Thermonuclear fusion triggered by collapsing standing whistler waves in magnetized overdense plasmas

Thermal fusion plasmas initiated by standing whistler waves are investigated numerically by two- and one-dimensional particle-in-cell simulations. When a standing whistler wave collapses due to the wave breaking of ion plasma waves, the energy of the electromagnetic waves transfers directly to the ion kinetic energy. Here we find that ion heating by use of standing whistler waves is operational even in multidimensional simulations of multi-ion species targets, such as deuterium-tritium (DT) ices and solid ammonia borane (${\mathrm{H}}_6\mathrm{BN}$). The energy conversion efficiency to ions becomes as high as 15% of the injected laser energy, which depends significantly on the target thickness and laser pulse duration. The ion temperature could reach a few tens of keV or much higher if appropriate laser-plasma conditions are selected. DT fusion plasmas generated by this method must be useful as efficient neutron sources. Our numerical simulations suggest that the neutron generation efficiency exceeds ${10}^9$ n/J per steradian, which is beyond the current achievements of the state-of-the-art laser experiments. Standing whistler-wave heating would expand the experimental possibility for an alternative ignition design of magnetically confined laser fusion and also for more difficult fusion reactions, including the aneutronic proton-boron reaction.

Figure 1

Schematic sketches of the force and density distributions of the electrons and ions during the collapse of a standing whistler wave. The horizontal axis is the spatial direction of $x$. (a) The density distribution of the electron fluid (green curve) is modified quickly in response to the appearance of the electromotive force (red curve) associated with the standing whistler wave. The force balance is established by the longitudinal electric field (black curve). (b) The longitudinal electric field accelerates the ions, which move toward the antinodes of the standing wave. The blue curve shows the ion-density fluctuation. (c) The electric field is sustained by the redistribution of the electron fluid as long as the standing wave exists. (d) The ions continue to be accelerated effectively by the constant electric field until the wave breaking takes place.

Figure 2

Time evolution of the position-momentum phase diagram for the deuterons during the formation and collapse of a standing whistler wave in the 2D fiducial run. The snapshot images are taken at (a) ${\omega }_{0}t=52$, (b) 83, (c) 114, and (d) 177. The fiducial parameters are the DT target density $n_{e0}/n_c=50$, the external magnetic field $B_{\mathrm{ext}}/B_c=20$, and the injected laser amplitude $a_0=4$. The color denotes the particle number that is integrated with the $y$ direction. The target plasma is located initially at $-7.5\le x/{\lambda }_0\le 7.5$. The left panel shows the half region of the target at $x<0$. The right panel is the magnified view of the central part of the target $0\le x/{\lambda }_0\le 1$.

Figure 3

Time evolution of the position-momentum phase diagram for the electrons in the 2D fiducial run. The model parameters and timing of the snapshots are the same as in Fig. 2.

Figure 4

(a) Time histories of the plasma energies in the 2D fiducial run (solid curves). The color indicates the different species of the target plasma, which are the electrons (red), deuterons (green), tritons (blue), and the total amount of all the ions (black). For comparison, the results of the 1D simulation with the same model parameters are also plotted by the dashed curves. The gray area stands for the laser pulse duration, where the injected laser irradiates the target surface ($0\le {\omega }_{0}t\le 100$). (b) Energy spectrum measured after the standing whistler-wave heating for the electrons (red), deuterons (green), and tritons (blue) in the 2D fiducial run (solid curves). The spectra are calculated at the end of the calculation ${\omega }_{0}t=1842$. The dashed curves are the reference results of the 1D case with the fiducial parameters.

Figure 5

(a) Dependence of the laser pulse duration on the conversion efficiency from the injected lasers to the target plasmas evaluated at ${\omega }_{0}t=2789$ by 1D simulations. The color indicates the different species, which are the electrons (red), deuterons (green), tritons (blue), and the total sum of the ions (black). The model parameters except for the pulse duration are identical for all the runs. The laser amplitude, field strength, and target density are the same as in the 2D fiducial run, that is, $n_{e0}/n_c=50, B_{\mathrm{ext}}/B_c=20$, and $a_0=4$. The target thickness adopted here is $L_x/{\lambda }_0=30$, which is twice of the fiducial run. (b) Dependence of the target thickness on the average energy of each plasma species. The ratio of the target thickness and laser pulse duration is fixed to be $c{\tau }_0/L_x=10/\left(3\pi \right)$ or ${\omega }_0{\tau }_0=(20/3)(L_x/{\lambda }_0)$ in all the cases. The model parameters are the same as those in the models shown by (a) except for the target thickness and pulse duration.

Figure 6

Time evolution of the laser field penetrating an overdense target for (a) collisional and (b) collisionless cases. The solid curves show the envelope of the tangential electric field of the laser. The different colors indicate the different snapshot timings taken at ${\omega }_{0}t=119$ (red), 402 (green), and 684 (blue). The target thickness and pulse duration are assumed to be infinite. The gray area denotes the target location at $x>0$. The flat-top shaped laser pulse reaches the target surface at $t=0$. The fiducial parameters ($n_{e0}/n_c=50, B_{\mathrm{ext}}/B_c=20$, and $a_0=4$) are used. The grid resolution is $\mathrm{\Delta }x={\lambda }_0/1000$, and the particle number per grid is 100. The horizontal arrow stands for the pulse length of the whistler wave entered in the target $L_w$.

Figure 7

(a) Time profile of the DT neutron yield for various cases with different pulse duration of ${\omega }_0{\tau }_0=100$ (dot-dashed blue), 200 (solid black), 400 (dashed red), and 600 (dotted green). The ratio of the target thickness and laser pulse duration is fixed to be $L_x/{\lambda }_0=(3/20){\omega }_0{\tau }_0$. The neutron yield is normalized by the laser energy, so it is shown in the unit of n/J. (b) DT neutron yield for more efficient cases with different target thickness of $L_x/{\lambda }_0=30$ (solid black), 60 (dashed red), 90 (dotted green), and 120 (dot-dashed blue). For these models, the same pulse duration ${\omega }_0{\tau }_0=200$ is used, which means that the difference in the neutron yeild is originated from the different ratio of $c{\tau }_0/L_x$.

Figure 8

Spatial distribution of the neutron generation rate per unit volume and unit time for the cases of (a) $L_x/{\lambda }_0=30$ and (b) 90. The snapshot data are taken at $t=1$, 10, 20, and 30 ps after the laser irradiation. The puluse duration is ${\omega }_0{\tau }_0=200$ (${\tau }_0=106$ fs) for both cases. The other laser-plasma parameters are the fiducial values such as $n_{e0}/n_c=50, B_{\mathrm{ext}}/B_c=20$, and $a_0=4$. The generation rate is evaluated from the average density and temperature of the DT plasma in each 3-$\mu \mathrm{m}$ segments. The temperature is estimated by two-thirds of the peak energy in the energy spectrum $\epsilon f_{\epsilon }\left(\epsilon \right)$. The gray area is the initial location of the DT target.

Figure 9

Ion temperature predicted by Eq. (2) in the density and magnetic-field strength ($n_{e0}-B_{\mathrm{ext}}$) diagram. The color indicates the ion temperature of $T_i$ in the unit of keV, which is divided by the nondimensional laser intensity and ion charge, $Z{a}_0^2$. The contours are shown by solid curves at $T_i/\left(Z{a}_0^2\right)=0.1$, 1, and 10.

Figure 10

(a) Energy spectra in a case of imploded DT target heated by the standing whistler wave for the electrons (solid red), deuterons (dashed green), and tritons (dotted blue). The key parameters of this run are the plasma density $n_{e0}/n_c=1500$, magnetic-field strength $B_{\mathrm{ext}}/B_c=240$, and laser amplitude $a_0=20$. The laser pulse duration is ${\omega }_0{\tau }_0=200$. The spectra are measured at ${\omega }_{0}t=1842$ after the laser irradiation. Taking account of the extremely high cyclotron frequency, the grid resolution used in this run is $\mathrm{\Delta }x={\lambda }_0/2000$, and the particle number per grid is 100. (b) Energy spectra for a case of ammonia borane target. Each species is shown by different colors, which are the electrons (solid red), protons (dot-dashed black), boron ions (dashed green), and nitrogen ions (dotted blue). The simulation parameters of this run are the plasma density $n_{e0}/n_c=250$, magnetic-field strength $B_{\mathrm{ext}}/B_c=100$, laser amplitude $a_0=12$, and pulse duration ${\omega }_0{\tau }_0=200$. The grid resolution is $\mathrm{\Delta }x={\lambda }_0/1000$ and the particle number per grid is 260.

Figure 11

Neutron generation efficiency of the unit of n/(J sr) in the optimized model calculated from the 1D PIC simulation. The highest neutron yield among our simulation results is $2.2\times {10}^{10}$ n/J which corresponds to $1.8\times {10}^9$ n/J per steradian. The 1D analysis has an ambiguity on the laser spot size so three cases are showed with the different assumption for $d_0=10\mu \mathrm{m}, 100\mu \mathrm{m}$, and 1 mm by the diamonds. For comparison, the experimental achievements of laser-generated neutrons done by various laser facilities are plotted in the figure. The previous experiments are divided into two categories depending on the method of neutron generation, which are the beam fusion method (circles) [22, 23, 24, 25, 26, 27, 28] and the implosion method (squares) [29, 30, 31, 32, 33]. The names of laser facilities used in the experiments are labeled near the corresponding data points.