Transverse Beam Dynamics - indico.cern.ch
Transcript of Transverse Beam Dynamics - indico.cern.ch
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics
C. Biscari
Transverse Beam Dynamics
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Outlook
Charged particle and magnetic fields Linear approximation Dipoles and quadrupoles Lorentz equation Horizontal plane From Lorentz equation to Hillβs equation From Hillβs equation to Twiss parameter definition Matrix representation Storage ring β periodic solution Stability diagram Chromaticity Sextupole correction
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The accelerator from the particle point of view is a sequence of Drifts β No external fields β Particles go straight Magnetic fields β Particles are bent according to the magnetic rigidity Electric fields β Particles go straight, gain or loose energy
B E B Reference orbit
Particle trajectory
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Transverse dynamics
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Single particle dynamics in magnetic fields
Reference system
y
x
s
x : horizontal y : vertical s : longitudinal along the trajectory
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Typical Magnetic Fields
β’ Normal: gap appears in the horizontal plane
β’ Skew: rotate around beam axis by p/2n
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Properties of Typical Magnets
Dipoles: used for guiding the particle trajectories
Bx = 0
By = Bo = constant Quadrupoles: used to focus the particle trajectories
Bx = G y
By = -G x
G = constant
Sextupoles: used to correct chromatism and non linear terms
Bx = 2 S x y
By = S ( x2-y2)
S = constant
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Lattice in an accelerator
The first step in calculating a lattice is to consider only the linear components of it (quadrupoles and dipoles). Non linear effects and chromatic aberration corrections will be evaluated later. The trajectory of the reference particle (the particle with nominal energy and initial position and divergence set to zero) along the optics is calculated. All the other beam particles are represented in a frame moving along the reference trajectory, and where the reference particle is always in the center. Coordinate systems used to describe the motion are usually locally Cartesian or cylindrical (typically the one that allows the easiest field representation)
Lattice: Sequence of magnets interleaved with drifts (used for diagnostics, vacuum pumping, Injection, extraction, etc)
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Magnetic rigidity
2222
2
2
2
energy kinetic
energy total
momentum
1
1
pcmcE
mcEK
mcE
mvp
E
pc
c
v
Remember:
Constant magnetic field: B
r q
A charged particle (charge = q) will follow a circle of radius r
Lorentz force: πΉπΏ = ππ£π΅
Centrifugal force: πΉππππ‘π =πΎπππ£
2
π
πΉπΏ = πΉππππ‘π
π΅π =π
π
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Equations of motion
π΅π¦ π₯ = π΅0 +ππ΅π¦
ππ₯π₯ +
1
2!
π2π΅π¦
ππ₯2π₯2+. .
π΅π₯ π¦ =ππ΅π₯
ππ¦π¦ +
1
2!
π2π΅π₯
ππ¦2π¦2+. .
Magnetic field representation (consider only normal terms)
Letβs normalize to momentum
π΅π¦ (π₯)
π/π=
π΅0
π΅ππ+
π π₯
π/π+
1
2!
ππβ²
π/π+ β―
gπ =ππ΅π¦
ππ₯, πβ² =
ππ
ππ₯
Concentrate in horizontal motion: only vertical fields
Gradient and its derivative
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Equations of motion
Consider only linear terms:
π΅(π₯)
π/π=
1
π+ ππ₯ where π =
π
π/π=
1
π΅π
ππ΅π¦
ππ₯
In the ideal orbit, r = const dr/dt = 0
General trajectory: r = r+x, with x<<r
πΉ = ππ2
ππ‘2π₯ + π β
ππ£2
π₯ + π= ππ΅π¦π£
Taylor expansion 1
π₯+πβ
1
π(1 β
π₯
π) Since x<<r
π2
ππ‘2π = 0
ππ2π₯
ππ‘2β
ππ£2
π(1 β
π₯
π) = ππ΅π¦π£
ππ2π₯
ππ‘2β
ππ£2
π(1 β
π₯
π) = ππ£ π΅0 + π₯
ππ΅π¦
ππ₯
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Equations of motion
Pass from t to s as independent variable ππ₯
ππ‘=
ππ₯
ππ
ππ
ππ‘
π2π₯
ππ‘2=
π
ππ‘
ππ₯
ππ
ππ
ππ‘=
π
ππ π₯β²π£
ππ
ππ‘
= π₯β²β²π£2 +ππ₯
ππ
ππ£
ππ π£
Second term is zero
π₯β²β²π£2 βπ£2
π1 β
π₯
π=
ππ£π΅0
π+
ππ£π₯π
π
Divide by v2
π₯β²β² β1
π+
π₯
π2=
π΅0
π π +
π₯π
π π
π₯β²β² β1
π+
π₯
π2= β
1
π+ ππ₯
π₯β²β² + π₯1
π2β π = 0
For vertical plane 1
π2= 0, π = βπ
π¦β²β² + ππ¦ = 0
π₯β²β² + π₯1
π2β π = 0
π¦β²β² + ππ¦ = 0
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Solution of equations of motion
Putting
Horizontal : πΎ = 1
π2 β π
Vertical : πΎ = π If K is constant we get the differential equation of harmonic oscillator with spring constant K
π₯β²β² + πΎπ₯ = 0
π₯ π = π1 cos ππ + π2 sin ππ
π = πΎ
At π = 0 β π₯ = π₯0, π₯β² = π₯β²0
(where now x can represent both x or y)
π1 = π₯0 π2 =π₯β²0
πΎ ,
π₯ π = π₯0 cos πΎπ + π₯β²01
πΎsin πΎπ
π₯β² π = βπ₯0 πΎ sin πΎπ + π₯β²0 cos πΎπ
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Matrix formalism
π =cos πΎπΏ
1
πΎsin πΎπΏ
β πΎ sin πΎπΏ cos πΎπΏ
π₯π₯β² π 1
= ππ₯π₯β² π 0
We can write the equations in matrix formalism: coordinates at point s1 can be obtained knowing the coordinates at s0
Example: Drift Length: L K = 0
ππ·ππππ‘ = 1 πΏ0 1
π₯1 = π₯0 + πΏπ₯β²0
π₯β²1 = π₯β²0
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Matrix formalism
Focusing quadrupole: Length L , K > 0
π =cos πΎπΏ
1
πΎsin πΎπΏ
β πΎ sin πΎπΏ cos πΎπΏ
π₯1 = π₯0πππ πΎπΏ +π₯β²
0
πΎπ ππ πΎπΏ
π₯β²1 = βπ₯β²0 πΎπ ππ πΎπΏ + π₯β²0πππ πΎπΏ
Defocusing quadrupole: Length L , K < 0
π =cosh πΎ πΏ
1
πΎsinh πΎ πΏ
πΎ sinh πΎ πΏ cosh πΎ πΏ
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Thin lens approximation
In parallel with optics π =
cos πΎπΏ1
πΎsin πΎπΏ
β πΎ sin πΎπΏ cos πΎπΏ
L=> 0 KL constant sin πΎπΏ
πΎπΏ -> 1
π =1
πΎπΏβ« πΏ
f positive or negative depending on quad
M =1 0
βπΎπΏ 1 =
1 0
Β±1
π1
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Dipole
Sector magnet: Nominal particle trajectory is perpendicular to dipole entrance Horizontal plane: πΎ = 1 π2 β π
Vertical plane: πΎ = π
If π = 0, πΏ = ππ ππ» =
πππ π ππ πππ
β1
ππ πππ πππ π
ππ =1 ππ0 1
Magnet with field index: π β 0
Exercise β write the matrix
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System of lattice elements: Drifts (MD), quads (MQ), bendings (or dipoles) (MB)
Starting with π₯0
π₯β²0 The final position and divergence of the particle will be
π₯1
π₯β²1
π₯1
π₯β²1
= ππ·π β πππ β ππ·πβ1 β― β ππ΅1 β ππ·2 β ππ1 β ππ·1 βπ₯0
π₯β²0
Or simpler
π₯1
π₯β²1
= π(π 1, π 0) βπ₯0
π₯β²0
The mathematical representation of an accelerator lattice is a sequence of matrices
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Twiss parameters β Betatron tune
π₯β²β² + πΎπ₯ = 0 If K = constant /harmonic oscillator If K varies with s: Hillβs equation π₯β²β² + πΎ(π )π₯ = 0
The solution of the Hill equation is given by:
π₯ π = π π½ π πππ π π + π0
e and Ο0 integration constants Inserting 1 in the equation of motion it can be shown that the phase advance is related to by
π π = ππ
π½(π )
π
0
In storage rings (length of circumference = L) beta is periodic
π½ π + πΏ = π½(π ) One complete turn: phase advance in one turn: Betatron Tune
πΈπ,π =π
ππ
π π
π·π,π (π)
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Solution of Hillβs equation
π₯ = ππ½ cosπ
With Ο = π π + π0 and π½ depending on s
π₯β² = ππ π½
ππ cosπ β ππ½
ππ
ππ sinπ
π₯β² = βπΌπ
π½cosπ β ππ½
ππ
ππ sinπ
π₯ββ= ππ2 π½
ππ 2cosπ β π
π π½
ππ
ππ
ππ sinπ β π
π π½
ππ
ππ
ππ sinπ β ππ½
π2π
ππ 2sinπ β ππ½
ππ
ππ
2cosπ
πΌπΌ
πΌ = β1
2
ππ½
ππ
πΎ =1 + πΌ2
π½
with
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Substituting in the 2nd order equation, and equating to zero terms multipliying sin and cos we get:
β2 ππ π½
ππ
ππ
ππ β ππ½
π2π
ππ 2= 0
Dividing by π and differentiating π½:
ππ½
ππ
ππ
ππ + π½
π2π
ππ 2= 0
π½πβ² β² = 0
π½πβ² = ππ‘π = 1
πβ² = 1
π½ π π =
ππ
π½(π )
π
0
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Betatron oscillations
Betat ron oscillat ion
β’ Beta function :
β Describes the envelope of the betatron oscillation in an accelerator
β’ Phase advance:
β’ Betatron tune: number of betatron oscillations in one orbital turn
bx (s)
y(s) =1
b x (s)0
s
Γ² ds
Qx =y(0 | C)
2p=
ds
b x (s)Γ² /2p =
R
Γ‘b x Γ±
Nominal closed orbit Betatron oscillation
Particles oscillate around the closed orbit, a number of times which is given by the betatron tune. The square of the function by the emittance represents the envelope of the betatron oscillations
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Amplitude of an oscillation
π₯ π = π π½(π ) π½ π represents the envelope of all particle trajectories at a given position s in a storage ring
π₯ π = π π½ π πππ π π + π0
π₯β² π = βπ
π½(π )πΌ π cos π π + π0 + π ππ π π + π0
πΌ π = β1
2π½β²(π )
πΎ π =1 + πΌ2(π )
π½(π )
a, and are the Twiss parameters
Twiss parameters
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Emittance
Inserting in xβ eq.
e is a constant of motion, not depending on s. Parametric representation of an ellipse in x,xβ phase space defined by alfa, beta, gamma: Courant-Snyder invariant emittance Ξ΅
For a single particle, different positions in the storage ring and different turns:
π = πΎ π π₯2 π + 2πΌ π π₯ π π₯β² π + π½ π π₯β²2(π )
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Drawing particle position in different points of a
storage ring
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
x
xp beta = 1
alfa = -2
alfa = 0
alfa = 2
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
x
xp beta = 4
alfa = -2
alfa = 0
alfa = 2
alfa = 0; beta = 1; gamma = (1+alfa*alfa)/beta; eps = 10; for i =1 : npoints thdeg (i) = i*13; teta(i) = thdeg(i)*rad; x(i) = sqrt(eps*beta)*cos(teta(i)); xp(i) = -sqrt(eps/beta)*( alfa*cos(teta(i)) + sin(teta(i)) ); hold on end grid on plot(x,xp,'-r*')
Same area, same emittance, Different orientation
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Liouvilleβs theorem
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
x
xp
Each particle, in absence of non conservative forces, has a constant invariant. Under the influence of conservative forces the particle density in phase space is constant. Magnetic fields of dipoles and quadrupoles are conservative: In a beam the phase space is maintained constant
Beam size π₯πππ₯ = π½(π )π π₯β²πππ₯ = πΎ(π )π
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Evolution of particle trajectory related to evolution
of Twiss Parameters
π₯ π = π π½ π πππ π π + π0
π₯β² π = βπ
π½(π )πΌ π cos π π + π0 + π ππ π π + π0
Developing sin(a+b)
π₯ π = π π½ π (πππ π π πππ π0 β π πππ π π πππ0)
π₯β² π = βπ
π½(π )πΌ π cosπ π πππ π0 β πΌ π sinπ π π πππ0 + sinπ π πππ π0 + cosπ π π πππ0
Starting at π 0 = 0 π 0 = 0 πππ π0 =π₯0
ππ½0, π πππ0 = β
1
ππ₯β²0 π½0 +
πΌ0π₯0
π½0
Substituting above
π₯ π =π½ π
π½0cosπ π + πΌ0π πππ(π ) π₯0 + π½(π )π½0 π πππ(π ) π₯β²0
π₯β²(π ) =1
π½(π )π½0
(πΌ0 β πΌ π ) cosπ π β (1 + πΌ0πΌ π )π πππ(π ) π₯0
+π½ π
π½0cosΟ s β πΌ(π ) π πππ(π ) π₯β²
0
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 28
Particle coordinates evolution as a
function of Twiss parameters
These equations can be written under matrix formalism π½ π β π½π πΌ π β πΌπ π π β ππ
π₯ π π₯β² π
=
π½π
π½0cosππ + πΌ0π ππππ π½π π½0 π ππππ
1
π½π π½0
πΌ0 β πΌπ cosππ β (1 + πΌ0πΌπ )π ππππ
π½π
π½0cosππ β πΌπ π ππππ π₯β²
0
βπ₯0
π₯β²0
Knowing Twiss parameters in two points and phase advance in between it is possible to define the particle position in the second point from the first one with no need of knowing the elements in between
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 29
One turn matrix
MT π =cos2ΟQ + Ξ±osin2ΟQ Ξ²osin2ΟQ
βΞ³osin2ΟQ cos2ΟQ β Ξ±osin2ΟQ= cos2ΟQ β 1 + sin2ΟQ β J
With π = 1 00 1
πππ π½ = πΌ π½βπΎ πΌ
It can be easily demonstrated that
ππ = 1
Knowing the one turn matrix at a certain location and the total phase advance, and a are defined at that location
Exercise: write the expression of and a as a function of one-turn matrix terms
For one revolution: π½ π = π½0 πΌ π β πΌ0 π π β 2πππ₯,π¦
One turn matrix
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N-turns Matrix
ππ = (1 β πππ 2ππ + π½ β π ππ2ππ)π= 1 β πππ 2πππ + π½ β π ππ2πππ Motion bounded (stable) if the elements of MN are bounded:
πππ 2ππ β€ 1 Or
ππ(π) β€ 2
Stability condition
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 31
Transformation of Twiss Parameters Start from two positions, so and s
π₯π₯β²
= π βπ₯0
π₯β²0
π =πΆ ππΆβ² πβ²
Or π = π½π π₯β²
2 + 2πΌπ π₯π₯β² + πΎπ π₯
2 π = π½0π₯β²0
2 + 2πΌ0π₯0π₯β²0 + πΎ0π₯02
π₯0
π₯β²0
= πβ1 βπ₯π₯β²
π₯0 = πβ²π₯ β ππ₯β²
π₯β²0 = βπΆβ²π₯ + πΆπ₯β² Inserting in e and rearranging the terms on x and xβ
π½ π = πΆ2π½0 β 2ππΆπΌ0 + π2πΎ0 πΌ π = βπΆπΆβ²π½0 + ππΆβ² + πβ²πΆ πΌ0 β ππβ²πΎ0
πΎ π = πΆβ²2π½0 β 2πβ²πΆβ²πΌ0 + πβ²2πΎ0
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 32
And can be written in matrix notation: given the Twiss parameters Ξ±, Ξ², Ξ³ at any point in the lattice we can transform them and calculate their values at any other point in the ring if we know the transfer matrix.
π½πΌπΎ
=πΆ2 β2πΆπ π2
βπΆπΆβ² ππΆβ² + πβ²πΆ βππβ²πΆβ²2 β2πβ²πΆβ² πβ²2
π½π
πΌπ
πΎπ
Transformation of Twiss Parameters
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rms Emittance
The emittance is the area of the phase space occupied by all particles in a beam Each particle has its own βinvariant emittanceβ rms emittance represents the beam characteristics, and is defined as:
ππ₯πππ = π₯2 π₯β²2 β π₯ π₯β² 2
Rms values behave the same for all distributions in linear systems Most usual beam distributions are gaussian
Xβ
x
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Emittance and beam dimensions
β’ The emittance is the area of the phase space occupied by the particles
With the emittance and the Twiss parameters in a point of the
accelerator, the beam dimensions are obtained : sx,y e sβx,y
34
Ellipse area= pex
1
'
'
2
2
2
xxx
xx
xx
xx
xx
x
x
a
ea
e
e
222 '' xxxxx e
x
px
xxx es
s 'x = gxex
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Effect of a quad on the phase space
The phase space orientation indicates if the beam trajectories are focused or defocused (at rms values):
s0 s1 s2
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 36
Example of betatron functions in a
storage ring - ALBA
Focusing quads above the line Defocusing quads below, Dipoles on both sides
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Emittance in linacs and transfer lines
Adiabatic damping: The Courant-Snyder invariant emittance Ξ΅ decreases if we accelerate the particle. This is called βadiabatic dampingβ. Particle with momentum p0:
π02 = ππ 0
2 + ππ₯02 + ππ¦0
2
The slope of the trajectory is
π§β²β² =ππ§
ππ (π§ = π₯ or π¦)
Accelerating the particle: ps increases but pz does not change
π§β² + βπ§β² =ππ§
ππ + βππ =
ππ§
ππ 1 +βππ ππ
β π§β² 1 ββππ
ππ
And therefore
βπ§β² = βπ§β²βπ
π
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In the phase space (z,zβ):
For each particle i: π§π = ππ½ cos π + π0 π§β²π = βπ
π½sin π + π0
The emittance of the particle is:
π = π½π§β²π2 + πΎπ§π
2 The change in e applying Dzβ
βπ = 2π½π§πβ²βπ§π
β² = β2π½π§πβ²2 βπ
π= β2π
βπ
ππ ππ2 π + π0
Averaging over all particles:
βπ = βπβπ
π =>
ππ
π= β
ππ
π
π π = π0π0
π
The emittance decreases in linacs as the beam is accelerated. Normalized emittance is defined as the invariant part:
ππ = ππ½πΎ
Where now and are not the Twiss parameters, but: π½ =π£
π πππ πΎ =
1
1βπ½2
In absence of other phenomena, the normalized emittance does not change during the acceleration
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Off-Momentum Particles
Magnets are chromatic elements:
Dipoles: π =π
ππ΅β
βπ
π0=
βπ
π0 =
βπ
π0
Quads: πΎ =πΊ
π΅π=
πΊπ
π
βπ = βπ0
βπ
π0
βπΎ = βπΎ0
βπ
π0
Dp/p<<1 Off-momentum particles are not oscillating around design orbit, but around a different closed orbit (chromatic closed orbit). The displacement between the design and chromatic orbits is regulated by the dispersion function D(s). For particles with energy deviation the Hillβs equation has an extra term and is not homogeneous:
π₯β²β² + πΎ π π₯ =1
π
βπ
π0
The solution is the sum of the solution of the homogenous equation + a term of dispersion:
π₯ = π₯π»ππ + π· π βπ
π0
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Dispersion equation
The dispersion equation is:
π·β²β²(π ) + πΎ π π·(π ) =1
π
The dispersion in a storage ring is a closed orbit. Example: Dipole The solution of the dispersion equation is given by the homogeneous equation solution plus a particular one of the non homogenous:
π· π = π·0 cosπ
π+ π·β²0π sin
π
π+ π 1 β cos
π
π
π·β² π = βπ·0
πsin
π
π+ π·β²
0 cosπ
π+ sin
π
π
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 41
Or in matrix formalism, representing the particle coordinates including the energy deviation:
π₯π₯β²
βπ π = π3π₯3
π₯0
π₯β²0βπ π
π3π₯3 =πΆ(π ) π(π ) π·(π )
πΆβ²(π ) πβ²(π ) π·β²(π )0 0 1
ππ πππ‘ππ =
πππ π ππ πππ π(1 β πππ π)
β1
ππ πππ πππ π π πππ
0 0 1
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 42
Also quadrupole focusing properties depend on particle energy: The focusing strength is
πΎ = 1
π΅π
ππ΅
ππ₯
Their implication on the dispersion function is of second order
0
0
0
000
:error1 kp
pkkkG
p
p
p
eG
pp
eG
p
ek p
DDD
D
DD
But they produce chromaticity
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 43
Particles with different momentum have different betatron tunes (The higher the energy the lower the tunes) Chromatism is the spread in tune divided by the spread in momentum
xx,y =DQx,y
Dp / p= -
1
4pki
i
Γ₯ Libx,i
And is given by the sum of the contributions of all elements in the ring
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 44
Chromaticity correction
β’ The natural chromaticity produced by quadrupoles is always
negative. x has maxima at focusing quads (QF, kx >0) and
minima at defocusing ones (QD). Horizontal chromaticity is
dominated by the focusing quads, and x at their positions.
β’ Chromaticity must be corrected to avoid large tune spread
(driving resonances or collective effects like head-tail
instability)
44
Sextupoles placed where D is not zero, act on particles which are not on the nominal orbit, but on x = x0 + D Dp/p The highest the energy deviation, the largest x, the strongest the focusing(defocusing) force
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 45
DQx =1
4pbx,s0
mLsx con x = DDp
p
DQx
Dp / p=
1
4pbx,s0
mLsD
Sextupoles
β’ The field is quadratic :
β’ Normalized gradient:
β’ Kick:
β’ Change in tune due to sextupole:
45
Bx = m xy
By =1
2m x2 - y2( )
Dx' = -mx2, Dy' = 2mxy
m=1
Br
d2By
dx2[m-3]
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 46
Chromaticity correction
β’ Sextupole position are chosen so that the produced
chromaticity counteracts the natural one created by
quadrupoles
β’ Mainly where the dispersion is high and x e y are well
separated.
β’ Total chromaticity is then:
46
xx =DQx
Dp / p= -
1
4pbx,ikiLq
i
Γ₯ +1
4pbx,imLsD
i
Γ₯
Quadrupoles Sextupoles
Accelerator Physics - UAB 2014-15 C. Biscari - Lectures 4-5-6 Transverse Dynamics 47