Macroparticle Models

1. Macroparticle Models Eric Prebys, FNAL

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, νs0 (ie, no synchrotron motion) , so the longitudinal positions of the particles remain fixed Lecture 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 Lecture 18 -Macroparticle Models

4. If x1 is executing β oscillations, then so the second particle sees driving term If the two have the same betatron frequency, then the solution is particular solution homogeneous solution Lecture 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 behind Must keep wake functions as low as possible in design! Lecture 18 -Macroparticle Models

6. Strong Head-Tail Instability 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 system complex form Lecture 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 Lecture 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 Lecture 18 -Macroparticle Models

9. We now consider the tune differences cause by chromaticity revolution frequency tune If the momentum changes by slip factor chromaticity revolution angular frequency Lecture 18 -Macroparticle Models

10. Now we write the positions of our macroparticles as Make s the independent variable We calculate the accumulated phase angle Lecture 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 Lecture 18 -Macroparticle Models

12. will cancel “spring constant” term If we assume Lecture 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 Lecture 18 -Macroparticle Models

14. We look at our previous matrix Once more, stability requires Define 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. Lecture 18 -Macroparticle Models