Booster
 Each synchronous particle
has the nominal energy and
follow the nominal trajectory
 Magnetic field increases with
momentum and the per turn
change of the momentum is
53
Damping rings, Linear Collider School 2015
Phase stability on electron synchrotrons
 For electron synchrotrons, the relativistic  is very large and
as momentum compaction
is positive in most cases
 Above transition, an increase in energy is followed by lower
revolution frequency
 A delayed particle with respect to the synchronous one will
get closer to it (gets a smaller energy increase) and phase
stability occurs at the point P2 ( - fs)
54
Energy and phase relation
 The RF frequency and phase are related
to the revolution ones as follows
and
Damping rings, Linear Collider School 2015
 From the definition of the momentum
compaction and for electrons
c
 Replacing the revolution frequency change, the following
relation is obtained between the energy and the RF phase
time derivative
c
c
55
Longitudinal equations of motion
 The energy gain per turn with respect to the energy gain of
the synchronous particle is
 The rate of energy change can be approximated by
Damping rings, Linear Collider School 2015
 The second energy phase relation is written as
 By combining the two energy/phase relations, a 2nd order
differential equation is obtained, similar the pendulum
^
d æ R df ö
ceV
sin f - sin f s ) = 0
(
ç
÷+
dt è cac h dt ø 2pREs
56
Small amplitude oscillations
 Expanding the harmonic functions in the vicinity of the
synchronous phase
Damping rings, Linear Collider School 2015
 Considering also that the coefficient of the phase derivative
does not change with time, the differential equation reduces
to one describing an harmonic oscillator
with frequency
 For stability, the square of the frequency should positive and
real, which gives the same relation for phase stability when
particles are above transition
cos fs < 0 Þ p / 2 < fs < p
57
Longitudinal motion invariant
 For large amplitude oscillations the differential equation of
the phase is written as
Damping rings, Linear Collider School 2015
 Multiplying by the time derivative of the phase and
integrating, an invariant of motion is obtained
reducing to the following expression, for small amplitude
oscillations
58
Damping rings, Linear Collider School 2015
Separatrix
 In the phase space (energy
change versus phase), the
motion is described by
distorted circles in the vicinity
of fs (stable fixed point)
 For phases beyond  - fs
(unstable fixed point) the
motion is unbounded in the
phase variable, as for the
rotations of a pendulum
 The curve passing through
 - fs is called the separatrix
and the enclosed area bucket
59
Energy acceptance
 The time derivative of the RF phase (or the energy change)
reaches a maximum (the second derivative is zero) at the
synchronous phase
 The equation of the separatrix at this point becomes
Damping rings, Linear Collider School 2015
 Replacing the time derivative of the phase from the first
energy phase relation
 This equation defines the energy acceptance which depends
strongly on the choice of the synchronous phase. It plays an
important role on injection matching and influences strongly
the electron storage ring lifetime
60
Stationary bucket
 When the synchronous phase is
chosen to be equal to 0 (below
transition) or  (above transition),
there is no acceleration. The equation
of the separatrix is written
Damping rings, Linear Collider School 2015
 Using the (canonical) variable
and replacing the expression for the synchrotron frequency
^
C
W = ±2
c
qV Es
f
sin
2p h a c
2
^
C
Wbk = 2
c
. For f= , the bucket height is
2p
eV Es and the area A = 2 W df = 8W
ò
bk
bk
2p h a c
61
0
Adiabatic damping
 The longitudinal oscillations can be damped directly by
acceleration itself. Consider the equation of motion when the
energy of the synchronous particle is not constant
Damping rings, Linear Collider School 2015
 From this equation, we obtain a 2nd order differential
equation with a damping term
 From the definition of the synchrotron frequency the
damping coefficient is
62
Damping rings, Linear Collider School 2015
Outline – Phase space concepts
Transverse phase space and Beam
representation
Beam emittance
Liouville and normalised emittance
Beam matrix
RMS emittance
Betatron functions revisited
Gaussian distribution
63
Damping rings, Linear Collider School 2015
Transverse Phase Space
 Under linear forces, any
particle moves on ellipse in
phase space (x,x’), (y,y’).
 Ellipse rotates and moves
between magnets, but its
area is preserved.
 The area of the ellipse
defines the emittance
x´
x´
x
x

The equation of the ellipse is

with α,β,γ, the twiss
parameters
Due to large number of
particles, need of a statistical
description of the beam, and
its size
64
Beam representation
Beam is a set of millions/billions of particles (N)
A macro-particle representation models beam as a set of n
particles with n<<N
Distribution function is a statistical function
representing the number of particles in phase space
between


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
1.0
0.8
0.6
0.4
0.2
-4
4
-3
3
-2
2
-1
1
0
0
1
-1
2
-2
3
-3
4
-4
65
Liouville emittance


Emittance represents the phase-space volume occupied by
the beam
The phase space can have different dimensions


Damping rings, Linear Collider School 2015




2D (x, x’) or (y, y’) or (φ, Ε)
4D (x, x’,y, y’) or (x, x’, φ, Ε) or (y, y’, φ, Ε)
6D (x, x’, y, y’, φ, Ε)
The resolution of my beam observation is very large
compared to the average distance between particles.
The beam modeled by phase space distribution function
The volume of this function on phase space is the beam
Liouville emittance
66
Damping rings, Linear Collider School 2015
Vlasov and Boltzmann equations

The evolution of the distribution function is described by
Vlasov equation

Mathematical representation of Liouville theorem stating the
conservation of phase space volume
In the presence of fluctuations (radiation, collisions, etc.)
distribution function evolution described by Boltzmann
equation


The distribution evolves towards a Maxwell-Boltzmann
statistical equilibrium
67
2D and normalized emittance

When motion is uncoupled, Vlasov equation still holds for
each plane individually

The Liouville emittance in the 2D
phase space is still
conserved
In the case of acceleration, the emittance is conserved in the
but not in the
(adiabatic damping)
Considering that
Damping rings, Linear Collider School 2015



the beam is conserved in the phase space
Define a normalised emittance which is conserved
during acceleration
68
Beam matrix


Damping rings, Linear Collider School 2015


We would like to determine the transformation of the beam
enclosed by an ellipse through the accelerator
Consider a vector u = (x,x’,y,y’,…) in a generalized ndimensional phase space. In that case the ellipse
transformation is
Application to one dimension gives
and comparing with
provides the beam matrix
which can be expanded to more dimensions
Evolution of the n-dimensional phase space from position 1
to position 2, through transport matrix
69
Damping rings, Linear Collider School 2015
Root Mean Square (RMS) beam parameters

The average of a function on the beam distribution defined

Taking the square root, the following Root Mean Square
(RMS) quantities are defined

RMS beam size

RMS beam divergence

RMS coupling
70
RMS emittance




Damping rings, Linear Collider School 2015



Beam modelled as macro-particles
Involved in processed linked to the statistical size
The rms emittance is defined as
It is a statistical quantity giving information about the
minimum beam size
For linear forces the rms emittance is conserved in the case
of linear forces
The determinant of the rms beam matrix
Including acceleration, the determinant of 6D transport
matrices is not equal to 1 but
71
Beam betatron functions
 The best ellipse fitting the beam distribution is
Damping rings, Linear Collider School 2015
 The beam betatron functions can be defined through the rms
emittance
72
Gaussian distribution

The Gaussian distribution has a gaussian density profile
in phase space
for which

The beam boundary is
Damping rings, Linear Collider School 2015
Uniform (KV)
Gaussian
73