Accelerator Physics Transverse motion. Elena Wildner. Accelerator coordinates. Vertical. Horizontal. Longitudinal. It travels on the central orbit. 2. Rotating Cartesian Coordinate System. Two particles in a dipole field. Particle A. Particle B.
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Vertical
Horizontal
Longitudinal
It travels on the central orbit
2
Rotating Cartesian Coordinate System
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Particle A
Particle B
3
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Particle B
Particle A
displacement
2p
0
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Quadrupole magnets
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Magnetic field
Hyperbolic contour
x · y = constant
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By x
Bx y
Magnetic field
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Focusing Quadrupole (QF)
Force on particles
Defocusing Quadrupole (QD)
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QF
QF
QD
L2
L1
‘FODO’ cell
Or like this……
Centre of
second QF
Centre of
first QF
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x’
x
s
x
ds
dx
x = displacement
x’ = angle = dx/ds
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These transverse oscillations are called Betatron Oscillations, and they exist in both horizontal and vertical planes.
The number of such oscillations/turn is Qx or Qy. (Betatron Tune)
(Hill’s Equation) describes this motion
If the restoring coefficient (K) is constant in “s” then this is just SHM
Remember “s” is just longitudinal displacement around the ring
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Remember is still a function of s
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and its second derivative
into our initial differential equation
Sine and Cosine are orthogonal and will never be 0 at the same time
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, which after differentiating gives
and
gives:
Which is the case as:
since
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w
b
a
d
'
=
=

w
b
ds
2
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x’
x
and
= 3/2
= 0 = 2
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x’
x’
x
x
Area =· r1· r2
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x’
x’
x
x
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x’
emittance
beam
x
acceptance
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x2 = x1 + L.x1’
x1’
x1
x1’ small
L
}
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deflection
x1
x2
x1’
Remember By x and the deflection due to the magnetic field is:
x2’
}
Provided L is small
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and
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Number of betatron oscillations per turn
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, where μ =Δ over a complete turn
Over one complete turn
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1 = + d
New phase advance
Change in phase advance
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equals:
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QD
QF
Remember 1 = + d and dμ is small
dQ = dμ/2π
,but:
In the horizontal plane this is a QF
If we follow the same reasoning for both transverse planes for both QF and QD quadrupoles
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Let dkF = dk for QF and dkD = dk for QD
bhF, bvF = b at QF and bhD, bvD = b at QD
Then:
This matrix relates the change in the tune to the change in strength of the quadrupoles.
We can invert this matrix to calculate change in quadrupole field needed for a given change in tune
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Acknowldements to
for course material
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