Macroparticle Models
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Transcript of Macroparticle Models
Macroparticle ModelsEric Prebys, FNAL
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
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
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
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!
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
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
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
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
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
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
USPAS, Knoxville, TN, January 20-31, 2014Lecture 18 -Macroparticle Models 12
will cancel “spring constant” term
If we assume
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
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.