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Negative Mass Instability
Eric Prebys, FNAL
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 2
Consider two particles in a bunch. Below transition
turn n
turn n+1 Further apart
Above transition…
turn n
turn n+1 Closer together
That is, the particles behave as if they had “negative mass”
Consider a beam of uniform line density
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 3
The fields outside the beam are given by
Inside the beam, the enclosed current/charge scales as r2/a2, so
We now find the field along the beam axis using
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 4
Assume we have the beam propagating through a beam pipe of radius b
Note, λ not constant
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 5
Any perturbation is propagating with the beam, so
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 6
If the wall is perfectly conducting, then Ew=0, and we have
We’ll factor any perturbations in the line density into harmonic components
azimuthal location
frequency of oscillation
n=mode number. General solution will be a combination of these.
Mode will propagate with an angular frequency
phase velocity of perturbation around ring
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 7
Recall, λ is a charge density, so it must satisfy the continuity equation
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 8
Look for a solution of the form
Assume small
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 9
We can write the current in the form
We now consider an individual particle in the distribution
particle deformation
But the angular velocity of an individual particle around the ring is related to the period by
slip factor
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 10
So we can write
We’re now going to introduce the concept of “longitudinal impedance to characterize the energy lost per particle in terms of the total current, defined by
We’re only interested in the fluctuating part, so we writerecall
Combining, we get
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 11
Substitute
and we get
Recall
so
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 12
So we have
Motion will be stable if RHS is both real and positive, so
energy loss given by
fraction of total
set
but
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USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 13
so we have
imaginary and negative
so for motion to be stable, we want η<0
In other words, motion will only be stable below transition.
This is why unbunched beams are not stable above transition.