Download - Macroparticle Models
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Macroparticle ModelsEric Prebys, FNAL
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 2
A common approach to understanding simple instabilities is to break a bunch into two “macrobunches”
As an example, we will apply this to an electron linac. At high γ, νs0 (ie, no synchrotron motion) , so the longitudinal positions of the particles remain fixed
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 3
From the last lecture, we have
Consider the lowest order (transverse) mode due to the leading macroparticle
The force on the second macroparticle will then be
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 4
If x1 is executing β oscillations, then
so the second particle seesdriving term
If the two have the same betatron frequency, then the solution is
homogeneous solution
particular solution
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 5
Try
Plug this in and we find
grows with time!
This is a problem in linacs, which can cause beam to break up in a length
wake function ~half a bunch length behindMust keep wake functions as low as possible in design!
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Strong Head-Tail Instability
USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 6
In a machine undergoing synchrotron oscillations, this problem is alleviated somewhat, in that the leading and trailing macroparticles change places every ~half synchrotron period
In and unperturbed systemcomplex form
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 7
For the first half period, we plug in the term from the linac case
pull out sin() term
We can express this as a matrix. For the first half period, we have
After half a synchrotron period, we have
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 8
For the second half of the synchrotron period, we get.
For the second half of the synchrotron period, we get.
We proved a long time ago that after many cycles, motion will only be stable if
“strong head-tail instability” threshold
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 9
We now consider the tune differences cause by chromaticity
revolution frequencytune
If the momentum changes by
slip factorchromaticity
revolution angular frequency
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 10
Now we write the positions of our macroparticles as
Make s the independent variable
We calculate the accumulated phase angle
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 11
So we can write
We identify the angular term and ω and write out equation
Assume that the amplitude is changing slowly over time, we look at the first term
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 12
will cancel “spring constant” term
If we assume
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 13
Integrate
Now we can obtain the evolution over half a period with
Compare to our simple case where
We have added an imaginary part due to the chromaticity
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USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 14
We look at our previous matrix
Once more, stability requiresDefine eigenvalues
For low intensity
So any the imaginary part of κ will give rise to growth
In fact, we’ll see that other factors, make adding chromaticity important, particularly above transition.