1. Eric Prebys, FNAL

2. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 2 Our simple model can only go so far. Now we have to develop the tools to help us deal with real particle distributions in phase space. Phase density: inin inin out

3. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 3 Vlasov Equation

4. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 4 Apply a special case of the Vlasov Equation ring

5. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 5 Recall that in our discussion of the Distribution Function, we found that if the current is described by then frequency of oscillation mode Just as we did with the current, we will define the density to have a constant part and an oscillatory part.

6. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 6 Plug this into the Vlasov equation, and we get Convert

7. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 7 Combining, we get Integrating the LHS, we have Integrating the RHS and equating, we get dispersion relation

8. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 8 Application to the negative mass instability Unbunched beam with δ=0 yeah, I know. Deal with it Write this in terms of ω, we have Dispersion relation gives as before!

9. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 9 Now consider a more realistic beam with a momentum spread Drop subscript in terms of angular frequency The dispersion integral becomes where

10. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 10 So the dispersion relation becomes recall If ΔΩ has a negative imaginary part, then motion will be unstable.

11. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 11 Use trick recall

12. USPAS, Knoxville, TN, January 20-31, 2014 Lecture 20 - Evolution of the Distribution Function 12 From the dispersion relation. The unstable solution (I D (u 0 )<1) can only exist if So motion will be stable if More generally, motion will be stable if Form factor which depends on details of distribution “Keil-Schnell criterion”