Modeling the EMMA Lattice

1. Stephan I. Tzenov and Bruno D. Muratori STFC Daresbury Laboratory, Accelerator Science and Technology Centre Modeling the EMMA Lattice FFAG08, Manchester Stephan I. Tzenov FFAG08, Manchester Stephan I. Tzenov

2. FFAG08, Manchester Stephan I. Tzenov Contents of the Presentation The Hamiltonian Formalism The Stationary Periodic Orbit Paraxial Approximation for the Stationary Periodic Orbit Twiss Parameters and Betatron Tunes Longitudinal motion Painting the Horizontal Phase Space in EMMA Conclusions and Outlook

3. FFAG08, Manchester Stephan I. Tzenov The Hamiltonian Formalism The Hamiltonian describing the motion of a particle in a natural coordinate system associated with a planar reference curve with curvature K is Here A=(Ax, Az, As) is the electromagnetic vector potential, while the tilde variables are the horizontal and vertical deviations from the periodic closed orbit and their canonical conjugates, respectively. The longitudinal canonical coordinate Θ and its conjugate γ are Since the longitudinal quantities are dominant, one canexpand the square root in power series in the transversecanonical coordinates

4. FFAG08, Manchester Stephan I. Tzenov The Hamiltonian Formalism Continued… where dΔE/ds is the energy gain per unit longitudinal distance s, which in thin lens approximation scales as ΔE/Δs, where Δs is the length of the cavity. In addition, γeis the energy corresponding to the reference orbit.

5. FFAG08, Manchester Stephan I. Tzenov Stationary Periodic Orbit To define and subsequently determine the stationary periodic orbit, it is convenient to use a global Cartesian coordinate system whose origin is located in the centre of the EMMA polygon. To describe step by step the fraction of the reference orbit related to a particular side of the polygon, we rotate each time the axes of the coordinate system by an angle Θp=2π/Np, where Np is the number of sides of the polygon. Let Xe and Pe denote the horizontal position along the reference orbit and the reference momentum, respectively. The vertical component of the magnetic field in the median plane of a perfectly linear machine can be written as A design (reference) orbit corresponding to a local curvature K(Xe, s) can be defined according to the relation In terms of the reference orbit position Xe(s) the equation for the curvature can be written as Note that the equation parameterizing the local curvature can be derived from a Hamiltonian

6. FFAG08, Manchester Stephan I. Tzenov Stationary Periodic Orbit Continued… which is nothing but the stationary part of the Hamiltonian (1) evaluated on the reference trajectory (x = 0 and the accelerating cavities being switched off, respectively). In paraxial approximation Pe<<βeγe Hamilton’s equations of motion can be linearised and solved approximately. We have In addition to the above, the coordinate transformation at the polygon bend when passing to the new rotated coordinate system needs to be specified. The latter can be written as

7. FFAG08, Manchester Stephan I. Tzenov Paraxial Approximation for the Stationary Periodic Orbit The explicit solutions of the linearized Hamilton’s equations of motion can be used to calculate approximately the reference orbit. To do so, we introduce a state vector The transfer matrix Mel and the shift vector Ael for various lattice elements are given as follows: 1. Polygon Bend 2. Drift Space

8. FFAG08, Manchester Stephan I. Tzenov Paraxial Approximation for Stationary Periodic Orbit Cont… 3. Focusing Quadrupole 4. Defocusing Quadrupole

9. FFAG08, Manchester Stephan I. Tzenov Paraxial Approximation for Stationary Periodic Orbit Cont… Since the reference periodic orbit must be a periodic function of s with period Lp, it clearly satisfies the condition Thus, the equation for determining the reference orbit becomes Here M and A are the transfer matrix and the shift vector for one period, respectively. The inverse of the matrix 1 - M can be expressed as A very good agreement between the analytical result and the numerical solution for the periodic reference orbit has been found.

10. FFAG08, Manchester Stephan I. Tzenov Stationary Periodic Orbit with FFEMMAG Stationary periodic orbit for two EMMA cells at 10 MeV

11. FFAG08, Manchester Stephan I. Tzenov Twiss Parameters and Betatron Tunes The phase advance χu(s) and the generalized Twiss parameters αu(s), βu(s) and γu(s) are defined as The third Twiss parameter γu(s) is introduced via the well-known expression The corresponding betatron tunes are determined according to the expression

12. FFAG08, Manchester Stephan I. Tzenov Twiss Beta Function With FFEMMAG Twiss beta function for two EMMA cells at 10 MeV

13. FFAG08, Manchester Stephan I. Tzenov Betatron Tunes with FFEMMAG Dependence of the horizontal and vertical betatron tunes on energy

14. FFAG08, Manchester Stephan I. Tzenov Longitudinal motion A natural method to describe the longitudinal dynamics in FFAG accelerators is to use the Hamiltonian where and the reference γe is chosen as the one corresponding to the middle energy – in the EMMA case 15 MeV. The coefficients K1 and K2 are related to the first and second order dispersion functions P1 and P2 as follows However, a different although equivalent method to approach the problem is more convenient.

15. FFAG08, Manchester Stephan I. Tzenov Path Length and Time of Flight Path length and time of flight as a function of energy

16. Longitudinal motion Continued… Using again the Hamiltonian governing the dynamics of the reference orbit, we obtain Numerical results concerning the time-of-flight parabola suggest that the following approximation is valid Clearly, A=2Bγm. Here B>0 and γmcorresponds to the minimum of the time-of-flight parabola. The free parameters can be easily fitted from the time-of-flight data. Thus the longitudinal motioncan be well described by the scaled Hamiltonian FFAG08, Manchester Stephan I. Tzenov

17. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA The phase space ellipse is shown below. Some of the most characteristic points are marked from 1 to 7. First of all, it is necessary to check whether possible to handle these points within the existing aperture. Clearly, this possibility is energy dependent.

18. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 1 turn (42 cells). Beam energy is 10 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 1 turn.

19. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 3 turns (126 cells). Beam energy is 10 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 3 turn.

20. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 5 turns (210 cells). Beam energy is 10 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 5 turns.

21. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 1 turn (42 cells). Beam energy is 11 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 1 turn.

22. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 3 turn (126 cells). Beam energy is 11 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 3 turns.

23. FFAG08, Manchester Stephan I. Tzenov Painting the Horizontal Phase Space in EMMA Continued… Tracking results for 5 turn (210 cells). Beam energy is 11 MeV, painted emittance is 3 mm rad. Figure on the left shows the initial phase space ellipse. Figure on the right shows the filamented (whiskered) phase space after 5 turns.

24. FFAG08, Manchester Stephan I. Tzenov Conclusions and Outlook • Synchro-betatron formalism has proven to be very efficient to study the beam dynamics in non scaling FFAG accelerators. • We believe that with equal success it can be applied to scaling FFAG machines. • A new computer programme implementing the features of the approach presented here has been developed. • This code has been extensively used as an in-home tool to find a number of important engineering solutions. • Studies with the existing (FODO) lattices show a good reason to adopt at least 13 MeV as the minimum energy for painting the 3 mm radian emittance without increasing the existing vacuum apertures, or decreasing it at least 5 times. • Next stage of the code is to introduce vertical orbit distortions. • Inclusion of longitudinal dynamics is close to completion and the next step will be a start-to-end single particle simulation of EMMA.