1. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov1 Modeling the EMMA Lattice Stephan I. Tzenov and Bruno D. Muratori STFC Daresbury Laboratory, Accelerator Science and Technology Centre FFAG08, ManchesterStephan I. Tzenov

2. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 2 Contents of the Presentation The Hamiltonian Formalism The Hamiltonian Formalism The Stationary Periodic Orbit The Stationary Periodic Orbit Paraxial Approximation for the Stationary Periodic Orbit Paraxial Approximation for the Stationary Periodic Orbit Twiss Parameters and Betatron Tunes Twiss Parameters and Betatron Tunes Longitudinal motion Longitudinal motion Painting the Horizontal Phase Space in EMMA Painting the Horizontal Phase Space in EMMA Conclusions and Outlook Conclusions and Outlook

3. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 3 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=(A x, A z, A s ) 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 can expand the square root in power series in the transverse canonical coordinates

4. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 4 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, γ e is the energy corresponding to the reference orbit.

5. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 5 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 Θ p =2π/N p, where N p is the number of sides of the polygon. Let X e and P e 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(X e, s) can be defined according to the relation In terms of the reference orbit position X e (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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 6 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 P e <<β e γ e 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 7 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 M el and the shift vector A el for various lattice elements are given as follows: 1. Polygon Bend 2. Drift Space

8. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 8 Paraxial Approximation for Stationary Periodic Orbit Cont… 3. Focusing Quadrupole 4. Defocusing Quadrupole

9. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 9 Paraxial Approximation for Stationary Periodic Orbit Cont… Since the reference periodic orbit must be a periodic function of s with period L p, 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 10 Stationary Periodic Orbit with FFEMMAG Stationary periodic orbit for two EMMA cells at 10 MeV

11. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 11 Twiss Parameters and Betatron Tunes The phase advance χ u (s) and the generalized Twiss parameters α u (s), β u (s) and γ u (s) are defined as The third Twiss parameter γ u (s) is introduced via the well-known expression The corresponding betatron tunes are determined according to the expression

12. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 12 Twiss Beta Function With FFEMMAG Twiss beta function for two EMMA cells at 10 MeV

13. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 13 Betatron Tunes with FFEMMAG Dependence of the horizontal and vertical betatron tunes on energy

14. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 14 Longitudinal motion A natural method to describe the longitudinal dynamics in FFAG accelerators is to use the Hamiltonian where and the reference γ e is chosen as the one corresponding to the middle energy – in the EMMA case 15 MeV. The coefficients K 1 and K 2 are related to the first and second order dispersion functions P 1 and P 2 as follows However, a different although equivalent method to approach the problem is more convenient.

15. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 15 Path Length and Time of Flight Path length and time of flight as a function of energy

16. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 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=2Bγ m. Here B>0 and γ m corresponds 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 motion can be well described by the scaled Hamiltonian

17. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 17 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 18 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 19 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 20 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 21 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 22 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 23 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. Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov 24 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.