Development of 2D particle-in-cell code to simulate high current, low energy beam in a beam transport system

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1 PRAMANA c Indian Academy of Sciences Vol. 9, No. journal of October 7 physics pp Development of D particle-in-cell code to simulate high current, low energy beam in a beam transport system S C L SRIVASTAVA, S V L S RAO and P SINGH Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 85, India shashi uv@yahoo.com MS received 1 March 7; revised 1 May 7; accepted 5 July 7 Abstract. A code for D space-charge dominated beam dynamics study in beam transport lines is developed. The code is used for particle-in-cell (PIC) simulation of z-uniform beam in a channel containing solenoids and drift space. It can also simulate a transport line where quadrupoles are used for focusing the beam. Numerical techniques as well as the results of beam dynamics studies are presented in the paper. Keywords. Space-charge; particle-in-cell; beam dynamics; Poisson s equation; solenoids; quadrupole magnets. PACS Nos 9.7.Bd; 5.5.Rr; 1..cv 1. Introduction Newly proposed accelerators with applications to nuclear waste transmutation and spallation neutron sources require high intensity Linacs. To develop the technology of high intensity Linacs for our ADS programme, a MeV, 3 ma CW proton accelerator is being built at the Bhabha Atomic Research Centre, Mumbai. It consists of 5 kev ECR ion-source, low energy beam transport line (LEBT), 3 MeV radio frequency quadrupole (RFQ), medium energy beam transport line (MEBT) and MeV drift tube Linac (DTL) [1]. In the low energy section of such accelerators, beams are strongly subjected to the Coulomb repulsion and understanding the behaviour of such space-charge dominated beams is a challenging task. This requires a careful control of particle dynamics by suitably incorporating space-charge forces in the beam. Several codes (TRACED [], TRANSPORT [3] etc.) are available to study the beam dynamics and to match the beam parameters between two structures (for example, ion-source and RFQ). As these codes are formulated by linearizing the space-charge forces and equations of motion, they are expected to be accurate in the linear regime. These codes cannot therefore simulate the nonlinear space-charge effects in beam dynamics. There are numerous particle-in-cell (PIC) codes (PARMILA [], PARMTEQ [5], WARP3D [], BEAMPATH [7] etc.) 551

2 S C L Srivastava, S V L S Rao and P Singh developed in the field of accelerator physics to study dynamics of space-charge dominated beams. To our knowledge, only BEAMPATH [7] simulates the continuous beam in transport lines using PIC method. In this paper, we present an algorithm that forms the basis of the code, which has been developed to study various aspects of space-charge dominated continuous beam in a LEBT. In addition to tracking the particles, generation of different kinds of beam distributions is an integral part of this code. In future, we plan to incorporate, in order to study space-charge compensation, collisions of protons with neutral gas atoms and molecules using Monte Carlo technique in the code.. Transport of space-charge dominated beam Consider the propagation of an intense charged particle beam through a beam transport system consisting of different focusing elements. For the beam dynamics study, the PIC method [8] is used. The beam is represented as a combination of a large number of macroparticles with the same charge-to-mass ratio as that of the real beam. The simulation is performed in D phase space of particle transverse positions x, y and transverse velocities v x, v y. The macroparticles following certain kind of distribution will be tracked in the presence of external field and the field exerted by the beam. To calculate field exerted by the beam (self-field), we need to solve the Poisson equation and from that electric fields are calculated. Since motion of particles is influenced by the self-fields which depend on their spatial distribution inside the beam, we need to solve the Poisson equation at each and every step and this problem has to be solved self-consistently. During simulations the macroparticle is lost if it touches the boundaries. From the single particle Hamiltonian, equations of motion can be derived and in general they can be written as follows: dx dt = v x, dy dt = v y, dv x dt = q m (E + v B) x, dv y dt = q m (E + v B) y. (1) Integration is performed with fixed time step δt. In (1), the electric field is a combination of external and space-charge fields, while magnetic field is external only (at low energy, self-magnetic field is very small as compared to electric field). In the following sections, the numerical algorithms of the code and results are discussed. 3. Beam distribution generator Initially, the particles will be distributed in the D transverse phase space according to a distribution function. For the generation of particle distribution in transverse 55 Pramana J. Phys., Vol. 9, No., October 7

3 Development of D PIC code D phase space x, x, y, y, consider a class of distributions with elliptical symmetry. The distribution function n(x, x, y, y ) depends on total emittance ɛ which has a meaning of radius-vector in D phase space: n(x, x, y, y ) = dn dxdx = n(ɛ), () dydy ɛ = A x + ca y, (3) where c is the ratio of beam emittances [9]. For any value of c except 1 the area of the D projection (ellipse) of (3) will be unequal. Parameters A x, A y describe a family of ellipses. A x = A y = ( βx x + α xx βx ) + ( x βx ), () ( ) ( ) βy y + α yy y +, (5) βy βy where α x,y, β x,y and γ x,y are Courant Snyder parameters in the x and y planes. The equation ɛ = constant describes a hyperellipsoid surface in the phase space x, x, y, y. As the distribution function depends on ɛ, the phase space density, n, will be constant on one hyperellipsoid surface while it will vary from one surface to another. The distribution function is normalized under the following condition: n(x, x, y, y )dxdx dydy = 1. () For generating the phase space distributions, we have to generate the N values of x, x, y, y which correspond to a given distribution function n(ɛ). To generate various distributions, we need to know the distribution of ɛ, g(ɛ) = dn(ɛ). (7) dɛ Let us transform x, x, y, y to new coordinates r, θ, φ, ψ ( θ π, φ π, ψ π): βx x + α xx βx = r sin θ cos φ sin ψ, (8) x βx = r sin θ sin φ sin ψ, (9) c ( βy y + α yy βy ) = r cos θ sin ψ, (1) ( ) y c = r cos ψ, r = ɛ. (11) βy Pramana J. Phys., Vol. 9, No., October 7 553

4 S C L Srivastava, S V L S Rao and P Singh The phase space element transforms as dxdx dydy = r3 c sin ψ sin θdrdθdφdψ. (1) Then the number of particles in the phase space element is dn(r, θ, φ, ψ) = n(r) r3 c sin ψ sin θdrdθdφdψ. (13) Integration of (13) over angle variables gives the number of particles dn as a function of r, g(r) = dn(r) dr = π r3 n(r). (1) c The algorithm for generation of different distributions is based on the above equations and it is given below: 1. To simulate the distribution g(r) we will use the inverse transform method in which we take the integral distribution defined by (15) to find r = r(g) under the assumption that the values of G are uniformly distributed in the interval [,1]: G(r) = r g(r )dr. (15). For each value of r, two random numbers A x and A y are chosen such that they satisfy (3) and then the points x, x, y, y are calculated by the following equations: x = A x βx cos p, (1) ( x = A x α x cos p + sin p ), βx βx (17) y = A y βy cos q, (18) ( ) y = A y α y cos q + sin q, βy βy (19) where p and q are two random numbers which are uniformly distributed in [, π]. The steps 1 and will be repeated N times for generating N particles. The generated particles will follow the distribution n(ɛ). The above method is implemented in the program, which presently generates D Kapchinskij Vladimirskij (KV), waterbag and parabolic distributions. This needs emittance, and Courant Snyder parameters of the beam to generate the specified distributions. The definition of different distributions is given in table 1. The (x, x ) projections of all the three distributions are shown in figure 1. The representation of the emittance was found to be better than 1% in all the cases for 1 and more particles. 55 Pramana J. Phys., Vol. 9, No., October 7

5 Development of D PIC code Table 1. Definition of different phase space distributions. Distributions Definition c rδ(r π r KV c π r Waterbag c (1 π r Parabolic 5 3 r ) r Waterbag Distribution 3 5 KV Distribution r ) 5 (a) (b) 5 3 Parabolic Distribution (c) Figure 1. Projection of (a) KV, (b) waterbag and (c) parabolic distribution on x x plane for Twiss parameters βx = βy =.78 cm/rad, γx = γy =.171 rad/cm, αx = αy = 1.8, emittance ²x = ²y =.π cm mrad and c = 1.. Poisson solver The space-charge potential of the beam, U, for an instantaneous space-charge density distribution, ρ, is calculated from the solution of Poisson s equation U ρ(x, y) U + =. x y ² () For a z-uniform beam, this problem reduces to a D problem in x y coordinates. First, we distribute the space-charge of macroparticles among grid nodes, then solve Pramana J. Phys., Vol. 9, No., October 7 555

6 S C L Srivastava, S V L S Rao and P Singh the Poisson s equation on grid and finally after calculation of field on grid nodes, scale it at macroparticle positions. The simulation region is divided into uniform rectangular meshes. Charge of every particle with coordinates (x n, y n ) is distributed among the nearest four nodes utilizing area weighting method [8]. The charge density at node point, ρ ij, is given by ρ ij = N n=1 ( ρ xy 1 x ) ( n x i 1 y ) n y i, (1) h x h y where ρ xy is the space-charge density of an individual particle and h x, h y are the mesh sizes. ρ xy is related to the beam current I and to the speed of the beam in the z-direction ρ xy = I v z Nh x h y. () The Poisson s solver is implemented such that it can handle either Dirichlet or Neumann boundary condition in one direction and mixed boundary condition in the other direction..1 For Dirichlet and mixed boundary condition Suppose that U(x, y) satisfies mixed boundary conditions in the x-direction, i.e., a l U(x, y) + b l U(x, y) x = g l (y) (3) at x = x l and a similar equation for x = x h with new constants. Here, a l, b l, etc. are known constants, whereas g l is a known function of y. Furthermore, suppose that U(x, y) satisfies the following simple Dirichlet boundary conditions in the y- direction: U(x, ) = U(x, L) =. () Let us write U(x, y) as a Fourier series in the y-direction: U(x, y) = u j (x) sin(jπy/l). (5) j= The functions sin(jπy/l) are orthogonal, and form a complete set in the interval y [, L]. In fact, L L sin(jπy/l) sin(kπy/l)dy = δ jk. () Thus, we can write the source term as 55 Pramana J. Phys., Vol. 9, No., October 7

7 Development of D PIC code ρ(x, y) = ϱ j (x) sin(jπy/l), (7) j= where ϱ j (x) = L L ρ(x, y) sin(jπy/l) dy. (8) Furthermore, the boundary conditions in the x-direction become a l u j (x) + b l du j (x) dx = Γ l j (9) at x = x l and a similar equation for x = x h with new constants, where Γ l j = L L g l (y) sin(jπy/l) dy. (3) Using (5) and (7) in () and equating the coefficients of sin(jπy/l) (since these functions are orthogonal), we obtain d u j (x) dx j π L u j(x) = ϱ j (x), (31) for j =. Now, we can discretize the problem in the y-direction by truncating our Fourier expansion, i.e., by only solving the above equations for j = J, rather than j =. This is essentially equivalent to discretization in the y- direction on the equally-spaced grid-points y j = jl/j. The problem is discretized in the x-direction by dividing the domain into equal segments. Thus, we obtain u i 1,j ( + j κ ) u i,j + u i+1,j = ϱ i,j (h x ), (3) for i = 1 N x and j = J. Here, u i,j u j (x i ), ϱ i,j ϱ j (x i ) and κ = πh x /L. The boundary conditions (9) discretize to give u,j = Γ ljh x b l u 1,j a l h x b l, (33) u Nx+1,j = Γ hjh x + b h u Nx,j a h h x + b h, (3) for j = J. Equations (3) (3) constitute a set of uncoupled tridiagonal matrix equations (with one equation for each j value). These equations can be inverted, to give u i,j. Finally, U(x i, y j ) values can be reconstructed from (5).. For Neumann and mixed boundary condition If there is Neumann boundary condition in the y-direction as Pramana J. Phys., Vol. 9, No., October 7 557

8 S C L Srivastava, S V L S Rao and P Singh U(x, y = ) y = U(x, y = L) y then we can express U(x, y) in the form U(x, y) = =, (35) u j (x) cos(jπy/l) (3) j= so that it automatically satisfies the boundary conditions in the y-direction. Likewise, we can write the source term ρ(x, y) as where ρ(x, y) = ϱ j (x) cos(jπy/l), (37) j= since ϱ j (x) = L L ρ(x, y) cos(jπy/l) dy, (38) L L cos(jπy/l) cos(kπy/l) dy = δ jk. (39) Finally, the boundary conditions in the x-direction become a l u j (x) + b l du j (x) dx = Γ l j, () at x = x l and a similar equation for x = x h with new constants, where Γ l j = L L g l (y) cos(jπy/l) dy. (1) Note, however, that the factor in front of the integrals in (38) and (1) takes the special value 1/L for the j = harmonic. As before, we truncate the Fourier expansion in the y-direction, and discretize in the x-direction, to obtain the set of tridiagonal matrix equations and solve as before. Fast Fourier transform is implemented using FFTW libraries [1]. Electric field on the grid nodes are calculated using E i,j = U i,j ; () electric field at the particle position is calculated from field values at grid nodes using the same area weight method (1), as for charge. Since these fields are in a moving frame of reference, the perpendicular component of the field in laboratory frame will be modified by the relativistic factor γ. The parallel component, however, will remain unaffected. (For low energies, the correction is insignificant because γ is close to unity.) 558 Pramana J. Phys., Vol. 9, No., October 7

9 Development of D PIC code 5. Integration of particle trajectories The particle trajectories are integrated using leap-frog method to preserve the kinematic time reversal symmetry in the simulation. In the most general case, integration of particle trajectories in this code is carried out through the following steps: 1. The particle performs a half-step acceleration in an electric field: v n = v n + qδt m E n. (3). The particle velocity undergoes rotation in the magnetic field. This is implemented by employing Boris scheme [11]: v n = v n + v n T, () T = q B δt m, (5) v n+1 = v n + v n s, () s = T 1 + T. (7) 3. The particle performs again half-step acceleration in an electric field: v n+1 = v n+1 + qδt m E n. (8). Finally, particles are advanced with a velocity, v n+1 : x n+3/ = x n+1/ + v n+1 δt. (9) In all the above equations v and x possess only x- and y-components (not z) while B has its usual meaning.. Focusing fields In this section, we describe the implementation of two focusing devices in the code, namely, solenoids and quadrupoles..1 Solenoid In this code the hard edge model of solenoid is implemented, i.e. the effect of fringe field is considered at a point. To see the effect of fringe field on the motion of charged particle, we take as Gaussian surface a truncated square circular cylinder of radius R coaxial with the solenoid. One end is well inside the solenoid, where Pramana J. Phys., Vol. 9, No., October 7 559

10 S C L Srivastava, S V L S Rao and P Singh Gaussian Surface(Truncated Cylinder) B Circular Area Cylindrical Area Figure. Side view of Gaussian surface in the solenoid. the field is B in the axial direction, and the other is well outside, where the field is zero as in figure. Then over this surface, B d S =. (5) So πr B = πr B r dz. (51) Simply put, all the flux leaving the cylinder cap inside the solenoid must have come in through the sides of the cylinder. This means that there is an irreducible amount of integrated radial magnetic field through the fringe field region: B r dz = B R/. (5) The effect of the fringe field on a particle of charge q traveling with axial velocity component v z = dz/dt is to kick it sideways with a force or F θ = qv z B r = dp θ /dt (53) dp θ = qb r dz, (5) p θ = q B r dz = qb R/. (55) The negative (positive) denotes the entrance (exit) of the beam in the solenoid field. There is no change in p r and p z. Let us transform these changes in Cartesian coordinates. As p x p y p z = cos φ sin φ sin φ cos φ 1 p θ, (5) 5 Pramana J. Phys., Vol. 9, No., October 7

11 Development of D PIC code so p x = ± qb y, (57) p y = qb x. (58) At the beginning and at the end of the solenoid, the transverse velocity changes according to (57) and (58) but there will be no change in position as we have represented the field as a Dirac-delta distribution. Inside the solenoid, magnetic field is constant and so the equations of motion are dv x dt = q m (E x + v y B ), (59) dv y dt = q m (E y v x B ). (). Quadrupole The field of quadrupoles [9] is given by the following equations: B x = B y a and B y = B x a, (1) where a is the aperture of the quadrupole. The equations of motion (1) are now augmented with the magnetic field values given by (1). 7. Code validation and low energy beam transport simulation To validate the code, we have chosen to study the evolution of round uniform beam following KV distribution in the drift space. As the space-charge forces are linear, the envelope equation can be written as [9] r m = ɛ rm 3 + I I β 3 γ 3, () r m where β and γ are relativistic parameters. This equation is numerically evaluated for a 5 kev, 3 ma proton beam and then evolution is compared with that obtained by the PIC code. The difference in the two solutions (defined as difference of two envelope radius divided by envelope radius calculated from ()) is less than 1. The two solutions are plotted in figure 3 and are found to be in good agreement. We have used our code to simulate the LEBT for the MeV, 3 ma proton accelerator being built in BARC [1]. A comparison between the parameters obtained by TRACED and our code is presented. The LEBT is designed using TRACED with the following input Twiss parameters, ɛ =.π cm mrad, β x = β y =.78 Pramana J. Phys., Vol. 9, No., October 7 51

12 S C L Srivastava, S V L S Rao and P Singh..5 Envelope Radius (m)..3. PIC KV Eq z (m) Figure 3. Envelope radius as a function of distance traveled in drift space. cm/rad, γ x = γ y =.171 rad/cm and α x = α y = 1.8 and the output parameters are found to be ɛ =.π cm mrad, α x = α y = 1.8, β x = β y =.3 cm/rad and γ x = γ y =.59 rad/cm. The lattice parameters of the LEBT is given in table. As TRACED is an envelope tracking code, a constant emittance is obvious. With the same input parameters and lattice, we obtain the output parameters as ɛ x =.31π cm mrad, ɛ y =.π cm mrad, α x = 1.83, α y = 1.9, β x =.1 cm/rad, β y =.87 cm/rad and γ x =.9 rad/cm, γ y =.7 rad/cm. The particle trajectories calculated using these parameters are shown in figure. As the numerical simulations are sensitive to mesh size, we found that for this problem there is no significant change in the output for mesh size greater than All the calculations are therefore performed on a mesh size of Effect of different distributions on the beam envelope To bring out the capabilities of simulating the non-linear space-charge fields with this code, we have simulated the behaviour of beam with different particle distributions in the designed LEBT. Since waterbag and parabolic distributions are non-uniform in D (x y) projections, they will give rise to non-linear space-charge field. The effect of non-linearity is clearly seen in the form of increase in the output emittance values. They are found to be ɛ x =.31,.7 and.33π cm mrad, ɛ y =.,.5 and.31π cm mrad for KV, waterbag and parabolic distribution respectively in each plane. The phase space projection on x x at the input and output are shown in figure 5. The initial Twiss parameters and emittance values are taken to be the same as mentioned in 7. Furthermore, the effect of various distributions is clearly visible on the maximum beam radius as shown in figure. 5 Pramana J. Phys., Vol. 9, No., October 7

13 Development of D PIC code x (cm) Distance (cm) Figure. Trajectories of particles in LEBT in x z plane. Table. LEBT (using solenoids and drift) parameters. Element Length (cm) Drift Solenoid Drift Solenoid Drift Strength (T) Table 3. LEBT (using quadrupole and drift) parameters. Element Drift Quadrupole Drift Quadrupole Drift Quadrupole Drift Quadrupole Drift Length (cm) Strength (T/m) For comparison, we designed a beam transport system using magnetic quadrupole lenses instead of solenoids. The input/output parameters are chosen the same as in case of LEBT containing solenoids and the designed LEBT (beam) optical parameters using TRACED are given in table 3. On simulation with our code with these parameters for KV distribution, the output turns out to be: ²x =.1π cm mrad, ²y =.π cm mrad, αx =.93, Pramana J. Phys., Vol. 9, No., October 7 53

14 S C L Srivastava, S V L S Rao and P Singh Input KV Distribution Output (KV) 1 15 (a) Input Waterbag Distribution 5 Output (Waterbag) (c) (d) Input Parabolic Distribution 1 Output (parabolic) (b) (e) (f) Figure 5. Phase space projections of different distributions: (a) KV, (c) waterbag and (e) parabolic in x x plane at the input while (b), (d) and (f ) are at the output of LEBT in the corresponding order. αy = 1.1, βx = cm/rad, βy = 3.57 cm/rad, γx =.593 rad/cm and γy =.581 rad/cm. The emittances were poorer in cases of waterbag (²x =.59, ²y =.98), and parabolic (²x =.3, ²y =.39π cm mrad) distributions. The poorer emittances were due to non-linearity in the space-charge field. The maximum beam radius (1.1 cm) and beam radius at the end of the LEBT (.3 cm) are found to be in agreement with the values obtained by TRACED (1 cm,.3 cm respectively). The beam radius as a function of z is shown in figure 7. 5 Pramana J. Phys., Vol. 9, No., October 7

15 Development of D PIC code 7 5 x (cm) 3 1 KV distribution Waterbag distribution Parabolic distribution z (cm) Figure. Effect of different distributions on maximum envelope radius. 1 Beam radius (cm) 1 8 x envelope y envelope z (m) Figure 7. Variation of beam radius in the two transverse planes as a function of longitudinal distance in the LEBT. 9. Conclusion The D PIC program for studying the beam dynamics in LEBT system has been developed. The results are found to be in good agreement with theoretical analysis. The algorithm implemented has good accuracy and takes a reasonable amount of computer time. It is planned to incorporate collisions of protons with neutral gas atoms and molecules for charge compensation estimation using Monte Carlo technique. Acknowledgements We thank Drs V C Sahni, S Kailas and R K Choudhury for their keen interest in this work. We also thank Dr Kartik Patel, Rajni Pande, Tushima Basak and Shweta Roy for many useful discussions. Pramana J. Phys., Vol. 9, No., October 7 55

16 References S C L Srivastava, S V L S Rao and P Singh [1] P Singh et al, Pramana J. Phys. 8, 331 (7) [] K R Crandall and D R Rusthoi, Computer program TRACED, LANL Report, LA- 135 (198) [3] K L Brown and S K Howry, TRANSPORT/3, a computer program for design charged particle beam transport system, SLAC report No. 91 (197) [] B Austin, T W Edwards, J E O Meara, M L Palmer, D A Swenson and D E Young, MURA Report No. 713 (195) D A Swenson and J Stovall, LANL Internal Report MP-3-19 (198) J P Boicourt and Charles R Eminhizer (eds), AIP Conference Proceedings (New York, 1988) vol. 177, p. 1 [5] K R Crandall, T P Wangler and Charles R Eminhizer (eds), AIP Conference Proceedings (New York, 1988) vol. 177, p. [] A Friedman, D P Grote and I Haber, Phys. Fluids B, 3 (199) [7] Y K Batygin, Nucl. Instrum. Methods in Phys. Res. A539, 55 (5) [8] C K Birdsall and A B Langdon, Plasma physics via computer simulation (IOP, Bristol 1991) [9] M Reiser, Theory and design of charged particle beams (John Wiley & Sons, New York, 199) [1] Matteo Frigo and Steven G Johnson, The design and implementation of FFTW3, Proc. IEEE 93(), 1 (5) [11] J P Boris, Relativistic plasma simulation-optimization of a hybrid code, Proc. th Conf. Num. Sim. Plasmas, Naval Res. Lab., November Pramana J. Phys., Vol. 9, No., October 7