Transverse dynamics
Transverse dynamics: degrees of freedom orthogonal to the reference trajectory
x : the horizontal plane
y : the vertical plane
Erik Adli, University of Oslo, August 2014, Erik.Adli@fys.uio.no, v2.07
The reference trajectory
•
•
An accelerator is designed around a reference trajectory (often called
design orbit in circular accelerators)
This is the trajectory an ideal particle will follow and consist of
– a straight line where there is no bending field
– arc of circle inside the bending field
Reference trajectory

•
We will in the following consider transverse deviations from the reference
trajectory, and how to control the magnitude of these deviations
Bending field
•
Circular accelerators: piecewise circular orbits with a bending radius
– straight sections are needed for e.g. particle detectors
– in circular arc sections the magnetic field must provide the desired bending
radius
•
The accelerator design specifies a design bending radius, , for the dipole
field bending magnets

  
 
F  q( E  v  B )  FE  FB
1

We define 1/ as the
normalized dipole strength :
•
1



B: bending field [T]
p: particle momentum [GeV/c]
: design bending radius [m]
eB
p
eB
p

1

1
[ m ]  0 .3
B [T ]
p [ GeV / c ]
In a synchrotron, the bending radius is kept constant during acceleration by
Focusing field: quadrupole magnets
•
•
Reference trajectory: typically through centers of magnets and structures
Desired: a restoring force of the type Fx=-kx in order to keep the particles
close to the ideal orbit, in both planes
• A linear field in both planes can be
derived from the potential V(x,y) = gxy.
Four poles with magnet surfaces
shaped as hyperbolas.
Bx = -gy
By = -gx
Fx = -qvgx
Fy = qvgy
(focusing)
(defocusing)
• Forces are focusing in one plane and defocusing in the orthogonal plane
• Opposite focusing/defocusing by rotating the quadrupole by ninety degrees
Normalized magnet strengths
• Analogy to dipole strength: normalized quadrupole strength
Quadrupole :
k 
eg
 k[m
2
g [T / m ]
]  0 .3
p
p [ GeV / c ]
g: field gradient [T/m]
p: particle momentum [GeV/c]
k: design normalized
quadrupole strength [m-2]
Dipole :
1


eB
p

1

1
[ m ]  0 .3
B [T ]
p [ GeV / c ]
B: bending field [T]
p: particle momentum [GeV/c]
1/: design normalized dipole
strength [m-1]
Magnetic lenses  linear optics
•
Analogy: magnetic lenses (quadrupoles)  linear optics
•
Focal length f of a quadrupole:
f = 1 / kl
where l is the length of the quadrupole
•
Alternating focusing and defocusing lenses will together give total focusing
effect in both planes; “alternating gradient focusing”
The Lattice
• An accelerator is composed of bending magnets, focusing magnets
and usually also sextupole magnets
• The ensemble of magnets in the accelerator constitutes the
“accelerator lattice”
Bending field: dipole magnets
•
•
Dipole magnets provide uniform field in the desired region
Several design choices, including cos(q)" design that allows
opposite and uniform field in both vacuum chambers
LHC dipole magnet design
Example: lattice magnets
CLIC Test Facility 3
LHC superconducting dipole magnet
Transverse dynamics
Equations of motion and transfer matrices
Coordinates
•
Coordinate system:
•
x, y are small deviations from the reference trajectory
– x: deviations in the horizontal plane
– y: deviations in the vertical plane
•
r=+x
•
We will outline the steps needed to develop the equations of motion in terms of
deviations from the reference trajectory. For full derivation, see Wille (2000).
Linear equations of motion I
•
Equations of motion in a uniform magnetic field :
B: dipole field
g: quadrupole field gradient
( assuming g >> 1 )
•
•
In accelerators we use x(s) instead of x(t), and we denote dx/ds = x’(s).
Furthermore, using vx << v, we write
yielding the equation of motion as function of the accelerator coordinate s :
1
e
x¢¢ =
(B(s) - g(s)x)
r(s) mv
( Corresponding equations for y )
Linear equations of motion II
•
Further approximations, and dropping higher order terms :
1
1
1
x
=
» (1- ),
r r+x r
r
•
1
1
1
Dp
=
» (1- )
p p0 + Dp p0
p0
We introduce the normalized magnet strengths:
1/ r = eB/ p, k = eg/ p
•
By substituting the above we obtain the linearized trajectory equations:
1
1 Dp
x¢¢ - (k(s) - 2 )x =
r (s)
r (s) p0
– k: normalized quadrupole strength
– 1/: normalized dipole strength
– p0 is the reference momentum, Dp is deviation from ideal momentum
Hill’s equation
•
Magnet strength terms are dependent on the position along the reference
trajectory, s. We can then write eq. of motion as a Hill's equation:
–  1/2(s) – k(s)  K(s)
•
We assume first Dp=0, yielding the homogenous Hill’s equation:
– We write equations for x, analogous for y
•
x   K ( s ) x 
1 Dp
 p0
x   K ( s ) x  0
For a given magnet lattice: we take a piece-wise approach to solution:
– For K(s) = const>0, solutions is: x (s) = x cos( Ks) + x ¢ (1/
1
0
0
– K(s)=const <0: replace with hyperbolic functions
– It is helpful to put the coefficients into a matrix, the transfer matrix :
K )sin( Ks)
éx1ù é cos( K s)
éx 0 ù
(1/ K )sin( K s)ùéx 0ù
úê ú = M ê ú
ê ú=ê
cos( K s) ûëx ¢0û
ëx1¢û ë- K sin( K s)
ëx ¢0 û
M: drift space
•
The element with the simplest transfer matrix M: drift space between
magnets (no field), with length l :
1
M  
0
•
l

1
Written out this gives:
 x0  1
x
   M     
x 
 x0  0
l   x0 
 
1  x 0 

x  x 0  l x 0
x   x 0
•
This simply says that in a drift space x’ is unchanged, and x drifts
Quadrupole transfer matrix

cos( l k )
M 

  k sin( l k )
•
Full solution :
•
Thin lens approximation :

sin( l k ) 
k

cos( l k ) 
1
– Real quadrupole may be modeled as a infinitely thin
lens that focuses or defocuses, plus the drift space to
represent the length of the quadrupole
– valid if focal length f=1/kl >> l
•
Written out multiplication:
x  x0
x   x 0 
•
•
1
f
x0
-1/f is a focusing term
A defocusing quadrupole in x (rotated 90): -f  f
k [m-2]
l [m]
é 1
ê
Mthin = ê 1
ê f
ë
0 ù
ú
1 ú
ú
û
Dipole transfer matrix
•
Bending magnets introduce focusing
terms as well.
•
The solution of Hill’s equation
provides the focusing terms for a
idealized sector dipole :
•
The more common type of dipole
found in accelerators is a rectangular
dipole, which does not provide the
focusing term.
é
l
ê cos
r
ê
Msec = ê
1
l
ê - sin
r
êë r
l ù
r sin ú
r ú
l ú
ú
cos
r úû
é
l
ê 1 r sin
Mrect = ê
r
ê 0
1
ë
ù
ú
ú
ú
û
Hill’s equation:
solutions for an accelerator
• The transverse optics of an accelerator can be modelled as
composed of elements where K(s) = constant inside the element
• We solve Hill’s equation by applying the transfer matrix M, from
the start to the end of each element
• Inside dipoles: K(s) = 1/2
• Inside quadrupoles: K(s) = +/-k
• Where there is no field: K(s) = 0
• One can find the effect on a particle travelling through the whole, or
part, of the lattice by multiplying the M matrices for the various
components:
tot
dipole
F
dipole
D
M =M
M M
M ...
Quadrupole FODO doublet
•
•
A FODO quadrupole doublet consist of a focusing quadrupole followed by
a drift, a defocusing quadrupole and a drift
Using the thin lens approximation we can calculate the total matrix :
M doublet
•
1
 1
 f

0
1

1 0


1
l / 2 
 1
 
1 
 f
0
1

1 0



l
1

l / 2 
2f



l
1  

 2f2



4f

2
l
l 
1

2
2f
4 f 
l
l
2
FODO is focusing in both the horizontal and the vertical planes (since
changing plane equals f = -f )
Stability of a FODO structure
•
A FODO lattice yields focusing in both planes, however too short focal
length will give overfocusing, and an unstable trajectory:
Liming case,
L=4f
stability Þ f > L / 4
•
We may calculate rigorously the stability criterion of a periodic lattice using
Courant-Snyder analysis introduced in the next section
?
Transverse dynamics
Courant-Snyder framework
Previous section: a straight-forward mathematical framework for computing
single particle motion
The next slides : analyze the transverse motion in an accelerators using
standard accelerator terminology, allowing for analysis of a beam of particles
Particle motion: Hill's equation
•
We have calculated particle motion for a single particle by solving Hill’s
equation piece-wise and multiplying transfer matrices M(s). We now seek a
general solution of the the equation :
x ¢¢ + K(s)x = 0
•
Reminder: solution of Hill’s equation with K(s) =K  harmonic oscillator
x   Kx  0

x ( s )  A sin(
K s  0 )
Reformulation of Hill's equation: beta function
We define function varying along the lattice in the solution to the Hill’s equation;
the beta function, b(s).
x ¢¢ + K(s)x = 0
the following reformulation fulfills the original equation :
x(s) = eb (s) sin(f (s) + f 0 ),
if the following constraints are fulfulled :
s dt
f (s) = ò
s0
b
1
1 2
b (s)b ''- b ' +K(s)b 2 (s) = 1
2
4
The complete solution is given
by solving the betatron
equation (left).
The solution is thus a quasi-harmonic oscillator, with amplitude and phaseadvance dependent on s.
Transverse phase-space
•
The particle phase-space in the horizontal plane is spanned by x and x‘ :
x(s) = eb (s) sin(f (s) + f 0 )
ß
x'(s) =
•
e
b (s)
(cos(f (s) + f 0 ) +
b '(s)
2
sin(f (s) + f 0 ))
By eliminating the phase, , we get equation for an ellipse with area pe :
g (s)x(s)2 + 2a (s)x(s)x'(s)+ b (s)x'(s)2 = e
where we have defined :
a(s)  -(1/2)b'(s),
g(s)  (1+ a2(s))/ b(s),
b(s), a(s),g(s): Twiss parameters
(in the USA: called Courant-Snyder parameters)
e: single particle emittance
For a given single particle emittance, e,the phase-space ellipse area is an invariant
(with respect to the s coordinate).
The phase-space ellipse
g (s)x(s)2 + 2a (s)x(s)x'(s)+ b (s)x'(s)2 = e
Envelope of particle motion :
Particles with different phase and
equal emittance.
x(s) = eb (s)
Evolution of the phase-space ellipse
• Particles populate trajectories on ellipses with the shape determined
by the Twiss parameters, b,a,g.
• The Twiss parameters evolve according to the solution of Hill’s
equation, i.e. they depend on the position along the lattice, s
• As they evolve, the area of the ellipse, pe, remains constant
g (s)x(s)2 + 2a (s)x(s)x'(s)+ b (s)x'(s)2 = e
From A. Chao
Single particle propagation
In the Courant-Snyder framework, which information do we need to describe the
propagation of a single particle?
1) We need the Twiss parameters. These are given as solutions of the betatron
equation. How they propagate depends only on the lattice specification, K(s) :
2) In addition, the particle initial state must be specified, by the initial single
particle emittance, e, and its initial phase 0.
This is hardly a simplification of the single particle motion, however, we shall
now relate the Twiss parameters and the description of the entire beam.
Twiss parameters: initial conditions
For a non-circular lattice, the
solution of Hill’s equation, the Twiss
parameters, depends on the initial
Twiss parameters (the initial
beam), b(0) and a(0)=-1/2b(0).
Usually the lattice has design Twiss
parameters, and one aims to match
the incoming beam to the lattice
design.
Example to the right: evolution of
Twiss parameters where the initial
magnets are adjusted to match the
beam into a periodic focusing lattice
of FODO-type.
F D
Example from the CLIC Test
Facility at CERN
Betatron oscillations and phase advance
Solution of Hill’s equation :
x(s) = eb (s) sin(j (s) + j 0 ), j (s) =
Example solution in a periodic lattice :
•
•
•
•
ò
s
dt
s0
b
A particle will undergo betatron oscillations around the reference trajectory.
The lattice parameter b(s) defines the envelope for the particle motion
(s) is the particle phase-advance from point s0 to point s in a lattice
In a FODO structure the beta function is at maximum in the middle of the F
quadrupole and at minimum in the middle of the D quadrupole
In the figure : about 5 FODO cells for a full oscillation, yielding a phaseadvance per FODO cell of cell  70º.
The beam sigma matrix
Gaussian beams
Evolution of beams
The importance of this is that we can describe the transport of the
2nd order moments of the beam in the same way as we describe
the transformation of the Twiss parameters along the lattice. For
Gaussian beams and linear optics, this means that the Twiss
parameters uniquely defines the beam shape along the lattice.
Relation b(s)and M(s)
• We calculated earlier the evolution of single particles along an
accelerator lattice using the transfer matrices: x1= M10x0
• We can calculate the evolution of the beta function using transfer
matrices as well. We define a matrix with the Twiss parameters :
é b (s) -a (s) ù
ú,
B(s) = S(s) / e rms = ê
êë -a (s) g (s) úû
• The solution of Hill's gives: xTB-1 (s) x = const.
• Substituting x1 = Mx0 gives :
B1 = M B0M
1
0
1T
0
é x ù
x =ê
ú
ë x' û
Emittance preservation
•
•
•
•
We have shown that the rms emittance is a preserved quantity for
Gaussian beams, in drift and with linear magnetic lenses
More generally the phase-space area, emittance, is preserved if only
conservative forces do work on the particles (Liouville’s theorem)
Particle acceleration by an RF-field is not conservative, the emittance will
shrink if the beam is accelerated :
The emittance will shrink proportional to the beam energy increase, given by
bg,. We define a normalized emittance, conserved under acceleration :
NB: bg on this slide is the normalized
velocity and the Lorentz factor
respectively. They are not related to the
Twiss parameters.
eN,rms gb erms
Independent quantities in
each plane, x, y.
Periodic lattices
• Consider a periodic accelerator lattice, with Mss+L(s) from point s to
point s+L. For example, one full turn of a circular accelerator may be
the period.
• The Twiss parameters must obey the following condition :
B(s + L) = M B(s)(M
s+L
s
s+L T
s
)
é b (s) -a (s) ù
ú
, B(s) = ê
êë -a (s) g (s) úû
• The parameters are in this case uniquely defined by Mss+L(s) :
a = a (s) = a (s + L)
b = b (s) = b (s + L)
g = g (s) = g (s + L)
Existence of periodic solutions follows by Floquet Theory.
Circular accelerators: periodic lattice by default.
Twiss parameters: period lattice
Using the solution of Hill’s equation we may re-write the transfer matrix between
two locations in an accelerator in terms of Twiss parameters as :
For a periodic lattice, with period L, this reduces to :
where Fy(s2) y(s1) is the phase-advance from s1 to s2. The lattice Twiss
parameters may then be calculated from the periodic transfer matrix :
Example: thin-lens FODO
f
L
Example from A. Chao
Stability of a lattice
•
Absolute requirement: the lattice structure of an accelerator must be stable:
motion must be bounded
•
Let M(s) be the matrix for one periodic cell, M(s)N is the matrix for the whole
accelerator,. M(s)nN corresponds to n turns in the accelerator
Stability requires that the elements of M(s)nN remain bounded as n  
Applying this general criterion on the FODO cell gives the condition :
•
•
stability Þ f > L / 4
Liming case,
L=4f
Particle motion in a periodic lattice, tune
Reminder: even if beta function is
be periodic, the particle motion
itself is in general not periodic;
after one revolution the initial
phase 0 is altered, since the
particle has advanced by a
certain phase, turn. Phase
advance per turn may be given
as the number of periodic cells,
and the phase-advance per cell,
turn = Ncell cell
We define the tune, Q, as
. the number of betatron
oscillations per turn :
Q = Ncell cell /2p
(From M. Sands)
Tune diagram
• In order to avoid driving resonant instabilities in a circular accelerator, the
fractional part of the tune, Q, must not be mQx + nQy = N, where m,n,N may
be 0,1,2,3….(up to a number depending on the performance requirements).
• These instabilities originate from imperfections of real magnets; beam
undergoes small kicks, which if they add up coherently may blow up beam
size.
Integer and N-integer tunes must be avoided
Evolution of a beam
Rms beam size:
 (s) 
e rms b ( s )
Beam quality
Evolves with lattice
Summary: Transverse parameters
• Betatron oscillations: particle oscillations in the transverse planes,
due to the focusing, for example alternating gradient focusing
 b(s): beta function  square of envelope of the particle motion
– in periodic accelerators: defined uniquely by the lattice
• e: emittance, measurement of beam quality; conserved under
conservative forces
• Q = Ncell cell /2p: accelerator tune
– Integer and half-integer tunes etc. will lead to instabilities and
must be avoided
• An alternative expression for luminosity:
L  f
n1n 2
4 e xb
*
xe y b
*
y
Acknowledgements
• Part of the material presented here is based on Alex
Chao’s USPAS lecture notes and Volker Ziemann’s
lecture notes.