Syllabus and slides Syllabus and slides • Lecture 1: Overview and history of Particle accelerators (EW) • Lecture 2: Beam optics I (transverse) (EW) • Lecture 3: Beam optics II (longitudinal) (EW) • Lecture 4: Liouville's theorem and Emittance (RB) • Lecture 5: Beam optics and Imperfections (RB) • Lecture 6: Beam optics in linac (Compression) (RB) • Lecture 7: Synchrotron radiation (RB) • Lecture 8: Beam instabilities (RB) • Lecture 9: Space charge (RB) • Lecture 10: RF (ET) • Lecture 11: Beam diagnostics (ET) • Lecture 12: Accelerator Applications (Particle Physics) (ET) • Visit of Diamond Light Source/ ISIS / (some hospital if possible) The slides of the lectures are available at http://www.adams-institute.ac.uk/training Dr. Riccardo Bartolini (DWB room 622)
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Syllabus and slides Lecture 1: Overview and history of Particle accelerators (EW) Lecture 2: Beam optics I (transverse) (EW) Lecture 3: Beam optics II.
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Syllabus and slidesSyllabus and slides
• Lecture 1: Overview and history of Particle accelerators (EW)• Lecture 2: Beam optics I (transverse) (EW)• Lecture 3: Beam optics II (longitudinal) (EW)• Lecture 4: Liouville's theorem and Emittance (RB)• Lecture 5: Beam optics and Imperfections (RB)• Lecture 6: Beam optics in linac (Compression) (RB)• Lecture 7: Synchrotron radiation (RB)• Lecture 8: Beam instabilities (RB)• Lecture 9: Space charge (RB)• Lecture 10: RF (ET)• Lecture 11: Beam diagnostics (ET)• Lecture 12: Accelerator Applications (Particle Physics) (ET)• Visit of Diamond Light Source/ ISIS / (some hospital if possible)
The slides of the lectures are available at
http://www.adams-institute.ac.uk/training
Dr. Riccardo Bartolini (DWB room 622)
R. Bartolini, John Adams Institute, 1 May 2013 1/28
Lecture 4: Emittance and Liouville’s theorem
Hill’s equations (recap)
More on transfer matrix formalism
Courant-Snyder Invariant
Emittance
Liouville’s theorem
2/28
Linear betatron equations of motion (recap)
In the magnetic fields of dipoles magnets and quadrupole magnets the coordinates of the charged particle w.r.t. the reference orbit are given by the
Hill’s equations
0)(2
2
ysKds
ydy
x
sB
BssK z
x
)(1
)(
1)(
2
x
sB
BsK z
z
)(1
)(
No periodicity is assumed but for a circular machine Kx, Kz and are periodic
These are linear equations (in y = x, z). They can be integrated.
weak focussing of a dipole
quadrupole focussing
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Pseudo-harmonic oscillations (recap)
The solution can be found in the form
)s(cos)s()s(y yyy s
s yy
0
)'s(
'ds)s(
which are pseudo-harmonic oscillations
The beta functions (in x and z) are proportional to the square of the envelope of the oscillations
The functions (in x and z) describe the phase of the
oscillations
R. Bartolini, John Adams Institute, 1 May 2013
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Principal trajectories (recap)The solutions of the Hill’s equation can be cast equivalently in the form of principal trajectories. These are two particular solutions of the homogeneous Hill’s equation
0)('' ysky
which satisfy the initial conditions
C(s0) = 1;C’(s0) = 0; cosine-like solution
S(s0) = 0;S’(s0) = 1; sine-like solution
The general solution can be written as a linear combination of the principal trajectories
)(')()( 00 sSysCysy
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Principal trajectories vs pseudo harmonic oscillations
))s(sin)s((cos)s(
)s(C 00
)s(sin)s()s(S 0
We can express amplitude and phase functions
)s(C)s(S
)s(Sarctg)s(
00
and viceversa
)s(C)s(S
)s(C)s(S)s(S1)s(
00
200
2
0
or more simply2
0 )s(sin
)s(S1)s(
)s(sin
)s(cos)s(
)s('S
)s( 0
)s(cos)s()s(y
)s(cos)s()s(sin)s(
)s('y
in terms of the principal trajectories )s(S'y)s(Cy)s(y 00
Simple algebraic manipulations yield
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Principal trajectories (recap)As a consequence of the linearity of Hill’s equations, we can describe the evolution of the trajectories in a transfer line or in a circular ring by means of linear transformations
)('
)(
)(')('
)()(
)('
)(
0
0
sy
sy
sSsC
sSsC
sy
sy
This allows the possibility of using the matrix formalism to describe the evolution of the coordinates of a charged particles in a magnetic lattice
)(')('
)()(
sSsC
sSsCM 21
C(s) and S(s) depend only on the magnetic lattice not on the particular initial conditions
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Matrices of most common elementsTransfer lines or circular accelerators are made of a series of drifts and quadrupoles for the transverse focussing and accelerating section for acceleration.
Each of these element can be associated to a particular transfer matrix
10
d1M
)LKcosh()LKsinh(K
)LKsinh(K
1)LKcosh(
M
)L|K|cos()L|K|sin(|K|
)L|K|sin(|K|
1)L|K|cos(
M
1KL
01M
1L|K|
01M
Matrix of a drift space
Matrix of a focussing quadurpole
Matrix of a defocussing quadurpole
Thin lens approximationL 0, with KL finite
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Matrix formalism for transfer lines
For each element of the transfer line we can compute, once and for all, the corresponding matrix. The propagation along the line will be the piece-wise composition of the propagation through all the various elements
L Q2Q1
s1 s2
2
1
1221 /1*/1
/1
1/1
01
10
1
1/1
01
fLf
LfL
f
L
fM
2121
11
*
1
ff
L
fff
Notice that it works equally in the longitudinal plane, e.g.
1/1
0121 f
M33
sin1
s
s
mc
qVL
f
thin lens quadrupole associate to an RF cavity of voltage V and length L
R. Bartolini, John Adams Institute, 1 May 2013
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Matrix formalism and analogy with geometric optics
Particle trajectories can be described with a matrix formalism analogous to that describing the propagation of rays in an optical system.
The magnetic quadrupoles play the role of focussing and
defocussing lenses, however notice that, unlike an optical lens,
a magnetic quadrupole is focussing in one plane and
defocussing in the other plane.
Magnetic field of a quadrupole and Lorentz force
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A slightly more complicated example: the FODO lattice (I)
Consider an alternating sequence of focussing (F) and defocussing (D) quadrupoles separated by a drift (O)
The transfer matrix of the basic FODO cell reads
2
2
2 f4
L
f2
L1
f2
Lf4
L1L
f2
L1
102
L1
1f
101
102
L1
1f
101
M
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In terms of the amplitude and phase function the transfer matrix will read
where 0 , 0 and the phase 0 are computed at the beginning of the segment of transfer line
We still have not assumed any periodicity in the transfer line.
If we consider a periodic machine the transfer matrix over a whole turn reduces to (put = the phase advance in one turn)
sin)s(cos)s()s(
sin))s(1(cos))s((
sin)s()sin(cos)s(
)s('S)s('C
)s(S)s(CM
0
0
00
000
ss0
sincossin
sinsincosM
00
00ss 00
Matrix elements from principal trajectories and optics functions
0
20
01
This is the Twiss parameterisation of the one turn map
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Consider a circular accelerator with transfer matrix over one turn equal to M (one turn map). Using the Twiss parameterisation for M
JsinIcossincossin
sinsincosM
00
00
Stability of motion with the matrix formalism
00
00J
01 xMx 0n
n xMx
It can be proven that (see bibliography)
nsinncosnsin
nsinnsinncosM
00
00n
After n turns, the transformation of the particle coordinates will be given by the successive application of the one turn matrix n times
In order for the phase advance to be real and hence for the motion to be a stable oscillation, the one turn map must satisfy the condition
1|trM|2
1|cos|
R. Bartolini, John Adams Institute, 1 May 2013
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Example: the FODO lattice (II)
Using the Twiss parameterisation of the matrix or the FODO cell we have
sincossin
sinsincos
f4
L
f2
L1
f2
Lf4
L1L
f2
L1
M
2
2
2
hence
2
2
f8
L1trM
2
1cos
The stability requires
1|f8
L1||cos|
2
2
4
Lf
In a similar way we can compute the optics functions at the beginning of the FODO cell.
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Optics functions in a transfer line
While in a circular machine the optics functions are uniquely determined by the periodicity conditions, in a transfer line the optics functions are not uniquely given, but depend on their initial value at the entrance of the system.
We can express the optics function in terms of the principal trajectories as
0
0
0
22
22
SSC2C
SSSCCSCC
SCS2C
''''
''''
This expression allows the computation of the propagation of the optics function along the transfer lines, in terms of the matrices of the transfer line of each single element, i.e. also the optics functions can be propagated piecewise from
)(')('
)()(
sSsC
sSsCM 21
R. Bartolini, John Adams Institute, 1 May 2013
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Examples
0
0
02
100
10
21
s
ss
In a drift space
The function evolve like a parabola as a function of the drift length.
In a thin focussing quadrupole of focal length f = 1/KL
The function evolve like a parabola in terms of the inverse of focal length
1
01
KLM
0
0
0
2 12)(
01
001
KLKL
KL
10
1 sM
R. Bartolini, John Adams Institute, 1 May 2013
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Diamond LINAC to booster transfer line
Optics functions from the LINAC
(Twiss parameters of
the beam)
Booster optics functions at the injection point
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Transfer line example: Diamond LTB
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Betatron motion in phase space (recap)
The solution of the Hill’s equations
0y)s(Kds
ydy2
2
describe an ellipse in phase space (y, y’)
)s(cos)s()s(y
)s(cos)s()s(sin)s(
)s('y
area of the ellipse in phase space (y, y’) is /)y'yy2'y()s(A 222
R. Bartolini, John Adams Institute, 1 May 2013
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Courant-Snyder invariant (I)
Hill’s equations have an invariant
0)(2
2
ysKds
ydy .consty'yy2'y)s(A 222
This invariant is the area of the ellipse in phase space (y, y’) multiplied by .
This can be easily proven by substituting the solutions y, y’
into A(s). You will get the constant
A(s) is called Courant-Snyder invariant
)s(cos)s()s(y
)s(cos)s()s(sin)s(
)s('y
R. Bartolini, John Adams Institute, 1 May 2013
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Courant-Snyder invariant (II)
Whatever the magnetic lattice, the area of the ellipse stays constant (if the Hill’s equations hold)
At each different sections s, the ellipse of the trajectories may change orientation shape and size but the area is an invariant.
This is true for the motion of a single particle !
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Real beams – distribution function in phase space
A beam is a collection of many charged particles
The beam occupies a finite extension of the phase space and it is described by a distribution function such that
),z,p,y,p,x(x yx 1xd)s,x( 6 The beam distribution is characterised by the momenta of various orders
xd)s,x(x)s(x 6jj Average coordinates (usually zero)
xd)s,x()xx)(xx()xx()xx()s(R 6jjiijiij
The R-matrix also called -matrix describes the equilibrium properties of the beam giving the second order momenta of the distribution
R11 = bunch H size; R33 = bunch Y size; R55 bunch Z size; R66 = energy spread
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Gaussian beams
In many cases the equilibrium beam distribution is a Gaussian distribution
ji1
ij )xx()xx(R2
1
3e
Rdet)2(
1)s,x(
Usually the three planes are independent hence in each plane
2
x'xx2'x
'xx
22
eRdet2
1)'x,x(
The isodensity curves are ellipses
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-matrix for Gaussian beams
zzzyzx
yzyyyx
xzxyxx
with (assuming <x> = 0 and <x’> = 0)
2
2
xx'x'xx
'xxx
Direct computation using
xxxx
xxxxxx
The 6x6 –matrix can be partitioned into nine 2x2 submatrices
We can associate an ellipse with the Gaussian beam distribution. The evolution of the beam is completely defined by the evolution of the ellipse
2x
2x
2xxxxx )det(
The ellipse associated to the beam is chosen so that its Twiss parameters are those appearing in the distribution function, hence, e.g.
222x 'xx'xx xxx xx'x
yields
2
x'xx2'x 22
e2
1)'x,x(
R. Bartolini, John Adams Institute, 1 May 2013
24/28
Generic beams – rms emittanceFor a generic beam described by a distribution functions we can still compute the average size and divergence and the whole -matrix
2
2
xx'x'xx
'xxx
xxxx
xxxxxx
we associate to this distribution the ellipse which has the same second order momenta Rij and we deal with this distribution as if it was a Gaussian distribution
2x
2xxxxxx )det(
and since
The invariant of the ellipse will be 222x 'xx'xx
which is the rms emittance of the beam
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Beam emittance and Courant-Snyder invariant
We have seen that the beam distribution can be associated to an ellipse containing 66% of the beam (one r.m.s.)
In this way the beam rms emittance is associated with the Courant Snyder invariant of the betatron motion
This links a statistical property of the beam (rms emittance) with single particle property of motion (the Courant-snyder invairnat)
In this way the Courant-Snyder invariant acquires a statistical significance as rms emittance of the beam. Hence the beam rms emittance is a conserved quantity also for generic beams.
This is valid as long as the Hill’s equations are valid or more generally the system is Hamiltonian. As such the conservation of the emittance is a manifestation of the general theorem of Hamiltonian system and statistical mechanics known as the Liouville theorem
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Liouville’s theoremLiouville’s theorem: In a Hamiltonian system, i.e. n the absence of collisions or dissipative processes, the density in phase space along the trajectory is invariant’.
0f,Ht
f
t
p
p
f
t
q
q
f
t
f
dt
df n
1k
k
k
n
1k
k
k
Liouville theorem states that volume of 6D phase are preserved during the beam evolution (take f to be the characteristic function of the volume occupied by the beam). However if the Hamiltonian can be separated in three independent terms
),(H)p,y(H)p,x(H),,p,y,p,x(HH yxyx
The conservation of the phase space density occurs for the three projection on the
(x, px) plane (Horizontal emittance)
(y, py) plane (Vertical emittance)
(z, pz) = (, ) plane (longitudinal emittance)R. Bartolini, John Adams Institute, 1 May 2013
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Beam emittance and Liouville’s theorem
The Courant-Snyder invariant is the area of the ellipse phase space.
The conservation of the area is a general property of Hamiltonian systems (any area not only ellipses !)
The invariance of the rms emittance is the particular case of a very general statement for Hamiltonian systems (Liouville theorem)
This is valid as long as the motion is Hamiltonian, i.e.
No damping effects, no quantum diffusion, due to emission of radiation
no scattering with residual gas, no beam beam collisions
no collective effects (e.g. interaction with the vacuum chamber, no self interaction)