Numerical simulations to model laser-driven coil-capacitor targets for generation of kilo-Tesla magnetic fields

The experimental results used as reference in the following study are reported in Santos et al⁶ The experiment was conducted at the LULI pico 2000 laser facility with a 1.057 μm wavelength, 500 ± 30 J laser energy, 1 nsec flat-top long-pulse laser beam focused on a target made of two parallel disks (3.5 mm diameter, 50 μm thickness) with a 1 mm diameter hole on the first disk enabling to focus the laser pulse into the rear disk surface. This part of the target represents the capacitor and is connected to a coil shaped wire. The coil is a 300° arc, with an inner radius of 250 μm made by a squared section wire with 50 μm side length. In Ref. 6 (Figure 1) a scheme of the experimental setup with used diagnostic devices and a picture of the capacitor-coil target are reported. Different conductive materials (Al, Cu, Ni) have been used during the experiment, however in the present study the capacitor-coil target is made of copper since this material has the best characteristics in terms of density, as well as thermal and electric conductivity. A scheme of the capacitor-coil used in the simulations is shown in left-hand-side (LHS) of Figure 1. A laser is focused on the rear side of the full disk passing through the hole of the disk front side, thus creating supra-thermal electrons which escape the potential barrier. The front disk collects a fraction of these electrons establishing a very large potential difference between the two disks, basically behaving like a capacitor with a certain characteristic time before discharging and energizing the coil part. For simplicity the coil and the capacitor parts have been studied independently, since the capacitor part introduces a negligible delay in the current pulse. The capacitor mass is, however, important for thermal compensation in the whole target. Moreover, the coil part used in the experiment has a tilt of 22.5° with respect to the capacitor axis. That tilt is considered in the simulation by tilting the axes and the points where the field is analyzed (again to simplify the model). Figure 1 right-hand-side (RHS) shows the conductor geometry used in the simulations. Two codes were used for the simulations: COMSOL²⁷ and Opera 3D (Tosca).²⁸ 

Among all the diagnostics used in the experiment, the preferred device for the analysis proposed in the following is the B-dot probe, as it provides a voltage signal that can be easily used to retrieve the current pulse through the coil. The B-dot probe was set at 30 mm from the coil center and with an angle of 33° with respect to the horizontal axis. The B-dot signal and the magnetic field vs time estimated through data processing of the measured signal are reported in Ref. 6 (Figure 2). The magnetic field peak intensity has been used by Santos et al to calculate the current circulating in the coil using Finite Element Methods (FEM) analysis. Such method uses the coil current as a free parameter which is varied in order to match the same magnetic field value at the experimental B-dot position.

In the present work a different approach has been used, consisting in scaling the peak value of the voltage signal (≃ 5.5 V) from the B-dot position to the coil position, which gives a value of V_coil = 5974 V. Then, the coil resistance has been calculated (excluding the capacitor part) considering the geometrical dimensions of the conductor and the copper resistivity, resulting in R_coil = 0.0185 Ω. The current peak is given by the ratio between coil voltage and resistance: I = 3.216 × 10⁵ A. Such current value allows to calculate the magnetic field considering that the coil is not actually a closed loop. In this case the following formula can be used:

where μ₀ is the vacuum permeability, I is the coil current, ϕ = 300° is the coil arc angle and r_a$v$ = 0.275 mm is the average between the inner and outer coil radius. This value results to be more accurate that those obtained with classical equation B = μ₀I/2r_a$v$ = 898.77 T, as shown later.

Another important parameter that has to be calculated and used for signal processing and for transient simulations is the coil inductance, which, for a coil with section side s = 50 μm, is given by:

that corresponds to a total stored magnetic energy of:

Inductance and magnetic energy in equations 2 and 3 are calculated considering only the 300° coil without straight parts of the conductor. These results are used to model the coil in COMSOL and OPERA 3D. In both codes the geometry shown in RHS of Figure 1 has been modeled. The first analysis is performed in the static case and considering the peak current value. In COMSOL the current I = 3.216 × 10⁵ A is used to energize the coil, in OPERA the current density J = I/s² = 1, 286 × 10⁵ A/mm² is used. Results for the peak field at the coil center are shown in Table I.

As mentioned above, the coil has a tilt of 22.5° with respect to the capacitor axis, and the field distribution, as shown in Figure 2, shows negligible difference if the field is plotted along the central axis or along an axis tilted of 22.5°.

The field intensity at the B-dot position is also compared with the peak value measured during the experimental campaign of 0.5 mT as discussed in Ref. 6. Again, the comparison is performed considering a point 30 mm far from the coil center, on the xy plane and at 33° with the horizontal axis. Moreover, to take into account the real tilt in the experimental geometry, the field has been measured on a point which is also displaced by 22.5° in the longitudinal direction (z direction). Hence, this second point is ot of the xy plane. The corresponding results are in Table II. Figure 3 shows the spatial position of the two points. The agreement between OPERA and experimental results is evident, while a quite high discrepancy is present in COMSOL. This happens even if OPERA simulation is made with a geometry that violates the Ampere‘s Law, as the conductor loop is not closed and its terminal parts are within the bounding box and the results have an error of 0.0742 %. This is not the case of COMSOL as the conductor is attached to the boundary. OPERA 3D provides more accurate and reliable results as a simple conductor can be analyzed without solving the FEM model, while COMSOL result is affected by a relatively large mesh size in the detector position.

Stored magnetic energy, coil resistance, inductance and voltage are summarized in Table III. The spatial distribution of the magnetic field along a straight line connecting the coil center with the B-dot probe is reported in Ref. 6. Results obtained with COMSOL and OPERA are reported in Figure 4. In this case only the point in the xy plane has been considered.

In OPERA simulations the agreement with the theoretical value can be considered satisfactory. OPERA does not allow to estimate the coil parameters since it is not part of the finite element mesh, hence the coil resistance and voltage are not reported. The static case shows an evident agreement between the different simulation results. The conductor with the horizontal part for connection with the holed disk was also modeled (only by COMSOL) giving the same results as those proposed before; Figure 5 shows the field distribution on the transverse plane for the full conductor. It is also shown that for this case, as for the simplified scheme, if the plane is taken at the conductor surface, the field produced by the straight connection part is negligible with respect to the main field due to the vertical parts.

The capacitor has been separately modeled in order to study the discharge time. A simple treatment consists in taking into account the common area of the two disks and use the classical formula to have a capacitance C = 0.1 pF. This approach underestimates the capacitance since the fringe effects are not taken into account. The simplest formula for fringe effect in a circular disk parallel plate capacitor is:²⁹ 

where Rad = 3.5 mm is the disk radius and d = 0.73 mm is the separation distance. Considering the coil electric parameters provided in Table III, the characteristic time constant of the capacitor is:

being R the coil resistance. The analytical solution for the current through the resistor and the capacitor in series will be compared with the simulation results of the capacitor transient study. The first step is to model the capacitor for the static case (both in OPERA and COMSOL) and verify the analytical capacitance calculation. Spatial field distribution in both codes is shown in Figure 6, Table IV summarized results on electric energy We and capacitance C.

Again, there is a small difference in the electric energy estimation as the integration volume is slightly different in the two codes, anyway the capacitance estimation is in agreement for all the three methods used. The results on the transient simulation are compared with the analytical formula for RC circuits and Figure 7 shows perfect agreement. It is evident from the plot that the capacitor is discharged after 20 femtoseconds, and it is basically not affecting the current pulse that energize the coil, in terms of delay. The geometrical layout of the capacitor plate allows to collect a considerable number of electrons which are responsible for a huge charge unbalance between the two plates and also, as it will be shown later, its mass helps in limiting the overall thermal jump due to the coil Joule heating.

The results on capacitor presented previously allow to perform transient simulations considering only the coil part, which simplify the geometry as well as the simulation. Transient simulations have been performed converting the signal acquired with the B-dot probe reported in Ref. 6 into current flowing in the coil.

In the experiment reported in Ref. 6 an RB-130 sensor was used, and being placed at 30 mm from the coil center. The voltage generated on the probe is connected to the magnetic field via the equation:

being A_eq = 4 × 10⁻⁵m² the probe equivalent area provided by the manufacturer. The magnetic field plot versus time is given in Figure 8, Left Hand Side (LHS). The magnetic field can then be scaled by a factor given by the ratio between the field at the solenoid center and the field at the B-dot probe position. The current signal shape is shown on the Right Hand Side (RHS) of Figure 8, and it will be used to modulate the peak current value in transient simulations.

Applying the Faraday-Lenz law, the magnetic field B produced by the coil depends on the coil voltage:

where the coil voltage V (t), inductance L, area A and impedance Z are introduced. This equation can be used to calculate the voltage signal and, using the equivalent capacitor-coil circuit, the current is obtained as:

being R the coil resistance and neglecting the capacitor term. Current signal is shown in Figure 8 (RHS). It can be noted that the peak current is slightly overestimated, as the COMSOL scaling factor is used. It will be shown in the following that the coil explodes after about 10 ns, hence the signal detected at longer times is probably due to the irradiated field component. Dedicated simulations on the irradiated field (not reported here) show that the irradiated field component results to be 6 orders of magnitude smaller than the static component, in agreement with the magnitude of the signal detected for t > 10 nsec. Equation 8 validity is demonstrated in Refs. 30 and 31 where it is also shown its dependence on the electron temperature.

Taking into account the signal processing results and the static analysis on the coil and on the capacitor parts it is possible to analyze the capacitor-coil target from the electrical point of view. It can be seen as RLC series circuit and at given AC current it can resonate if the inductive Z_L = jωL and capacitive Z_C = −j/ωC terms of the electrical impedance are equal to each other. In the previous formula j is the imaginary unit, ω is the angular frequency of the current, L is the coil inductance and C is the capacitance of the plates. It implies that the circuit can resonate only in a frequency range defined by the central value $ω0=1/LC$ with a FWHM Δω = R/L, where R is the circuit resistance. The current signal flowing into the electrical circuit is just a pulse with a maximum width of 10 ns and no conventional AC current is established. Anyway it can be modeled as a half wavelength sinusoidal current with period of 20 ns and with associated angular frequency of 314 MHz. Taking into account the electric features of the capacitor-coil target, the resonant frequency should be centered in ω₀ = 63 GHz with FWHM Δω = 13 MHz, that means the frequency of the current pulse cannot allow the system to resonate.

Transient simulations have been performed only using COMSOL. The current signal in Figure 8 has been normalized to 1 and used to modulate the maximum current value I = 3.216 × 10⁵ A. In the simulations the current signal starts at t = −4 nsec. The magnetic field at the coil center as a function of time is shown in Figure 9.

As expected the magnetic field has the same shape of the current signal, which is in agreement with equation 1. The same shape is present in the field at the B-dot position, as shown in Figure 10 where the results for both the previous mentioned points are compared. The comparison between the obtained field and the measured signal shown in Ref. 6 shows a good agreement in terms of field shape and duration. Field underestimation has been explained in the previous section.

Magnetic transient study gives also, as a result, a constant value of resistance equal to the static case (no thermal effects are here considered) and the voltage signal has a similar shape as the signal detected with the B-dot probe in Ref. 6. The magnetic energy peak value is close to those calculated in the static case and it has the same trend as the magnetic field, as expected. The inductance has a constant value with some fluctuations (in the range of 0.0010 nH) due to numerical computation errors. Discrepancies between magnetic energy peak value and inductance in the static and transient cases are due to the difference mesh size (coarser in the transient case) and to the discretization (quadratic in the static case and linear in the transient one) used, which give less accurate results in the time dependent case. This choice is due to the fact that transient simulations are extremely expensive in terms of calculation resources, computational time and storage space.

Since the magnetic field is extremely strong it is important to model forces produced in the coil, for validating and explaining the experimental data as well as for benchmarking numerical simulations that will be used also for the design of a more complex magnet in a magnetic trap configuration. The magnetic force (Lorentz force) can be expressed as F = B_condIl and the force per volume unit as F_d = B_condIl/V ol, being l = 2.7 mm the conductor length, V ol = 0.00694 mm³ its volume, I the peak current and B_cond = 1500 T is the average field in the conductor. The B_cond value is retrieved from simulation and used only for the analytical calculation. This is a simplified approach but still sufficient to validate the force calculation in the simulations. In fact, only the radial force for this conductor is relevant since it produces the coil expansion, while for a multi-turn conductor (solenoid) it is important to consider also the longitudinal force which is responsible for the coil compression. Table V shows value of the force (modulus and components) from the analytical formula and the simulations when the field at the coil center reaches its maximum value of 815 T, i. e. t = −0, 5 nsec. Simulations are performed considering the usual current pulse as input, hence the field evolution is realistically simulated.

The slight overestimation in the analytical case is due to the approximation in the conductor length and to the approximation made for the magnetic field in the conductor. Figure 11 (LHS) shows the Lorentz force contribution and direction, modulus and components. The time dependence is shown in Figure 11 (RHS).

Force is analytically calculated considering the conductor as a straight wire, hence there is no cancellation effects, the same results is obtained if the modulus of the force is calculated. Cancellation effect is here relevant where the conductor geometry has a left/right symmetry, for example the straight parts of the conductor have a large opposite forces (on the order of 9.5 × 10⁴ N) that nearly cancel out. The upper part of the circular section has nothing which can compensate its force on the opposite side (the conductor has no top/bottom symmetry), hence the Fy component has a very large magnitude. Obviously, there is no longitudinal components, the Fz = 73 N calculated in COMSOL is probably a numerical error.

The coil expansion due to the magnetic force have been studied. Figure 12 shows the simulation results at different time frame and can be compared with experimental results obtained with shadow-graphic technique available in Ref. 31.

In the simulation the expansion is less regular and has a maximum value of 0.45 mm. This is affected by numerical errors which are quite consistent as the domains in the simulation are discretized using linear elements. Experimental data on coil expansion have been retrived from the data published in Ref. 31 and compared with simulations in Figure 13. The expansion velocity have also been determined studying the dynamics over a relatively long time frame (from the laser shot to the coil explosion). The trend of expansion velocity is also reported in Figure 13 and it is in quite good agreement with the informations proved in Ref. 31.

There is a quite good agreement between simulation and experimental data on coil expansion. This is important as it can be used to demonstrate that the most important effects on the coil expansion and on the following explosion come from the magnetic field, and Joule heating and radiation (X-ray) damages are less crucial.

Joule effect can be analytically estimated considering that the current pulse has a current value of I_FWHM = 200 kA and a time duration of t_FWHM = 2.7 nsec. Hence the power associated with this current is $P=RIFWHM2=7.39×108W$ which correspond to an electric energy Q = Pt_FWHM = 1.99 J. If only the coil is considered, taking into account its volume, the estimated thermal jump is:

being CS_Cu the copper specific heat and M = 9.2 × 10⁻8 kg the conductor mass. If the whole target is considered, one can calculate:

being M_Tot = 8.31 × 10⁻⁶ kg. It is evident that the whole mass of the capacitor-coil target strongly reduces the heating effects and the change in the conductor resistance is quite small (it can be calculated to be R_h = 0.063Ω). These results are also confirmed by a transient simulation in which the thermal jump is estimated. Figure 14 shows the thermal jump in the two cases as a function of time. It can be noted that thermal jump for the coil itself is much higher than the one calculated from analytical results. In simulations the change in resistance due to temperature is considered which causes a power peak a few orders of magnitude higher than in the analytic calculation (compatible with a resistance of 6 Ω, as can be calculated considering copper thermal coefficient). For the same reason, a slightly higher thermal jump is also found for the full target case, however the result shows that the change in conductor resistance is not crucial.

The case with the conductor itself has the same geometry as in the previous simulations. In order to simplify the simulation for the full target geometry, the capacitor mass is introduced as a simple copper block set on the conductor terminals, moreover the simulation requires a very precise treatment of the skin effect and a very fine discretization, hence only a quarter of the whole geometry is included in the simulation. Figure 15 shows the geometry implemented in the simulation and color scale represents the temperature after 10 nsec from the start of the current pulse. It has to be specified that possible local phase transitions at the skin-depth level are not taken here into account. This phenomenon is possible, as discussed in Ref. 31, and can contribute to the bigger expansion measured during the experiment. Only molecular dynamics simulation can give a precise description combining both magnetic force and phase transition, but this study is not in the scope of the present work. We can reasonably state that the expansion due to the thermal jump is not predominant if compared with the magnetic expansion, as shown in Figure 13.

The previous analysis of the state of the art of megagauss disruptive coils energized with long pulse laser can be of interest for different applications such as ion beam transport³² or plasma confinement. Here we propose a conceptual scheme based on a multi-turn coil trap to confine proton and boron ions in the energy range of interest for the fusion reaction (p⁺ + 11B → 8.5 MeV + 3α).³³ In order to increase the density and time necessary for enhancing the reaction rate,³⁴ ions in the energy range of few MeV/u should be contained in a volume with a radius smaller than 1 mm, considering the Boron plasma expands in a region of hundreds of microns in the interaction laser time.^2,4 Figure 16 shows the curvature radius of proton and boron ions for different magnetic field strength.

It is evident from the plot that a magnetic field of about 300 T is necessary to keep particle of interest for p+B reaction (H⁺ with E = 1 MeV and ¹¹B with E = 3 MeV) in a space region of about 1 mm in radius. Moreover, assuming that the plasma of ions to be confined is produced by a laser propagating in a direction which is perpendicular to the magnetic field and taking into account that laser-produced plasma has a quite high angular aperture, a magnetic field able to ensure radial confinement, i.e. the previous mentioned 300 T, with a configuration able to ensure longitudinal confinement is also necessary to maximize the efficiency of the system considering the destructive feature of the magnet. A possible solution is achieved considering a classical magnetic mirror, typically used to confine plasma in ECR ion sources.¹⁶ The magnetic mirror trap scheme is shown in Figure 17.

The trap requires a B_min = 300 T while the maximum field can be evaluated considering a proton with energy of 1 MeV and 15° of angular aperture with respect to the target surface. The orthogonal component of its velocity is $v⊥$ = 3.58 × 10⁹ mm/s and the mirror ratio is:

being $v$ the modulus of the velocity. The maximum magnetic field is then given by:

Two solenoids able to produce the required maximum field (considering the current peak value used in the previous section) with enough spatial extension have been analytically designed in a first step and then reproduced in the particle tracking code SIMION with the aim to verify the confinement efficiency, setup is shown in Figure 18.

The target is set between the two solenoids and laser incidence direction is perpendicular, or almost perpendicular to the B_z field direction on the horizontal axis. This setup, with the mirror ratio achieved, ensures that particles produced within a cone of about 75° with respect to the laser incidence direction are confined. As it can be seen from the setup in Figure 18 lower panel, the system works both as a longitudinal and radial trap. The main function is the radial trapping, even if this configuration is used in ECR ion sources for longitudinal trapping. Figure 19 shows a detail of a particle produced in the laser target interaction and how it moves in absence of field in the region of interest (left panel) and a comparison of several particles produced within a cone of 15° half-angle aperture with respect to the target normal.

It should be noted that with no field, a particle remains in the region of interest (4 × 4 mm domain in the transverse plane) for 0.3 nsec. If the field is present ions are more dense in the central part of the domain (1.5 × 1.5 mm region) for several nanoseconds, as it can be seen in Figure 20. It is evident that particles within the 15° half-angle cone are confined for a relatively long time, and the best time confinement is obtained if the particles are produced in the trap as soon as the field ramps up, namely at 3 − 4 nsec after the beginning of the current signal. Protons are not exactly produced at the trap center, the source is in (x, y, z) = (0.05, 0.05, 0) mm and they are produced at different times in the interval [0.3, 25] nsec, the proton production time (or time of birth - TOB). This result demonstrates the possibility to keep particle confined for a reasonable long time.

Details on the trap magnetic design with a complete analysis as in the single turn case are in progress.

With the aim of enhance α particle production rate in laser-induced pB reaction a complete and detailed study of an already experimentally tested coil-capacitor target have been performed in order to both validate the theoretical model and confirm the experimental result. The results presented in the paper confirm the robustness of the model and give further explanation to the experimental results available in literature. The proposed analysis gives a detailed description of the magnetic field, both in the static and transient case, and shows that, under the used approximations, the magnetic expansion force is responsible for the coil explosion, and other phenomena are less important. A more precise description of the coil-capacitor evolution can be obtained with molecular dynamics simulations able to highlight possible local phase transition in the skin-depth layer of the conductor; however this is out of the scope of this work. It is also demonstrated that the capacitor part is just able to collect the moving charges but does not introduces any relevant delay in the current pulse. This part is crucial for thermal compensation. Moreover, a simple analytic consideration allows to show that the capacitor-coil target cannot resonate with the current pulse produced during the laser interaction. These results are important in order to optimize the design of more complex structures, in fact the same technique is also used to provide a preliminary design a magnetic mirror to enhance the alpha particle yield produced in the pB nuclear fusion reaction. A detailed study of the trap in a standard configuration (double solenoid) and also using 2D spiral coils is in progress.

This work is founded by ELI-Beamlines Contract n ° S14 − 187, under LaserGen (CZ.1.07/2.3.00/30.0057), under the Ministry of Education, Youth and Sports of the Czech Republic (ELI-Beamlines reg. N ° CZ.1.05/1.1.00/02.0061), the institution Fyzikalni Ustav, AVCR, v.v.i. and under the project, co-financed by the European Social Fundand the state budget of the Czech Republic. Authors wish to thank Dr. D. Manura for the help in writing the LUA code necessary to implement time dependent simulation in SIMION, COMSOL support for useful advices and discussion in implementing mechanical and thermal simulations, Cobham Technical Services, in particular Klaus, for advices and suggestions.