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PARTICLE-IN-CELL SIMULATION STUDY OF SINGLE AND MULTISPECIES BEAM P. Sing Babu*, A. Goswami and V. S. Pandit Variable Energy Cyclotron Centre, 1- AF, Bidhannagar, Kolkata-700 064, India Abstract A self-consistent two dimensional macroparticle model has been developed and used to study the dynamics of space-charge-dominated single and multispecies beams in a solenoid based transport system. We have performed the simulations for both emittance-dominated and spacecharge-dominated regimes of the beam. For multispecies beam, particle-in-simulation results shows the formation of hollow density profiles of subdominant species around the primary beam and separation is more in the case of initial K-V distribution of all the species. The evolution of the beam radius and emittance growth of each species has been investigated. INTRODUCTION The beam envelope equations are generally used for studying the average beam behaviour but it is not possible to have any information on emittance growth due to nonlinear forces. In general case, where nonlinear forces from both the applied and space-charge fields are present, the analytical investigation is very difficult. For the selfconsistent description of an intense beam, particle-in-cell (PIC) simulation methods are widely used [1-7]. In order to understand the detailed dynamics of space charge dominated beams with different density distributions, a two-dimensional PIC code has been developed and used to investigate the evolution of phase space distribution, rms emittances, rms size, and centroid in aligned and misaligned focussing system. The typical proton fraction from the microwave ion sources operating presently is of the order of 60-85% of the extracted beam. The presence of unwanted species alters the dynamics of the primary beam during the transport particularly in the cases where the beam intensity is high. In the PIC method we have included the effect of unwanted species. The PIC code has also used to get detailed information about the evolution of the beam distribution and emittance growth caused due to nonlinear effects in the case of multispecies beams. The formation of beam hollow of unwanted species is observed due to nonlinear space charge effect. MACROPARTICLE MODEL We consider a continuous, space-charge-dominated beam propagating in a beam transport system consisting of solenoid magnets. In the PIC method we consider both types of misalignments of solenoids, translational as well as tilt with respect to the symmetry axis. In this section we have discussed the PIC method for multispecies beam. In PIC simulation each species of the beam is represented by a large number of macroparticles. For a -------------------------------------------------- *[email protected] particular species, each macroparticle represents many ions keeping charge to mass ratio q j m j of a single ion. At first, the space-charge fields of each species are calculated in the beam frame where the motion of particles is non-relativistic. These fields are then Lorentztransformed to the laboratory frame [8]. The equations of motion of macroparticles of each species are integrated using a second order leap-frog method [1]. The transverse calculation region is divided into n x n y uniform rectangular meshes to calculate charge density required for the Poisson solver. For a particular species, the charge of each macroparticle is distributed among the four nearest nodes of the computational mesh, using the area-weighting method. The Poisson equation is then solved using the fast Fourier transform method to obtain the space-charge potential on the computational grids with perfect conducting wall boundary conditions. The components of the electric field at the grid points are evaluated numerically by differentiation of the potential at the grid points using the centered difference scheme. After that, these electric field components are Lorentztransformed to the laboratory coordinate system [8]. The process is repeated for all the other species to get the components of the electric and magnetic fields in the laboratory frame. The space-charge fields at the position of macroparticles from the fields at grid points are obtained utilising an interpolation technique and the same area weighting scheme as used for the calculation of the charge density. For the case of beam propagation through the solenoid focussing fields, it is convenient to work in the Larmor frame of reference. The Larmor frame corresponding to different beam species are different. In 2D PIC simulation, we require the distribution of macroparticles of each species in the 4D phase space x, x , y , y . In the present study, we have considered five different distributions: K-V, waterbag, parabolic, Gaussian and semi-Gaussian [4, 6]. RESULTS The Variable Energy Cyclotron Centre at Kolkata is developing a high current proton cyclotron. The injection system consists of a 2.45 GHz microwave ion source and two solenoid magnets of physical length of 40cm to transport and match the proton beam at the entrance of spiral inflector [9]. Generally, microwave ion sources produce proton fraction only 60-85% of the beam. The other major unwanted species are H2+ and H3+ beams. In the simulation, the transverse calculation region having dimension 12.8 cm ×12.8 cm has been divided into uniform rectangular meshes with 128 meshes in each direction. We have used a step size of s 1 mm in the axial direction of the beam propagation. The number of macroparticles used for each species is 77000. Typical values of normalised emittances (in πmmmrad) for p, H2+ and H3+ species used in the simulation are 0.8, 0.4 and 0.27 respectively. The beam energy is 80 keV. the growth is very rapid and the emittance growth vs. for different combinations of current and emittance are similar. Simulation of single species beam Figure 1 shows the evolution of rms emittance for 10 mA beam. The beam is loaded initially according to five different distributions: K-V (KV), waterbag (WB), parabolic (PA), semi-Gaussian (SG) and Gaussian (GA). We have seen that the variation of rms beam sizes as a function of axial distance is almost similar for all the distributions. It is interesting to note from Fig. 1 that the rms emittance for all the distributions shows an oscillating pattern except the KV distribution. There is an increase in the emittance which is very small in the case of waterbag and parabolic distributions and relatively more for the GA and SG distributions. The emittance for all the distributions increases slowly in the initial part of the beam line, reaches its first maximum at s ~112 cm just before the first waist and then decreases with s. It again starts increasing from the entrance of the second solenoid and reaches to its maximum before the second waist. The reduction in the emittance between the region s 0 cm to s 40 cm for the semi-Gaussian distribution is due to the relaxation. Figure 2 shows the phase space distributions of KV and GA beam at s = 0 and s =277cm. Figure 2: Phase space distributions at s = 0 and s = 277cm for K-V (KV) and Gaussian (GA) distribution. Figure 3: The growth in the emittance as a function of intensity parameter for Gaussian distribution. Simulation of multispecies beam Figure 1: Evolution of rms emittance εrms(s)/ εrms(0) for different beam distributions. We have studied the emittance evolution of Gaussian distribution for different values of by varying current from 1mA to 40mA with three values of rms emittances (in mmmrad) 3.82, 7.64 and 15.27. The dimensionless 1 intensity parameter 1 8I 0 a 2 n2rms / I is the ratio of the space charge force to the external focusing force defined in smooth focusing approximation [10]. Here n rms rms is the normalised rms emittance and a 2 X rms (s) is the average of the envelope size over the path length. The average emittance growth as a function of is shown in Fig. 3. It is clear that for higher values of Figure 4(a) shows the evolution of the beam envelope of the different species for 10 mA total beam current. The peak magnetic fields of the solenoids S1 and S2 are 2.8 kG and 2.7 kG, respectively. A slit of radius 5 mm is used at s 135 cm for the selection of proton beam. In the simulation p, H2+ and H3+ beams are K-V distributed initially and fractions (in %) are [60, 20, 20] respectively. The evolution of emittances of p, H2+ and H3+ beams is shown in Fig. 4(b). There is a very little growth in the emittance of proton beam, however the situation is not same in the case of H2+ and H3+ beams. It can be readily seen that there is a sharp increase in the emittances of H2+ and H3+ beams after s 100 cm. The main contributor to this emittance growth is the nonlinear space-charge force of the converging intense proton beam. At the slit major portions of the H2+ and H3+ beams are rejected together with the substantial reduction in their emittances. We see a sharp increase in the emittance of H2+ and H3+ after the slit in the region where the proton beam size is small. The emittance of these species remains almost constant in the region where the proton beam size is comparable to the beam sizes of H2+ and H3+ and again it increases as the proton beam converges. The growth in emittance of H2+ beam is mainly due to the nonlinear space-charge force of proton beam, whereas the growth in the emittance of H3+ beam is due to the nonlinear space-charge forces of proton and H2+ beams. PIC simulation shows the formation of hollow density profile of H2+ and H3+ beams around the proton beam, which becomes more distinct after the second solenoid. We have performed simulations for different species fraction in total beam current of 10 mA. Figure 5 shows the real space as well as phase space distributions of p, H2+ and H3+ beams at s 257 cm, just before the second waist of the proton beam, where the distributions of the H2+ and H3+ beams are well separated from the proton beam. The fractions (in %) of p, H2+ and H3+ beams used in the simulation are [80, 15, 5]. It is readily seen from the figure that H2+ and H3+ beams are separated from proton beam and phase space distributions of H2+ and H3+ beams are highly distorted for KV distribution. When the initial distributions of all the species are Gaussian, phase space distributions of all the species are highly distorted. CONCLUSION Figure 4: The evolution of (a) rms beam sizes and (b) rms emittances of p, H2+ and H3+ beams for I 10 mA. A circular slit of radius 5 mm is placed at s 135 cm. A self-consistent PIC method has been developed and utilized to study the dynamics of space-charge-dominated single and multispecies beams in a solenoid based transport system. We have performed the simulations for both emittance-dominated and space-charge-dominated regimes (0.3< <0.99) of the beam. It is shown that for higher values of the growth is very rapid and the emittance growth vs. for different combinations of current and emittance are similar. The growth in the emittance is seen to be more in the converging region of the beam where the strength of nonlinear space-charge force is comparatively stronger. In the case of multispecies beam, the evolution of rms emittance of proton is very small whereas the rms emittance growth of subdominant species H2+ and H3+ beams is large due to nonlinear space charge force. It is found that H2+ and H3+ species, which initially have K-V distributions, develop hollow density profiles around the proton beam, downstream the transport line. REFERENCES Figure 5: Real space and phase space distributions of H2+ and H3+ at s 257 cm with initial K-V (KV) and Gaussian (GA) distribution. [1] R.W. Hockney and J.W. Eastwood, Computer Simulation Using Particles (Taylor & Francis, New York, 1988). [2] M. A. Dorf, R. C. Davidson and E. A. Startsev, Phys. Plasmas 18, 043109 (2011). [3] P. Sing Babu, A. Goswami and V. S. Pandit, Phys. Plasmas 19, 080702 (2012). [4] Y. K. Batygin, Nucl. Instrum. Methods Phys. Res., Sect. A 539, 455 (2005). [5] P. Sing Babu, A. Goswami and V. S. Pandit, Phys. Plasmas 20, 083110 (2013). [6] S. M. Lund, T. Kikuchi, R. C. Davidson, Phys. Rev. ST Accel. Beams 12, 114801 (2009). [7] P. Sing Babu, A. Goswami and V. S. Pandit, Nucl. Instrum. Methods Phys. Res., Sect. A 724, 54 (2013). [8] R. C. Davidson and E. A. Startsev, Phys. Rev. ST Accel. Beams 7, 024401 (2004). [9] V. S. Pandit et.al., Journal of Physics: Conference Series 390 (2012) 012066. [10] D. Stratakis, R. A. Kishek, I. Haber, S. Bernal, M. Reiser, and P. G. O’Shea, Phys. Rev. ST Accel. Beams 12, 064201 (2009).