particle trajectory computation for accelerators and related technologies

2007 International Nuclear Atlantic Conference - INAC 2007 Santos, SP, Brazil, September 30 to October 5, 2007 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: 978-85-99141-02-1

PARTICLE TRAJECTORY COMPUTATION FOR ACCELERATORS AND RELATED TECHNOLOGIES Carlos A. F. Jorge1 and Reinaldo J. Jospin1 1

Instituto de Engenharia Nuclear (IEN / CNEN - RJ) Caixa Postal 68550 21945-970 Rio de Janeiro, RJ [email protected] [email protected]

ABSTRACT This work reports an alternative method for single non-relativistic charged particle trajectory computation in 2D electrostatic or magnetostatic fields. This task is approached by analytical computation of particle trajectory, by parts, considering the constant fields within each finite element. This method has some advantages over numerical integration ones: numerical miscomputation of trajectories, and stability problems can be avoided. Among the examples presented in this paper, an interesting alternative approach for positive ion extraction from cyclotrons is shown, using strip-foils. Other particle optics devices can benefit of a method such the one proposed in this paper, as beam bending devices, spectrometers, among others. This method can be extended for particle trajectory computation in 3D domains.

1. INTRODUCTION An R&D work has been carried out in IEN/CNEN for computational electromagnetics and charged particle trajectory computation, initially intended for application to the CV-28 Cyclotron [1], to support possible optimization of its operation for medical radioisotopes production. An effort has been directed towards code development for these purposes. An alternative method for single non-relativistic charged particle trajectory computation in 2D electrostatic or magnetostatic fields has been developed. Instead of numerical integration of trajectory in generic field distributions [2, 3], as long as linear triangular finite elements [4] are used for fields approximation, particle trajectory can be computed analytically and traced by line segments, within each finite element. The theoretical development of this method is described in details in [5]. Some examples illustrate the method, showing stripping extraction of both negative and positive ions from cyclotrons. This last approach is very interesting for positive ion cyclotron upgrade, such as CV-28 Cyclotron, since it avoids all the technological challenges related to negative ion cyclotron design or upgrade.

2. STRIPPING EXTRACTION SYSTEMS FOR CYCLOTRONS Extraction systems based on strip-foils were invented in 1964, with first experimental results reported some years later [6]. The ratio of specific charge change after and before stripping extraction is given by K = ηa /ηb [6], where: • Specific charge η of an ion means the ratio of charge state / mass number, Z/A; • Charge state is the net charge of an ion; • ηb is the specific charge before stripping; • ηa is the specific charge after stripping.

An ion moving in a magnetic field has its trajectory bended, with radius given by Eq. (1). When passing through a strip-foil, negative ions lose electrons, becoming positive ions. Thus rotation direction changes, and ions are directed towards cyclotron's exit [6, 7]. Extraction efficiency in this case is near 100 %. When a positive ion passes through a strip-foil, its charge state Z increases [6, 8], as well as its charge q, reducing curvature radius ρ. The positive ion, thus, performs at least one short loop before exiting the cyclotron [6, 8, 9]. Light positive ions can be extracted with high efficiency, near 100 % [6, 10].

ρ=

mv qB

(1)

where: • q, m and v are particle's charge, mass and velocity magnitude, respectively; • B is magnetic induction. Some examples of negative and positive ion that can be extracted by stripping are shown: 1- H− Hydrogen has specific charge η = 1, and is stripped with K = −1, [6, 9]. 2- 2H− Deuteron has specific charge η = 0.5, and is stripped to 2H+ with K = −1, [6, 9]. 3- Molecular Hydrogen H2+ has specific charge η = 0.5, and when stripped, dissociates into two Protons, with K = 2, [6, 9]. 4- He+ has specific charge η = 0.25, and is stripped to He++, with K = 2.

3. PARTICLE TRAJECTORY COMPUTATION 3.1. Element Identification For trajectory computation, the element that contains the particle have to be identified at each step, given particle's coordinates, which has been implemented based on Lohner's algorithm [11] for 2D triangular finite element meshes. 3.2. Trajectory Computation Particle's acceleration is assumed as constant within each element, where field is constant. Velocity and coordinates' initial values are the ones at the beginning frontier. Formulation for particle's coordinates and velocity are shown in Eq. (2) and (3), respectively, while Eq. (4) shows equation of a frontier's support line of a generic finite element, say for frontier 12. For the other frontiers, 23 and 31, cyclic permutation of indexes has to be performed.

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ri = ri −1 + v i −1 ∆t + (1 2 )a e ∆t 2

(2)

v i = v i −1 + a e ∆t

(3)

 y − y1  x y −x y  x + 2 1 1 2 y =  2 x2 − x1  x2 − x1 

(4)

where: • ri is particle's updated coordinate, with initial value (on the initial frontier) ri–1, • vi is particle's updated vector velocity, with initial value vi–1, • ae is particle's acceleration within each element, • (x1, y1) and (x2, y2), are Cartesian coordinates of element's nodes 1 and 2, respectively. Equation (2), decomposed in Cartesian components results in two equations in x and y which, combined with Eq. (4), form a system of three equations and three unknowns, which can be solved for the time variable ∆t, or for the coordinates variables x and y. As explained in [5], the first approach leads to better solutions for the electrostatic field case, while the later leads to better solutions for the magnetostatic field case. The particular solutions for each case are shown in the following, more details can be found in [5]. 3.2.1. Electrostatic field case A second order equation results in time ∆t variable, as shown in Eq. (5).

1 ( y 2 − y1 )a xe − (x 2 − x1 )a ey ∆t 2 + ( y 2 − y1 )v x i −1 − (x 2 − x1 )v y i −1 ∆t + 2 + ( y 2 − y1 )xi −1 − ( x 2 − x1 ) y i −1 + x 2 y1 − x1 y 2 = 0

[

[

]

]

(5)

which is normalized before solution, and where: • a xe and a ey are Cartesian components of acceleration ae, •

v x i −1 and v y i −1 are Cartesian components of initial velocity vi–1.

Solving for the time interval ∆t, a total of six solutions results, the correct one corresponding to the lower non-zero positive solution. Once identified, the time ∆t is used to advance the particle up to the next frontier with Eq. (2), and to update particle's velocity with Eq. (3). 3.2.2. Magnetostatic field case In this case, the solution is obtained for the coordinates x and y, by considering the circular formula, with trajectory approached by circular arcs within each element. Circular formula is shown in Eq. (6), and acceleration in Eq. (7).

[

x = xC ± ρ 2 − ( y − yC ) q a = v×B m

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∴

2

]

12

a xe = ωC v y i −1  e a y = −ωC v x i −1

(6)

(7)

where: • ρ is the Larmor radius (the radius described by the particle in the circular arc); • (xC, yC) is the trajectory center's coordinates; • ωC is the cyclotronic frequency (particle's angular velocity). After some algebraic manipulation, a second order equation in coordinate xi results, as shown in Eq. (8).

[(x

2

2

2

]

− x1 ) + ( y 2 − y1 ) xi2 +

{

}

+ 2 ( y 2 − y1 )[x2 y1 − x1 y 2 − yC ( x2 − x1 )] − xC ( x2 − x1 ) xi + 2

(8)

+ [x2 y1 − x1 y 2 − yC ( x2 − x1 )] + x ( x2 − x1 ) − ρ ( x2 − x1 ) = 0 2

2 C

2

2

2

After its solution, coordinate yi is obtained with Eq. (4). The correct overall solution, among the six candidates, corresponds to the one that gives the lowest distance to the initial point (xi–1, yi–1).

4. RESULTS

A typical sector-focused cyclotron [12, 13] has been simulated in 2D, with characteristics: • Magnetic poles with four sectors (four hills and four valleys); • Straight sectors (not spiraled); • Pole diameter: 2 m; Hard-edge approximation has been adopted for the magnetic induction, as usually done in preliminary simulations [14], and closed orbit approximation has been considered [15], with orbit parameters set to: • Azimuthally averaged magnetic induction Bav = 1 T; • Average particle's radius ρav = 0.8 m; • Average particle's velocity vav = 0.8 m.s−1. Magnetic induction on hills Bh and in valleys Bv have been set to hypothetical values around Bav: Bh = 1.5 T, and Bv = 0.5 T. Cyclotron's center is placed at coordinates (x, y) = (0., 0.) m, and particle's initial data are: • Initial coordinates: (x0, y0) = (0., 0.83572354) m; • Initial velocity: (vx0, vy0) = (0.8, 0.) m; • Trajectory's initial center (xC0, yC0) = (0., 0.3) m. Thus trajectory's radius and center vary, in the range: • ρh = 0.53 m (on the hills), and ρh = 1.6 m (on the valleys); Figure 1a and Fig. 1b show result for a closed orbit at extraction radius ρav = 0.8 m., while Fig. 2a and Fig. 2b show orbits for stripping extraction: Fig. 2a for negative ion with K = −1, and Fig. 2b for positive ion with K = 2. Strip-foil is positioned at angle 22.5º, angle's origin at right horizontal, counterclockwise. Hills are shown in brown, and valleys in light blue.

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a)

b) Figure 1. a) Mesh and orbit for closed orbit simulation; b) Closed orbit at extraction radius.

a)

b) Figure 2. a) Stripping extraction of negative ion; b) Stripping extraction of positive ion.

5. CONCLUSIONS

Particle trajectory computation and tracing is shown for magnetostatic fields, obtained with the method developed. Results for trajectory in electrostatic fields can be found in [5]. The first cyclotron simulation shows a closed orbit at extraction radius, agreeing with theoretical predictions. Simulations of stripping extraction of negative and positive ions have been performed, leading also to expected results. The presented method can be applied to more complex and realistic examples, using 2D computed fields instead of hard-edge approximation. It can also be extended to 3D particle trajectory computation, by considering intersection of trajectory with element's faces, instead of frontier lines.

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ACKNOWLEDGMENTS

We would like to acknowledge Ms. Y. C. P. Migliano and other people from IEAv/CTA’s staff, and also Dr. J. J. Barroso de Castro from LAP/INPE, for providing visits of first author to IEAv/CTA, and to LAP/INPE, in 2004; thanks and to Prof. J. P. A. Bastos from GRUCAD/UFSC, and to Prof. J. R. Cardoso and other people from LMAG/USP’s staff.

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