Fusion energy offers a promising path to solving the global energy crisis. To initiate nuclear fusion of deuterium-tritium (DT) fuel, extremely high energy densities are required, which poses major challenges to both physics and engineering.

The Double-Cone Ignition (DCI) scheme proposed by Professor Jie Zhang is depicted by Fig. 1. Two shells of DT fuel are embedded in the head-on-head guiding cones. Multiple nanosecond laser pulses uniformly irradiate the capsule, compress and accelerate the DT fuel to eject out of the cone and collide at the center, forming a high-density and uniform plasma. Through careful shaping of the laser pulse and the low-entropy compression, the plasma remains in a quantum-degenerate state. Picosecond petawatt laser pulses then irradiate a heating cone perpendicular to the collision axis, generating ~MeV relativistic electron beam (REB) injected into the central plasma. These electrons rapidly deposit energy, locally heat the plasma to ignition temperatures and form a "burning" hot spot. Considerable energy is released from the hot spot by alpha particles produced by the DT fusion reaction, further heating the surrounding fuel and leading to full ignition.

Graphic description: (a) Schematic of the DCI scheme. (b) Schematic of the transport and heating process of the REB in the high density plasma.

During the overall ignition scenario, the energy deposition efficiency of the REB within the high-density plasma determines whether a hot spot can be generated, which is greatly crucial for the successful ignition. To enhance the deposition efficiency, the divergence of the REB should be well controlled.

On the transport of the REB in the high-density plasma, the self-generated magnetic field was used to be overlooked for granted. However, the situation is quite different concerning the plasma in state of degeneracy. In contrast to the classical ideal plasma model, the resistivity $\eta$ in the degenerated plasma bears the same raising trend with temperature T. According to the Faraday's law,

$$\frac{\partial\mathbf{B}}{\partial t} = - \mathbf{\nabla} \times \mathbf{E} = - \eta\mathbf{\nabla} \times \mathbf{j -}\left( \frac{\partial\eta}{\partial T}\mathbf{\nabla}T \right) \times \mathbf{j}$$

when the REB transports in the degenerated plasma, the $\eta$ term and the $\frac{\partial\eta}{\partial T}$ term both contribute to the growth of the magnetic field B. In contrast, in classical regime the two terms have different sign and compete with each other. Therefore, for the DCI scheme, the initially degenerated plasma facilitates the rapid growth of the self-generated magnetic field.

Although the plasma is rapidly heated up by the REB and turns non-degenerated, the already generated magnetic field has been intense enough to change the trajectories of the fast electrons. As shown in Fig. 2, the magnetic field confines the initially diverging electrons to the center, increasing forward current density and in turn further enhancing the growth of the magnetic field.

Graphic description: The generation of the self-generated magnetic field and the self-organized pinching effect.

To verify this self-organized pinching effect, three-dimension simulations are conducted with the hybrid Particle-In-Cell code LAPINS, which self-consistently contain the quantum degeneracy effect. A $1.5\ MeV$ REB with diverging angle of $30{^\circ}$ is injected into a $300\ g/cm^{3}$, $200\ eV$ degenerated DT plasma. As is shown in Fig. 3, the evolution of the self-generated magnetic field undergoes three phases:

1. < 2 ps: Plasma remains degenerate, leading to rapid magnetic field growth.
2. 2–5 ps: Plasma heats up and exits degeneracy, slowing field growth.
3. > 5 ps: Self-organized pinching accelerates the field growth again.

Compared to cases where electromagnetic fields were artificially turned off, the presence of self-generated magnetic field leads to a two-fold enhancement of electron energy deposition efficiency.

Graphic description: (a-d) The evolution result of the self-generated magnetic field in the PIC simulations by LAPINS code, in which the black arrows indicate the direction of the current density. (e) The evolution of the maximum of the self-generated magnetic field

This finding was published in High Power Laser Science and Engineering, (Y.-H. Li, D. Wu, J. Zhang, "Pinching relativistic electrons in the quantum degenerate plasmas to enhance the fast heating," High Power Laser Sci. Eng. 13, 03000e34).

In conclusion, we reveal the self-organized pinching on the transport of the REB in the quantum degenerated high-density plasma, and its considerable enhancement on the energy deposition efficiency. This finding holds significant implications for the fast ignition schemes that employ the REB as a fast heating mechanism. During the compression of the DT fuel, the entropy should be controlled to ensure that the high-density plasma remains in state of degeneracy before the injection of the REB.