Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov1 Modeling the EMMA Lattice Stephan I....

Modeling the EMMA LatticeModeling the EMMA Lattice

Stephan I. Tzenov and Bruno D. Muratori Stephan I. Tzenov and Bruno D. Muratori STFC Daresbury Laboratory,STFC Daresbury Laboratory,

Accelerator Science and Technology CentreAccelerator Science and Technology Centre

FFAG08, Manchester Stephan I. Tzenov

Tuesday, 02 September 2008 FFAG08, Manchester Stephan I. Tzenov 2

Contents of the PresentationContents 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 EMMAPainting the Horizontal Phase Space in EMMA Conclusions and Outlook Conclusions and Outlook


The Hamiltonian FormalismThe 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 can expand the square root in power series in the transverse canonical coordinates


The Hamiltonian Formalism Continued… 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, γe is the energy corresponding to the reference orbit.


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


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


Paraxial Approximation for the Stationary Periodic OrbitParaxial Approximation for the Stationary Periodic OrbitThe 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


Paraxial Approximation for Stationary Periodic Orbit Cont…Paraxial Approximation for Stationary Periodic Orbit Cont…3. Focusing Quadrupole

4. Defocusing Quadrupole


Paraxial Approximation for Stationary Periodic Orbit Cont…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.


Stationary Periodic Orbit with FFEMMAGStationary Periodic Orbit with FFEMMAG

Stationary periodic orbit for two EMMA cells at 10 MeV


Twiss Parameters and Betatron TunesTwiss 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


Twiss Beta Function With FFEMMAGTwiss Beta Function With FFEMMAG

Twiss beta function for two EMMA cells at 10 MeV


Betatron Tunes with FFEMMAGBetatron Tunes with FFEMMAG

Dependence of the horizontal and vertical betatron tunes on energy


Longitudinal motionLongitudinal motionA 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.


Path Length and Time of FlightPath Length and Time of Flight

Path length and time of flight as a function of energy


Longitudinal motion Continued… 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 γm 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

222 Pdsd

20 BA

cos32

20

32

0 aIIH


Painting the Horizontal Phase Space in EMMAPainting the Horizontal Phase Space in EMMAThe 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.


Painting the Horizontal Phase Space in EMMA Continued…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.



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.



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.



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.



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.



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.


Conclusions and OutlookConclusions 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.

Tuesday, 02 September 2008FFAG08, Manchester Stephan I. Tzenov1 Modeling the EMMA Lattice Stephan I....

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