3D Simulation of Heavy-Ion Beams for Inertial Confinement Fusion Drivers

Heavy-ion induction accelerators are being developed in the United States as the principal candidate ``drivers'' for inertial confinement fusion (ICF) power production. They also represent an attractive driver option for a high-yield microfusion research facility. Heavy-ion drivers are attractive in both cases because of favorable reliability, repetition rate, efficiency, lifetime, cost, and the favorable fusion chamber geometry they allow.

The requirements for high-gain ignition of the fusion targets set the driver parameters. Current concepts employ about 10 beams of singly charged ions with masses of 100 to 200 amu and energies of 1 to 10 GeV, depositing a total of about 5 MJ on the target. Each beam at the target has a current of 1 to 10 kA and a main pulse duration of 10 ns (preceded by a longer, less intense ``foot''). Focusing the beam from a distance of 3 to 10 m onto the target with a focal-spot radius of about 3 mm requires that the transverse temperature of the beam be kept below 1 keV.

The space-charge-dominated beams in an induction accelerator are effectively non-neutral plasmas; therefore, computational modeling of these beams uses techniques related to those used in both the accelerator and plasma physics communities. Because the beam resides in the accelerator for relatively few plasma oscillation periods, particle-in-cell (PIC) simulation techniques are especially effective and have proved valuable in the design and analysis of both ongoing experiments and future machines and in the study of basic physics issues. The WARP code includes both fully 3D (WARP3d) and axisymmetric (WARPrz) electrostatic PIC simulation models. Studies of an electrostatic quadrupole (ESQ) injector and a small recirculating induction accelerator with WARP3d are described in this article.

WARP3d uses a number of novel techniques to improve accuracy and efficiency. These include a capability for subgrid-scale placement of internal conductor boundaries and a technique for rapidly but accurately following particles through a long sequence of sharp-edged accelerator ``lattice elements,'' using a relatively small number of time steps. WARP3d can simulate ``bent'' accelerator lattices, which iss how the code gets its name.

The goal of the Heavy Ion Fusion Accelerator program being pursued at LBNL, in collaboration with LLNL, is to address critical beam physics and technology issues of a heavy-ion fusion accelerator using beams of full driver-scale radius and line-charge density. The injector developed as a part of that program uses a sequence of ESQ lenses with a superposed voltage gradient along the axis. The net effect is to both confine the beam transversely and accelerate it. The beam is emitted from a hot-plate source and is accelerated through a diode section into the main ESQ region. Figure 28 shows the layout of the injector.

A major issue in the injector design is degradation of beam quality (brightness) by the nonlinear multipole components of the focusing fields and the so-called ``energy effect.'' The energy effect arises from the axial velocity spread of particles in the ESQ, which is due to the transverse variation of the potential. This effect is present in all electrostatic focusing systems, but is generally insignificant since the focusing potentials are a small fraction of the beam energy. However, at the low-energy end of an ESQ injector, the focusing potential is about half the beam energy. Proper treatment of the beam dynamics requires detailed 3D simulation with a realistic lattice model.

For design of the injector optics, the steady-state beam behavior was of primary interest. Runs were made in a quasi-steady-state mode, taking several particle time steps between each field solution and continuing until the injector was filled with beam and the system converged to a time-independent state. These 3D simulations of the injector required less than three minutes on a single processor of the NERSC CRAY Y-MP C90. Taking advantage of the ESQ's two-plane symmetry, the dimensions of the field grid were typically 40x40x348. Typically, 100,000 particles and 500 time steps were used. The savings made by this approach are apparent when compared to fully time-dependent simulations of transient behavior which require several hours of C90 CPU time.

These simulations led to an understanding of the mechanisms of beam quality degradation in an ESQ, and a number of corrective measures were identified. The most effective of these was increasing the diode voltage, which directly reduces the energy effect, decreases the size of the beam, and so reduces the influence of nonlinear fields.

Using WARP3d simulations as a guide, a scaled-down version of the injector was built at LBNL as a proof-of-principle experiment. Figure 29 shows an example of the excellent agreement between the experiment and simulation results.

The final design of the full-size ESQ injector was also carried out with the help of WARP3d simulations. The completed injector exceeds its design requirements. Figure 30 shows phase space representations of the simulations and experiment.

**Figure 28:** The layout of the ESQ injector. The beam is emitted from the source and accelerated to the right through the four quadrupole lenses. The voltages are arranged so that the beam is both strongly focused and accelerated.
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**Figure 29:** Comparison of experimental and simulated transverse beam parameters at the end of the scaled ESQ for a varying diode voltage. The emittance is a measure of beam quality; high quality beams have low emittance.
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**Figure 30:** Comparison of experimental and simulation transverse phase space representations at the end of the full-size ESQ injector. The top compares the results at the design parameters. The bottom compares the results with an increased diode voltage; in this non-optimal case, the representations are significantly distorted from a straight line.
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A recirculating induction accelerator, or ``recirculator,'' offers the prospect of reduced cost relative to a conventional linear accelerator because the accelerating and focusing elements are re-used many times per shot. The circumference of the recirculator can be much less than the length of the equivalent linear accelerator, and the accelerating cores can be smaller because there is no need to accelerate the beam at the maximum gradient. A small recirculator is being developed at LLNL to explore the beam dynamics of a full-sized fusion driver in a scaled manner. This research is being carried out as a sequence of experiments that will lead to the construction of a complete prototype.

The key dimensionless parameters that characterize the beam are similar to those of a driver-scale ring. The planned ring will have a circumference of 14.4 m, made up of 40 ``half lattice periods,'' each with a permanent magnetic quadrupole for transverse confinement, an electric dipole for beam bending, and an induction acceleration gap (except in the insertion/ extraction region). A beam of singly charged potassium ions will be accelerated from 80 to 320 keV over 15 laps, at a current increasing from 2 to 8 mA. With its low temperature, the beam is strongly influenced by the self-electric fields arising from its own space charge.

Although a full-scale driver will use magnetic dipole elements for bending the beam, electric dipole plates were chosen for the prototype to minimize costs. However, the size of the plates is constrained by space considerations and voltage-holding requirements. As a result, they introduce significant nonlinear fields. Using WARP3d, the plate shape was adjusted to minimize 3D field nonlinearities and their influence on beam quality.

WARP3d simulations were used to follow the beam for all 15 laps. Typically, the 2-ms-long beam is made up of 100,000 simulation particles. The field grid is a moving window covering four half lattice periods. It has a 32x16 transverse grid (taking advantage of vertical symmetry), and a 128-cell axial grid. Figure 31 shows a beam of length two half lattice periods in the lattice.

The simulations show an initial transverse heating of the beam on the order of 50 percent as the beam ``matches'' itself from the straight insertion line to the bent recirculator lattice. This heating is due to radial separation of particles with differing axial velocities in the bends, which leads to nonlinearities in the space charge forces. The nonlinearities thermalize, producing an increase in temperature. The observed growth agrees well with theoretical analysis.

After the initial heating, little further degradation in beam quality is seen over the 15 laps. The overall degradation is acceptable.

WARP3d simulations played an important role in the design and successful operation of the ESQ injector for the Heavy Ion Fusion Accelerator program at LBNL. For the small-recirculator experiments at LLNL, WARP3d simulations showed that a beam can be accelerated through 15 laps using electric-dipole bending, which gave us sufficient confidence to adopt this reduced-cost approach. The goal of the heavy-ion-fusion modeling effort is effective simulation of experiments and, ultimately, of a full-scale fusion driver.

**Figure 31:** A beam in the lattice as represented in WARP3d. The figure shows a sampling of the particles that make up the beam. The locations of the focusing and bending elements are identified. The bend regions are sections where polar coordinates are used to model the bent accelerator. For the purposes of the plot, the bent regions were straightened out.
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NERSC 3 Greenbook

3D Simulation of Heavy-Ion Beams for Inertial Confinement Fusion Drivers

NERSC 3 Greenbook