Orbit displacement arises from dipole fields that establish the ideal ... B y. (B )
design. 1+ x. 2 p o p. B y = B o + оB x d. 2 x ds. 2. +. 1. 2. 2p o p p. +. B. (B )
design.
Unit 9 - Lecture 18
Deviations from the design orbit
William A. Barletta Director, United States Particle Accelerator School Dept. of Physics, MIT
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Off- momentum particles & Momentum dispersion
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Momentum dispersion function of the lattice Off-momentum particles undergo betatron oscillations about a new class of closed orbits in circular accelerators Orbit displacement arises from dipole fields that establish the ideal trajectory + less effective quadrupole focusing
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Start with the equation of motion We have derived
By d x + x 2 = 1+ 2 ds (B) 2
2 x
Using p = (B)
d x + x 2 = 1+ 2 ds (B) design 2
By
2 x po p
Consider fields that vary linearly with transverse position
By = Bo + Bx Then neglecting higher order terms in x/ we have
1 p po 1 p d 2 x 1 2 po p po B x = + +
2 2 ds p p p (B) design p US Particle Accelerator School
Equation for the dispersion function Define D(x,s) such that x = D(x,s) (p/po) Look for a closed periodic solution; D(x,s+L) = D(x,s) of the inhomogeneous Hill’s equation 1 po d 2 D 1 2 po p po B D = + + ds2 2 p p (B) design p K(s)
For a piecewise linear lattice the general solution is D a = D out c
b D e + d D in f
or
a b D = D c d 1 out 0 0
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e D f D 1 1 in
Solution for D The solution for the homogeneous portion is the same as that for x and x’ The values of M13 and M23 for ranges of K are
[ (
e Bo cosh pK
) ]
K l 1
e p K
1 eBo l l 2 p
[
(
e Bo 1 cos K l pK
[ (
Bo sinh
K l
)]
eBo l p
)]
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e p K
[ (
Bo sin K l
)]
What is the shape of D? In the drifts D'' = 0 D has a constant slope
For focusing quads, K > 0 D is sinusoidal
For defocusing quads, K < 0 D grows (decays) exponentially
In dipoles, Kx(s) = G2 D is sinusoidal section “attracted to” D = 1/G =
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SPEAR-I dispersion
From: Sands SLAC - pub 121
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The condition for the achromatic cell We want to start with zero dispersion and end with zero dispersion This requires S
ds Ia = a(s) =0 (s) 0 and S
ds =0 Ib = b(s) (s) 0
In the DBA this requires adjusting the center quad so that the phase advance through the dipoles is US Particle Accelerator School
Momentum compaction Consider bending by sector magnets The change in the circumference is p C = + D d po
d
Therefore C = C
D ds p = ds p o
D p po
or
D 1 = t
For simple lattices t ~ Q ~ number of cells of an AG lattice US Particle Accelerator School
Total beam size due to betatron oscillations plus momentum spread. Displacement from the ideal trajectory of a particle First term = increment to closed orbit from off-momentum particles Second term = free oscillation about the closed orbit
x total
p =D + x po
Average the square of xtotal to obtain the rms displacement 2 (s) p 2 2 x (s) = + D (s) po
in a collider, design for D = 0 in the interaction region US Particle Accelerator School
Chromatic aberrations The focusing strength of a quadrupole depends on the momentum of the particle
1 1 p f
==> Off-momentum particles oscillate around a chromatic closed orbit NOT the design orbit Deviation from the design orbit varies linearly as
p x D = D(s) p The tune depends on the momentum deviation Expressed as the chromaticity
Q Qx /Qx or x = Qx = p / po p / po
Qy /Qy Q or x = Qy = p / po p / po
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Example of chromatic aberation
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Chromatic aberration in muon collider ring
1
1
1
-1
-1
-1
X`[mrad]
Y`[mrad]
X`[mrad]
View at the lattice0.005 beginn
-0.005
Y [cm]
View at the lattice0.005 end
Y [cm]
View at the lattice0.005 end
-0.005
X [cm]
View at the lattice0.005 end
1
1
X [cm]
1
-0.005
-0.005
Y`[mrad] -0.005
Y [cm]
View at the lattice0.005 beginn
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From: Alex Bogacz and Hisham Sayed presentation
-1
-1
-1
Y`[mrad]
Y`[mrad]
-0.005
Y [cm]
View at the lattice0.005 end
Chromatic closed orbit The uncorrected, “natural” chromaticity is negative & can lead to a large tune spread and consequent instabilities Correction with sextupole magnets
natural =
Design orbit On-momentum particle trajectory
1 4
(s)K(s)ds 1.3Q Design orbit Chromatic closed orbit Off-momentum particle trajectory
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Measurement of chromaticity Steer the beam to a different mean radius & different momentum by changing rf frequency, fa, & measure Q p f a = f a p
Since
Q =
and
p r = Dav p
p p
dQ = f a df a
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Chromaticity correction with sextupoles
US Particle Accelerator School From: Wiedemann, Ch. 7, v.1
Sextupole correctors Placing sextupoles where the betatron function is large, allows weak sextupoles to have a large effect Sextupoles near F quadrupoles where ßx is large affect mainly horizontal chromaticity Sextupoles near D quadrupoles where ßy is large affect mainly horizontal chromaticity
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Coupling Rotated quadrupoles & misalignments can couple the motion in the horizontal & vertical planes A small rotation can be regarded a normal quadrupole followed by a weaker quad rotated by 45° By Bx Bs,x = x and Bs,y = y y x This leads to a vertical deflection due to a horizontal displacement
Without such effects Dy = 0 In electron rings vertical emittance is caused mainly by coupling or vertical dispersion US Particle Accelerator School
Field errors & Resonances
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Integer Resonances Imperfections in dipole guide fields perturb the particle orbits Can be caused by off-axis quadrupoles
==> Unbounded displacement if the perturbation is periodic The motion is periodic when mQx + nQy = r M, n, & r are small integers
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Effect of steering errors The design orbit (x = 0) is no longer a possible trajectory Small errors => a new closed orbit for particles of the nominal energy Say that a single magnet at s = 0 causes an orbit error = Bl (B )
Determine the new closed orbit
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After the steering impulse, the particle oscillates about the design orbit At s = 0+, the orbit is specified by (xo, x'o) Propagate this around the ring to s = 0- using the transport matrix & close the orbit using ( 0, )
x o 0 x o M + = x o x o specifies the new closed orbit
xo 1 0 = (I M) x o
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Recast this equation As ()ring = Q, M can be written as M ring
cos(2Q) + sin(2Q) , sin(2Q) =
sin(2Q), cos(2Q) sin(2Q)
After some manipulation (see Syphers or Sands) 1/ 2 (s) 1/ 2 (0) x(s) = cos( (s) Q) 2sin Q
As Q approaches an integer value, the orbit will grow without bound US Particle Accelerator School
The tune diagram The operating point of the lattice in the horizontal and vertical planes is displayed on the tune diagram The lines satisfy mQx + nQy = r M, n, & r are small integers Operating on such a line leads to resonant perturbation of the beam Smaller m, n, & r => stronger resonances US Particle Accelerator School
Example: Quadrupole displacement in the Tevatron Say a quad is horizontally displaced by an amount Steering error, x' = /F where F is the focal length of the quad
For Tevatron quads F 25 m & Q = 19.4. Say we can align the quads to the center line by an rms value 0.5 mm For = 0.5 mm ==> = 20 rad If = 100 m at the quad, the maximum closed orbit distortion is xˆ quad =
20 μrad 100 m = 1 mm 2 sin (19.4 )
The Tevatron has ~ 100 quadrupoles. By superposition 2 xˆ = N 1/quad xˆ quad = 10 mm for our example
Steering correctors are essential! US Particle Accelerator School
Effect of field gradient errors Let
Kactual(s) = Kdesign(s) +k(s)
where k(s) is a small imperfection k(s) => change in (s) => Q Consider k to be non-zero in a small region at s = 0 ==> angular kick y' ~ y y = ky s
(1)
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Sinusoidal approximation of betatron motion Before s = 0 -
s y = bcos n
(2)
At s = 0 + the new (perturbed) trajectory will be s y = (b + b)cos + n
where
b + b sin = y n
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Sinusoidal approximation cont’d If y' is small, then b and will also be small ==>
n y b
Total phase shift is 2Q; the tune shift is
(1) & (2)
n ks phase shift
Principle effect of the gradient error is to shift the phase by ks Q = n 2 2
The total phase advanced has been reduced US Particle Accelerator School
This result overestimates the shift The calculation assumes a special case: = 0 The particle arrives at s = 0 at the maximum of its oscillation
More generally for 0 The shift is reduced by a factor cos2 The shift depends on the local value of
On successive turns the value of will change the cumulative tune shift is reduced by < cos2 > = 1/2 ==>
Q =
1 (s)( ks) 4
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Gradient errors lead to half-integer resonances For distributed errors Q =
1 4
(s)k(s)ds
Note that ~ K-1/2 ==> Q 1/ K1/2 Q k ==> Q/Q k 2 k/K (relative gradient error) Or Q ~ Q (B'/ B') Machines will large Q are more susceptible to resonant beam loss Therefore, prefer lower tune US Particle Accelerator School
Tune shifts & spreads Causes of tune shifts Field errors Intensity dependent forces • Space charge • Beam-beam effects
Causes of tune spread Dispersion Non-linear fields • Sextupoles
Intensity dependent forces • Space charge • Beam-beam effects
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Spread & shifted tune
Example for the RHIC collider
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Stopbands in the tune diagram Think of the resonance lines as having a width that depends on the strength of the effective field error Also the operation point has a finite extent Resonances drive the beam into the machine aperture
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In real rings, aperture may not be limited by the vacuum chamber size Resonances can capture particles with large amplitude orbits & bring them in collision with the vacuum chamber ==> “virtual” or dynamic aperture for the machine Strongly non-linearity ==> numerical evaluation Momentum acceptance is limited by the size of the RF bucket or by the dynamic aperture for the offmomentum particles. In dispersive regions off-energy particles can hit the dynamic aperture of the ring even if p is still within the limits of the RF acceptance
Cornell ILC-DR
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