1. Hamiltonian Description of the Dynamics of Particles in Non Scaling FFAG Machines
Stephan I. Tzenov
STFC Daresbury Laboratory,
Accelerator Science and Technology Centre
FFAG 2007, Grenoble, France
12 – 17 April 2007

2. Contents of the Presentation
Introduction from First Principles
The Synchro-Betatron Formalism and the Reference Orbit
Accelerated Orbit and Dispersion
The Betatron Motion and Twiss Parameters
The Reference Orbit and Acceleration
The Phase Motion
Conclusions and Outlook
12 – 17 April 2007
FFAG 2007, Grenoble, France

3. Introduction from First Principles
The milestone for analytical description of beam dynamics in both isochronous cyclotrons and FFAG machines is the Hamiltonian:
where
Expanding the square root in the small relative coordinate and momentum variables, we have the Hamiltonian decomposition:
12 – 17 April 2007
FFAG 2007, Grenoble, France

4. Introduction Continued…
The coefficients a's have a very simple meaning:
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FFAG 2007, Grenoble, France

5. Introduction Continued…
while the coefficients b’s can be expressed in terms of a’s as follows:
Here the prime denotes differentiation with respect to the longitudinal variable s.
In other words, this implies that provided the vertical component of the magnetic field and its derivatives with respect to the horizontal coordinate x are known in the median plane, one can in principle reconstruct the entire field chart.
12 – 17 April 2007
FFAG 2007, Grenoble, France

6. The Synchro-Betatron Formalism and the Reference Orbit
The vertical component of the magnetic field in the median plane of the machine can be expressed as
where
The flutter function is periodic in the azimuthal variable, so that it can be expanded in a Fourier series:
12 – 17 April 2007
FFAG 2007, Grenoble, France

7. The Synchro-Betatron Formalism Continued…
Let us introduce the relative variation of the vertical component of the magnetic field in the median plane under deviation from a fixed radius R
Here we have denoted
12 – 17 April 2007
FFAG 2007, Grenoble, France

8. The Synchro-Betatron Formalism Continued…
In the case of N-fold symmetry, the flutter function consists of a structural part including harmonics of the type n = kN and the rest (a non structural part), which is usually considered as a perturbation.
Let us now write the first several coefficients a’s
A design (reference) orbit with a reference momentum and reference angular velocity is defined according to the relation
12 – 17 April 2007
FFAG 2007, Grenoble, France

9. The Synchro-Betatron Formalism Continued…
Using the explicit form of the mean field and the fact that it is a linear function in radius (in the case of EMMA), we obtain
where for brevity we have denoted
Typical range of angular velocity is from GHz – MHz for energies in the range 10 MeV – 20 MeV, respectively. In terms of the azimuthal variable as an independent variable, we can rewrite the Hamiltonian as follows
12 – 17 April 2007
FFAG 2007, Grenoble, France

10. The Synchro-Betatron Formalism Continued…
Here we have introduced the following notations:
The reference orbit can be defined via a canonical transformation
FFAG 2007, Grenoble, France
12 – 17 April 2007

11. The Synchro-Betatron Formalism Continued…
The equations determining the reference orbit read as
How the above formalism work for EMMA? The vertical component of the magnetic field in the median plane can be written as
where d is the offset from the benchmark radius in the defocusing qudrupoles. Further,
With the above in hand, one can express the Fourier decomposition of the flutter and its derivatives, the deviation from a fixed radius, deviation from the ideal isochronous field, etc.
12 – 17 April 2007
FFAG 2007, Grenoble, France

12. Accelerated Orbit and Dispersion
Consider a canonical transformation specified by the generating function
aiming at cancelling the first term in the linear part of the Hamiltonian. We have
The latter defines the so-called "accelerated orbit" rather than equilibrium orbit, because it is superimposed on the reference orbit, which is NOT a closed curve. The equations determining the accelerated orbit are
The new Hamiltonian can be written as
12 – 17 April 2007
FFAG 2007, Grenoble, France

13. Accelerated Orbit and Dispersion Continued…
Taking into account additionally the reference orbit, and assuming that the deviation of the energy with respect to the reference one is small, we can cast the above Hamiltonian in the form
FFAG 2007, Grenoble, France
12 – 17 April 2007

14. Accelerated Orbit and Dispersion Continued…
The last canonical transformation aimed at cancelling the linear part of the Hamiltonian is
Taking into account that the dispersion function D satisfies the set of equations
the new Hamiltonian can be expressed as
12 – 17 April 2007
FFAG 2007, Grenoble, France

15. The Betatron Motion and Twiss Parameters
The total Hamiltonian describing the betatron oscillations can be obtained by adding terms quadratic in coordinates and momenta encountered in the third order Hamiltonian to the second order Hamiltonian. The result is:
where
A generic Hamiltonian of the above type can be transformed to the normal form
12 – 17 April 2007
FFAG 2007, Grenoble, France

16. The Betatron Motion and Twiss Parameters Continued…
by means of a canonical transformation specified by the generating function:
The old and the new canonical variables are related through the expressions
The phase advance and the generalized Twiss parameters are defined as
The corresponding betatron tunes are determined according to the expression
12 – 17 April 2007
FFAG 2007, Grenoble, France

17. The Reference Orbit and Acceleration
We will solve explicitly the equations determining the accelerated orbit in the case where the accelerating force exerted by the cavities is considered in the kick approximation. We can write
Here is the cavity voltage, is the RF frequency, Nc is the number of cavities and is the corresponding cavity phase. We introduce a new variable
It is straightforward to write the transfer map for the interval just before entering the k-th cavity to the location just before entering the (k+1)-st cavity. It has the form
12 – 17 April 2007
FFAG 2007, Grenoble, France

18. The Reference Orbit and Acceleration Continued…
The reference orbit map is a generalized Standard Map, which is known to exhibit the so-called accelerator modes. To find the accelerator modes, we expand the relevant variables in a formal small parameter according to the relations:
Order by order one finds:
Performing further analysis, one can find the amplitude equation governing the nonlinear acceleration regime. It is of the form:
Highly nontrivial problem, which is now in progress.
12 – 17 April 2007
FFAG 2007, Grenoble, France

19. The Phase Motion
Clearly, the phase motion depends only on the horizontal degree of freedom. The total Hamiltonian after extracting the accelerated orbit can be written as
The equations of motion with account of the accelerated orbit only read as
A more rigorous treatment is achieved if the dispersion effects are included in the analysis.
All these are now in progress. It should be mentioned that the phase motion in FFAG accelerators is HIGHLY NONTRIVIAL.
12 – 17 April 2007
FFAG 2007, Grenoble, France

20. Conclusions and Outlook
A close parallel between isochronous cyclotrons and FFAG accelerators has been established.
The transversal motion in both machines is very similar and can be studied from a unified point of view.
However, the longitudinal dynamics differs. While in isochronous cyclotrons the source of phase motion is the deviation of the real magnetic field from the isochronous one, in FFAG machines the phase motion is an essential part of the dynamics.
A computer code FFEMMAG to simulate the linear dynamics in FFAG machines is now under development. It will calculate the reference orbit of EMMA, the median plane footprint, the lattice functions and dispersion.
Future work also includes dynamic aperture module, resonance crossing part, as well as a module to simulate space charge effects, which have proven to be very essential in EMMA.
12 – 17 April 2007
FFAG 2007, Grenoble, France

21. Acknowledgements
Bruno Muratori is acknowledged as a principal collaborator in the development of the FFEMMAG computer code, as well as for his enthusiasm during the learning process dedicated to the principles and operation of FFAG machines.
Special thanks for many illuminating and fruitful discussions are due to:
Neil Bliss
Neil Marks
Susan Smith
James Jones
12 – 17 April 2007
FFAG 2007, Grenoble, France