USPAS - Transverse Measurements [presentation slides]

33 Pages·2004·0.369 MB·English

Measuring Lots of (transverse) Good
Things: Betatron Tunes, Beta
Functions, Chromaticity, Dispersion...
Ken Jacobs
SRC Accelerator Development Group
USPAS sponsored by
the University of Wisconsin, Madison
Beam Instrumentation Laboratory at the SRC
June 21 - July 2, 2004
1
Outline
Fundamental accelerator properties, measured “by hand”:
• Betatron tunes • Chromaticity
– Harmonics, aliasing – Corrected
– Distinguishing horizontal – Natural
from vertical
• Dispersion
• Beta functions
– RF frequency variation
– Quadrupole variation
• Beam Based
– Orbit kick
alignment
References:
Measurement and Control of Charged Particle Beams, by M.G. Minty and F.
Zimmermann, Springer, 2003.
Beam Measurment, edited by s-i. Kurokawa, S.Y. Lee, E. Perevedentsev,
and S. Turner (Proceedings of the Joing US-CERN-Japan-Russian School
on Particle Accelerators), World Scientific, 1999.
2
Betatron Tune Spectrum
Start simple: Consider a single point particle of charge q circulating at
velocity v in a ring of circumference C.
0
Rotation period T = C/v .
0 0
ω π
Rotation frequency f = /2 = 1/T .
0 0 0
A beam current monitor at a fixed location in the ring would see a current
∞
( ) ∑ ( )
I t = q δ t − nT
0
n=−∞
This is a “comb function” in the time domain:
amplitude
-4 -3 -2 -1 0 1 2 3 t/T
0
3
Fourier transform to the frequency domain. The beam current monitor
signal is a comb function in frequency space,
∞
( ) ∑ ( )
I ω = qω δ ω − nω
0 0
n=−∞
amplitude
-4 -3 -2 -1 0 1 2 3 ω/ω
0
Viewing this signal on a spectrum analyzer, you would see lines at the ring
rotation frequency, and all its harmonics.
4
Now include transverse betatron oscillations of amplitude A around a
ω νω ν
closed orbit located at x . The betatron frequency is = , where
0 β 0
is the tune. A monitor sensitive to the dipole moment of the beam (e.g.,
a beam position monitor) would see the time domain signal
( ) ∞
( ) iω t ∑ ( )
D t = q x + Ae β δ t − nT
0 0
n=−∞
The Fourier transform of this is
∞
[ ( ( ))]
( ) ∑ ( )
D ω = qω x δ ω − nω + Aδ ω − nω + ω
0 0 0 0 β
n=−∞
The spectrum now looks like
amplitude
ω
0
ω ω
0 β
ω ω
With such a spectrum in hand, you could measure and ,
0 β
and hence the fractional part of the tune.
5
But what about those negative frequencies? A spectrum analyzer maps
negative frequencies to positive frequencies, so the observed spectrum
looks like
amplitude
ω
0
ω
Now, which is ? There are ω ?
β β
pairs of tune lines between the
ω ?
β
rotation harmonics.
And if the pick-up you are using is sensitive to both x and y planes,
there will be a pair of x and a pair of y tune lines between rotation
harmonics.
6
Rotation harmonics and tune lines
7
Decrease 2Q1A (a horizontally focusing quad)
8
Decrease 2Q2B (a horizontally defocusing quad)
9
Measuring Beta Functions
Conceptually…
The magnitude of the beta function sets the scale for how
the stored beam responds to a perturbation.
Bigger β (cid:198) Bigger response
Some possible perturbations:
• Gradient (quadrupole) (cid:198) Tune shift
• Closed orbit kick (steering dipole) (cid:198) Orbit shift
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