Negative Mass Instability

Eric Prebys, FNAL

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

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

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 5

Any perturbation is propagating with the beam, so

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

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 8

Look for a solution of the form

Assume small

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

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

USPAS, Knoxville, TN, Jan. 20-31, 2014Lecture 16 -Negative Mass Instability 11

Substitute

and we get

Recall

so

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

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.