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.