2.1 Growth rate of self-generated magnetic field in dense, collision-dominated plasma

When an REB with the current density of ${\boldsymbol{j}}_{\mathrm{f}}$ is injected into the DT plasma along the $z$ -axis, the macroscopic electrical neutrality condition should be fulfilled, leading to the drifting of background particles. Since the DT ions have a mass thousands of times larger than the mass of electrons, the drifting of background ions by the interaction with the REB is relatively negligible, and the drifting current of background electrons is ${\boldsymbol{j}}_{\mathrm{b}}=-{\boldsymbol{j}}_{\mathrm{f}}$ . Since the thermal drift motion of the background electrons is much slower than that of the injected relativistic electrons ( ${v}_{\mathrm{e}}/{v}_{\mathrm{f}}\sim {10}^{-4}$ ), the $-{\boldsymbol{v}}_{\mathrm{e}}\times \boldsymbol{B}$ and $-\nabla {P}_{\mathrm{e}}/{en}_{\mathrm{e}}$ terms in the generalized ohmic law are neglected, and an approximation of the electric field of $\boldsymbol{E}=\eta {\boldsymbol{j}}_{\mathrm{b}}$ is presented, where $\eta$ is the resistivity of the background electrons. In the practical case, ${\boldsymbol{j}}_{\mathrm{f}}$ of the injected REB is spatially nonuniform, and according to Faraday’s law, a magnetic field is generated with the growth of $\partial \boldsymbol{B}/\partial t=-\mathbf{\nabla}\times \left(\eta {\boldsymbol{j}}_{\mathrm{b}}\right)=\mathbf{\nabla}\times \left(\eta {\boldsymbol{j}}_{\mathrm{f}}\right)$ . For a cylindrically symmetrical REB, neglecting the microscopic perturbations, the plasma is also assumed to be heated up symmetrically. Therefore, by noting $\mid {\boldsymbol{j}}_{\mathrm{f}}\mid$ as $j(r)$ and $\eta$ as $\eta (r)$ , the azimuthal derivation $\partial /\partial \theta$ vanishes. For simplicity, we assume that ${\boldsymbol{j}}_{\mathrm{f}}$ is only along the $z$ -axis with a gradient along the $r$ -axis. Under these assumptions, the growth rate of the azimuthal magnetic field can be expressed as follows:

(1) $$\begin{align}\frac{\partial {B}_{\theta }}{\partial t}=-\frac{\partial }{\partial r}\left(\eta j\right)=-\left(\eta \frac{\partial j}{\partial r}+j\frac{\mathrm{d}\eta }{\mathrm{d}T}\frac{\partial T}{\partial r}\right).\end{align}$$

According to Equation (1), the growth of the magnetic field is closely dependent on the competition between $\eta$ and $j\left(\mathrm{d}\eta /\mathrm{d}T\right)\left(\partial T/\partial r\right)$ , assuming that the background ions and electrons are in thermal equilibrium with the same $T$ . For the first term, the power law of $\eta \propto {T}^{\alpha }$ is applied, according to the degenerate limit of the Lee–More model and the classical Spitzer model, with $\alpha =1$ and $-3/2$ respectively representing the degenerate and classical state. For the second term, $\partial T/\partial r$ is determined by the heating efficiency:

(2) $$\begin{align}{c}_{\mathrm{v}}\frac{\partial T}{\partial t}={P}_{\mathrm{heat}}(j)=-\frac{j}{e}\frac{\mathrm{d}{E}_{\mathrm{k}}}{\mathrm{d}z},\end{align}$$

where ${c}_{\mathrm{v}}$ is the heat capacity of background plasma and ${P}_{\mathrm{heat}}$ is the heating power of the REB, related to the beam current density $j$ and the stopping power $\mathrm{d}{E}_{\mathrm{k}}/\mathrm{d}z$ . In this way, $\partial T/\partial r$ is connected to $\partial j/\partial r$ . A more intensive heating current density leads to the elevation of local temperature, inducing a gradient in $T$ with the same direction as $j$ .

Detailed analysis of the stopping power $\mathrm{d}{E}_{\mathrm{k}}/\mathrm{d}z$ in Appendix A demonstrates that for an REB with the intensity of ${10}^{20}\;\mathrm{W}/{\mathrm{cm}}^2$ in the $300\;\mathrm{g}/{\mathrm{cm}}^3$ DT plasma, the collisional component in the stopping power is dominant. For the sake of simplicity and clarity in analytical derivation, the ohmic component in Equation (A1) is omitted in the calculation, focusing on the fundamental physics of the collisional cases. In this way, ignoring the minute changes of $\ln {\Lambda}_{\mathrm{f}}$ , the stopping power $\mathrm{d}{E}_{\mathrm{k}}/\mathrm{d}z$ is independent of $T$ , and hence Equation (2) is promptly simplified into a concise form $T={T}_0+ jt/C$ , where ${T}_0$ is the initial temperature and $C$ is approximately regarded as a constant. In this way, $\partial T/\partial r$ is equal to $\left(t/C\right)\left(\partial j/\partial r\right)$ . Let ${\eta}_0=\eta \left({T}_0\right)$ , and the growth rate of the magnetic field is calculated as follows:

(3) $$\begin{align}\frac{\partial {B}_{\theta }}{\partial t}=-\frac{\partial j}{\partial r}{\eta}_0{\left(1+\frac{jt}{CT_0}\right)}^{\alpha -1}\left[1+\left(\alpha +1\right)\frac{jt}{CT_0}\right].\end{align}$$

The effect of quantum degeneracy is important for the growth of the magnetic field. According to Equation (1), the growth of the self-generated magnetic field is related to both $\eta$ and $j$ . During the heating process, the temperature in the centre region with the highest intensity of the REB undergoes a more rapid increase. Therefore, in the degenerate plasma, $\eta \propto T$ also holds its peak value in the centre region, with the same trend of gradient as $j$ , and the growth of the magnetic field is consequently promoted. By substituting $\alpha =1$ into Equation (3), the growth of the magnetic field in the degenerate becomes the following:

(4) $$\begin{align}\frac{\partial {B}_{\theta }}{\partial t}=-\frac{\partial j}{\partial r}{\eta}_0\left(1+2\frac{jt}{CT_0}\right),\end{align}$$

which means that $\partial {B}_{\theta }/\partial t$ is linear with respect to $t$ , indicating a quadratic growth of the magnetic field.

As for the classical $\eta \propto {T}^{-3/2}$ case, the two terms on the right-hand side (RHS) of Equation (1) possess different signs, and the growth of ${B}_{\theta }$ depends on the competition between $\eta$ and $T\left(\mathrm{d}\eta /\mathrm{d}T\right)$ . Numerically, Equation (3) becomes the following:

(5) $$\begin{align}\frac{\partial {B}_{\theta }}{\partial t}=-\frac{\partial j}{\partial r}{\eta}_0{\left(1+\frac{jt}{CT_0}\right)}^{-5/2}\left(1-\frac{jt}{2{CT}_0}\right).\end{align}$$

When $jt/2{CT}_0>1$ , it is observed from Equation (5) that the sign of $\partial {B}_{\theta }/\partial t$ is reversed. However, the ${\left(1+ jt/{CT}_0\right)}^{-5/2}$ term decreases towards zero quickly with the power of $-5/2$ , which retards the decline of ${B}_{\theta }$ .

2.2 Numerical calculation of the self-generated magnetic field

In this section, a numerical calculation with practical parameters is presented, following the theoretical analysis in the previous section.

Firstly, a proper expression of $\eta$ is discussed. In many researches, to avoid divergence of the Spitzer model in the cool region, a constant scale value ${\eta}_{\mathrm{c}}$ was introduced by ${\eta}^{-1}={\eta}_{\mathrm{Sp}}^{-1}+{\eta}_{\mathrm{c}}^{-1}$ ^[ Reference Davies, Green and Norreys^35, Reference Gus’kov and Kuchugov^36]. However, this approach eliminates the gradient of $\eta$ with respect to $T$ in the degenerate regime. In this investigation, to align with Lee and More’s model^[ Reference Lee and More^37], with $\alpha =1$ for degenerated cases and $\alpha =-3/2$ for classical cases, a piece-wise approximation of $\eta$ is applied as Equation (B1), as shown in Appendix B.

In the pursuit of rigour, this study has considered the influence of degeneracy effects on the heat capacity ${c}_{\mathrm{v}}$ . As is commonly known, ${c}_{\mathrm{v}}$ in Equation (2) consists of the ion and electron heat capacities. Both heat capacities are equal to the classical value $\left(3/2\right)n$ in classical case, where $n={n}_{\mathrm{i}}={n}_{\mathrm{e}}$ is the number density of ions or electrons for fully ionized fuel. In the strongly degenerate state, however, the heat capacity is calculated by ${c}_{\mathrm{v}}=n\left(3/2+\mathrm{d}{\overline{\varepsilon}}_{\mathrm{e}}/\mathrm{d}T\right)$ , where ${\overline{\varepsilon}}_{\mathrm{e}}=\int \varepsilon f\left(\varepsilon \right)\mathrm{d}\varepsilon$ is the average energy of the background electrons and $f\left(\varepsilon \right)$ is the Fermi–Dirac distribution.

Now all the pre-works have been done, and we numerically solve Equation (1). After deriving Equation (3), with the symbolic constant $C$ being calculated along with Equations (A1) and (B1), $T$ is then determined as a function of $t$ and $j$ . Subsequently, this time-varying $T$ is substituted back into Equation (B1) to calculate $\eta$ at a specific moment, as well as its derivative to temperature $\mathrm{d}\eta /\mathrm{d}T$ . By incorporating $\eta$ , $\mathrm{d}\eta /\mathrm{d}T$ and $\partial T/\partial r$ into Equation (1) and integrating both sides over time, the magnetic field ${B}_{\theta }$ is finally solved.

A specific model is constructed for analysis, which refers to typical parameters for ignition^[ Reference Zhang, Wang, Yang, Wu, Ma, Jiao, Zhang, Wu, Yuan, Li and Zhu^20, Reference Li, Yang, Xu, Zhang, Zeng, Chen, Ma, Wu and Zhang^38– Reference Tao, Zheng, Jia, Yan and Zheng^40]. The density of the DT plasma is chosen as a representative value of $300\;\mathrm{g}/{\mathrm{cm}}^3$ , with the Fermi temperature to be ${T}_{\mathrm{F}}=629.17\;\mathrm{eV}$ . The beam number density follows a bell-shaped distribution ${n}_{\mathrm{f}}(r)={n}_{\mathrm{f}}(0)\exp \left(-{r}^2/{\sigma}^2\right)$ along the $r$ -axis, with a peak value of ${n}_{\mathrm{f}}(0)=2.5\times {10}^{22}\;{\mathrm{cm}}^{-3}$ and a waist of $\sigma =20\;\mu \mathrm{m}$ . Consequently, the current density is in form of $j(r)\approx {n}_{\mathrm{f}}(0) ec\exp \left(-{r}^2/{\sigma}^2\right)$ . For the simplicity of calculating the current density $j$ , we assume an idealized exponent kinetic energy spectrum $f(E)\propto \exp \left(-E/{E}_{\mathrm{k}_0}\right)$ for the REB, with ${E}_{\mathrm{k}_0}=1.5\;\mathrm{MeV}$ , and the corresponding beam intensity integrating over the energy spectrum $E$ is $I(0)\approx {n}_{\mathrm{f}}(0){E}_{\mathrm{k}_0}c=2\times {10}^{20}\;\mathrm{W}/{\mathrm{cm}}^2$ . The profile of $j$ is assumed to be invariant, while the lateral movement will be discussed in the following.

According to Equation (3), the growth rate $\partial {B}_{\theta }/\partial t$ is self-similar in scale to $jt/{CT}_0$ , and Figure 2 shows a typical calculation of ${B}_{\theta }$ and $T$ at $r=\sigma$ . In the high-density regime, the primary contribution to heating the DT fuel is the collision effect between the fast electrons and background electrons, and $T$ is increased near-linearly. The green and yellow regions in Figure 2 roughly separate the degenerate and non-degenerate states by the Fermi temperature ${T}_{\mathrm{F}}$ . Notably, although the degenerate state lasts for merely a short time, less than $1\;\mathrm{ps}$ , compared with the approximately $10\;\mathrm{ps}$ full fast heating process, the quadratic growth of ${B}_{\theta }$ in the early degenerate stage is quite crucial. As the plasma is heated up and turns non-degenerate, the growth of ${B}_{\theta }$ is decreasing as expected. However, the already generated ${B}_{\theta }$ in the initial degenerate stage is still sustained for a long while, which leads to self-organized pinching of the REB.

Figure 2 Theoretical calculation of ${B}_{\theta }$ and $T$ at the position of $r=\sigma$ , for a bell-shaped REB with $I(0)=2\times {10}^{20}\;\mathrm{W}/{\mathrm{cm}}^2$ in $300\;\mathrm{g}/{\mathrm{cm}}^3,{T}_0=200\;\mathrm{eV}$ DT plasma. The green and yellow background colours indicate the degenerate and non-degenerate states, respectively.

2.3 Threshold for self-organized pinching

In the practical fast heating process, attention should be paid to the self-organized pinching of the REB. Due to the degenerate effect, the self-generated EB field rises rapidly to deflect and concentrate the REB. Consequently, the current density $j$ increases while the waist of the REB decreases, and hence $\partial j/\partial r$ in Equation (3) increases, accelerating the growth of the magnetic field. In turn, the boosted magnetic field further pinches the diverging fast electrons towards the centre. It should be clarified that the approximately $10\;\mathrm{ps}$ time span of fast heating is insufficient for hydrodynamic effects of the background plasma to occur; therefore, this self-organized pinching is purely EB in nature.

To determine the threshold at which the magnetic field will inevitably evolve into the pinching process, we outline an estimation of the required magnetic fields, in which the Larmor radius of fast electrons plays a key role, ${r}_{\mathrm{L}}=\gamma {m}_{\mathrm{e}}v/ eB$ , where $\gamma$ is the Lorentz factor of the electron and $v$ is the corresponding velocity. For an REB out of the guiding gold cone tip, its divergence is modelled as a Gaussian function $f\left(\theta \right)\propto \exp \left(-{\theta}^2/{\theta}_0^2\right)$ characterized by ${\theta}_0$ ^[ Reference Robinson, Strozzi, Davies, Gremillet, Honrubia, Johzaki, Kingham, Sherlock and Solodov^6, Reference Debayle, Honrubia, d’Humieres and Tikhonchuk^41]. It is essential that the transverse extension of the magnetic field along the $r$ direction, denoted as $L$ , should be at least equivalent to ${r}_{\mathrm{L}}\left(1-\cos {\theta}_0\right)$ at a particular instant. Once the inequality

(6) $$\begin{align}{\int}_{\kern-5pt 0}^L{B}_{\theta }(r)\mathrm{d}r>\gamma \frac{m_{\mathrm{e}}v}{e}\left(1-\cos {\theta}_0\right)\end{align}$$

for pinching is fulfilled, most of the diverging electrons are deflected by the EB field, causing them to converge towards the beam centre. For a rough estimation, substituting $L=10\ \mu \mathrm{m}$ , ${\theta}_0=30{}^{\circ}$ and $\gamma =3.9$ (corresponding to $1.5\;\mathrm{MeV}$ ) into Equation (6), a $10\;\mu \mathrm{m}$ scale of magnetic field with $B>90\;\mathrm{T}$ is required, which is available under the parameter of theoretical calculation.

3.1 Introduction to LAPINS code and simulation settings

To examine the theoretical analysis, we conducted 3D3V hybrid PIC simulations with the LAPINS code^[ Reference Wu, He, Yu and Fritzsche^42– Reference Wu, Yu, Zhao, Hoffmann, Fritzsche and He^45]. LAPINS is a hybrid PIC code compatible with simulation of degenerate and dense plasmas over a large scale by improving the interactions of collision and the EB field. Compared with conventional PIC codes, LAPINS modifies the kinetic equation to Boltzmann–Uhling–Uhlenbeck equations, where the Pauli exclusion principle is considered in collision terms. With this approach^[ Reference Wu, Yu, Fritzsche and He^44], Fermi–Dirac statistics are implemented to degenerate particles. Different from other hybrid PIC codes, in LAPINS code, the hydro method is only applied to solve the EB field, while the background electrons and ions are still treated as discrete particles. Hence, the collisions among ions, background electrons and injected electrons are self-consistently included through the Monte Carlo schemes ^[ Reference Nanbu and Yonemura^46– Reference Pérez, Gremillet, Decoster, Drouin and Lefebvre^48]. Benchmarks for the correctness of the collision module and range of fast electrons are presented in Appendix C.

The major configurations for simulation are listed as follows, referring to Refs. [Reference Zhang, Wang, Yang, Wu, Ma, Jiao, Zhang, Wu, Yuan, Li and Zhu20,Reference Li, Yang, Xu, Zhang, Zeng, Chen, Ma, Wu and Zhang38–Reference Tao, Zheng, Jia, Yan and Zheng40]. Due to the special sharp density edge in the DCI scheme, we assume that the REB is directly injected into the high-density isochoric plasma in our simulations. The density of DT plasma is $300\;\mathrm{g}/{\mathrm{cm}}^3,$ corresponding to a number density of ${n}_{\mathrm{e}}={n}_{\mathrm{i}}=7.2\times {10}^{25}\;{\mathrm{cm}}^{-3}$ , and the initial temperature is ${T}_0=200\;\mathrm{eV}$ . The REB is injected into the $300\;\mathrm{g}/{\mathrm{cm}}^3$ , $200\;\mathrm{eV}$ degenerate DT plasma. A complete distribution of the broad-spectrum, diverging and bell-shaped REB is written as follows:

(7) $$\begin{align}{n}_{\mathrm{f}}\left(r,{E}_{\mathrm{k}},\theta \right)\propto 2\pi r\exp \left(\!-\frac{r^2}{\sigma^2}\right)\exp \left(\!-\frac{\mid {E}_{\mathrm{k}}-1.5\;\mathrm{MeV}\mid }{1\;\mathrm{MeV}}\!\right)\exp \left(\!-\frac{\theta^2}{\theta_0^2}\!\right),\end{align}$$

where $\sigma =20\;\mu \mathrm{m}$ and ${\theta}_0=30{}^{\circ}$ . The total energy carried by the REB and injected into the system is $40\;\mathrm{kJ}$ , and the injection power is constant in the $10\;\mathrm{ps}$ simulation span. The fast electrons are real particles injected into the system at each time step, randomly assigning the r-position, kinetic energy and momentum according to Equation (7) and the total energy requirement. Therefore, the transport and stagnation process of the injected electrons is self-consistent.

The size of simulation region is $120\;\mu \mathrm{m}\times 120\;\mu \mathrm{m}\times 100\;\mu \mathrm{m}$ with Cartesian coordinates, with the length of each meshgrid being $1\;\mu \mathrm{m}$ along the $x$ - and $y$ -axes, and $0.5\;\mu \mathrm{m}$ along the $z$ -axis. The fields adhere to the periodic boundary condition, and the background ions and electrons adhere to the thermal boundary condition. In particular, the injected electrons apply the absorbing boundary. The time step $\mathrm{d}t$ is set to be $1.67\;\mathrm{fs}$ . The DT ions are equivalent to a virtual species of particle with a mass number of $A=2.5$ , and the nuclear fusion module is turned off. The particle per cell (ppc) values of background ions and electrons are both $64$ . The REB is injected at $z=0$ from the left-hand boundary of the simulation region. In each time step, the ppc of fast electrons generated in the boundary cell is 8. To ensure computation efficiency, an upper limit of $400$ is imposed on the ppc of fast electrons. A newly developed merging algorithm^[ Reference Gonoskov^49] is applied to the injected electrons, which simultaneously ensures the conservation of energy, momentum and current density, while decreasing the number of virtual particles to achieve greater computing speed.

3.2 Simulation results

The simulation results are presented in Figure 3. As shown in Figure 3(a), significant self-generated EB fields are generated in the high-density plasma. Clear illustration of the temporal evolution of the magnetic field profile is presented in Figures 4(a)–4(d). The ${B}_{\theta }$ exhibits a significant trend of pinching towards the centre like a pair of folded wings, with the field intensity in the ‘mid-wing’ region exceeding $200\;\mathrm{T}$ . The extraordinarily large field intensity near the injection edge $z=0$ results from the overly idealized REB shape, whereas in reality the stacking and transverse spread of low-energy electrons would lessen the field. Hence, to focus on the self-organized pinching and to check with theoretical calculation, the maximum value of ${B}_{\theta }$ is only collected within the region of $z>20\;\mu \mathrm{m}$ and $r<30\;\mu \mathrm{m}$ , as is marked by the black dotted box in Figure 3(a).

Figure 3 Snapshots of the transport of REB in DT plasma at $t=10\;\mathrm{ps}$ . (a) Deposition of injected electrons, vector schematic of self-generated magnetic field and profile of ${B}_{\theta }$ . (b)–(d) Profile of the distribution of deposited, injected electrons, electron temperature and ion temperature at $x=0$ . For comparison, the bottom half part of (b)–(d) shows the non-EB simulation result, which turns off the EB field. To demonstrate the pinching of the self-generated EB field, the contours in (b) indicate the intensity distribution of the magnetic field ${B}_{\theta }$ , and the black quivers indicate the direction of the REB’s current density $-{\boldsymbol{j}}_{\mathrm{f}}/\mid j\mid$ .

Figure 4 (a)–(d) Snapshots of the self-generated magnetic field ${B}_{\theta }$ at $t=2\;\mathrm{ps}$ , $5\;\mathrm{ps}$ , $8\;\mathrm{ps}$ and $10\;\mathrm{ps}$ in the EB case. The black arrows show the direction of the REB’s current density $-{\boldsymbol{j}}_{\mathrm{f}}/\mid j\mid$ . (e)–(h) Full width at half maximum of the REB’s current density $j$ along the $z$ -axis at the corresponding moment. The black dotted lines indicate the linear fitting result, with the slopes shown in Table 1.

Table 1 Linear fitting slope $\mathrm{d}\left({\mathrm{FWHM}}_{\mid j\mid}\right)/\mathrm{d}z$ according to Figures 4(e)–4(h). The region for least-square fitting is from $z=10\;\mu \mathrm{m}$ to $z=80\;\mu \mathrm{m}$ .

The growth line of maximum ${B}_{\theta }$ is illustrated in Figure 5(a). Due to the divergence of the REB, the equivalent current density along the $z$ -axis is lower than that in the one-dimensional (1D) theoretical calculation. However, the simulation results match the theoretical trend well. In the first $1\;\mathrm{ps}$ , when the peak temperature of the background electrons is still lower than $1\;\mathrm{keV}$ and most of the system is in a degenerate state, ${B}_{\theta }$ increases quadratically, consistent with the prediction of Equation (3), and a considerable magnetic field of about $100\;\mathrm{T}$ is generated just in this instant. Then, the growth of ${B}_{\theta }$ goes over the inflection point and slows down, as the plasma is heated up and becomes non-degenerated. Up to $t=5\;\mathrm{ps}$ , the line of ${B}_{\theta }$ in Figure 5(a) shows a high degree of similarity with the $r=\sigma$ line in Figure 2(b). However, as the magnetic field increases to be strong enough to deflect diverging electrons to the centre, the self-organized pinching effect occurs to concentrate the REB, leading to a boosted increase in the EB field, with ${B}_{\theta }$ increasing from $150\;\mathrm{T}$ to nearly $250\;\mathrm{T}$ in the subsequent $5\;\mathrm{ps}$ of the heating process.

Figure 5 (a) Maximum value of ${B}_{\theta }$ within the statistical region shown in Figure 3(a). The stages of degenerate state, non-degenerate state and pinching are indicated by the green, yellow and blue background colours, respectively. (b) Detailed profile of ${B}_{\mathrm{max}}$ in the first $2\;\mathrm{ps}$ and the corresponding quadratic fitting line (the black dotted line). (c) Profile of $\mid j\mid$ at $t=10\;\mathrm{ps}$ in EB and non-EB simulations along the $r$ -axis, normalized by the maximum value of the blue solid line. (d) Deposit efficiency of the REB in EB and non-EB simulations. (e) Maximum values of ${T}_{\mathrm{i}}$ and ${T}_{\mathrm{e}}$ of plasma in EB and non-EB simulations.

This magnetic field effectively concentrates the deposition of injected electrons. The top half of Figure 3(b), representing the EB case, shows a significantly denser deposition compared to the bottom half, which represents the non-EB case. The black arrows in Figure 3(b) represent the unit vector $-{\widehat{\boldsymbol{j}}}_{\mathrm{f}}=-{\boldsymbol{j}}_{\mathrm{f}}/\!\!\!\mid j\mid$ , indicating the flow direction of the REB. It is noted that in the top half the arrows predominantly point forward due to the collimating effect of the self-generated EB field, whereas in the bottom half, the arrows diverge as initially configured. The direction of the REB’s current density $-{\widehat{\boldsymbol{j}}}_{\mathrm{f}}$ in the EB case is also plotted in Figures 4(a)–4(d). At $t=2\;\mathrm{ps}$ , $-{\widehat{\boldsymbol{j}}}_{\mathrm{f}}$ shows the typical divergence of the REB, as in the initial settings. However, as the magnetic field ${B}_{\theta }$ grows over $100\;\mathrm{T}$ at $t=5\;\mathrm{ps}$ , the diverging trend is restrained, and $-{\widehat{\boldsymbol{j}}}_{\mathrm{f}}$ is gradually collimated to the beam centre. When the simulation duration exceeds $8\;\mathrm{ps}$ , most of the $-{\widehat{\boldsymbol{j}}}_{\mathrm{f}}$ arrows are pointed forward along the $z$ -axis.

The divergence of the REB can be characterized by the profile of current density $\mid j\mid$ along the z-axis. To attain the multiple relationship clearly, $\mid j\mid$ is normalized by the maximum of the counted profiles. As is shown in Figure 5(c), $\mid j\mid$ has a broader profile in the non-EB case, and decreases rapidly as $z$ increases deeper into the plasma, which indicates that the injected electrons have been scattered and the REB is diverging. In contrast, due to the pinching of the self-generated EB field, the REB in the EB case still retains a considerable intensity even after transporting deeply to $z=50\;\mu \mathrm{m}$ , and the waist expansion is minimal. In comprehensive consideration, the divergence of the REB during transport is quantified by $\mathrm{d}\left({\mathrm{FWHM}}_{\mid j\mid}\right)/\mathrm{d}z$ , obtained through a least-squares fit from $z=10\;\mu \mathrm{m}$ to $80\;\mu \mathrm{m}$ , as shown in Figures 4(e) and 4(f). Due to the diverging angle ${\theta}_0$ in the initial settings and the collisional scattering, as shown by the red lines in the non-EB case, ${\mathrm{FWHM}}_{\mid j\mid }$ broadens as $z$ increases. However, when considering the effect of the EB field, the profile of ${\mathrm{FWHM}}_{\mid j\mid }$ changes significantly. Even at $t=2\;\mathrm{ps}$ in Figure 4(e), the blue line of the EB case exhibits a slower growth rate than the red line, which indicates a smaller diverging angle. At $t=5\;\mathrm{ps}$ , ${\mathrm{FWHM}}_{\mid j\mid }$ in the EB case no longer increases over z, and the REB has been collimated. As for $t$ exceeding $8\;\mathrm{ps}$ , the slope $\mathrm{d}\left({\mathrm{FWHM}}_{\mid j\mid}\right)/\mathrm{d}z$ turns negative, which means that the waist of the REB is shrinking over z, and the REB is pinched by the self-generated magnetic field.

3.3 Analysis of the deposit efficiency of the REB and the enhancement of fast heating

The pinching of the REB significantly impacts the deposit of injected electrons. Figure 5(d) illustrates the deposit efficiency of the REB, which is defined as the ratio of the number of injected electrons deposited in the heating region to the total number of injected electrons in the REB. A high deposit efficiency indicates that more injected electrons deposit their energy in a localized region, making the most use of the energy of the REB. According to the profiles of electron and ion temperature shown in Figures 3(c) and 3(d), the heating region of the DT plasma has been estimated as a cylinder, with the lateral radius $r=\sqrt{x^2+{y}^2}$ being less than $20\;\mu \mathrm{m}$ . It can be noted in Figure 5(d) that in both simulations, the deposit efficiency decreases over time in the beginning stage. However, the blue line representing the EB case increases after $t=3.5\;\mathrm{ps}$ , which means that more injected electrons are concentrated to the heating region by the large EB field and pinching. At $t=10\;\mathrm{ps}$ , the deposit efficiency is merely $2.46\%$ for the non-EB case, while it is $10.78\%$ for the EB case, showing a significant difference of more than $8\%$ . Accordingly, the pinching is promotive for the concentrated deposit of the REB to form a localized hot spot for ignition.

To directly demonstrate the effect of pinching on the ignition, the electron and ion temperatures of the DT plasma are also discussed. The top half of Figures 3(c) and 3(d), representing the EB case, shows a larger and hotter heated region, which extends deeper along the $z$ -axis. The increase of the maximum values of ${T}_{\mathrm{e}}$ and ${T}_{\mathrm{i}}$ is shown in Figure 5(e). It can be observed that ${T}_{\mathrm{i}}$ is always lower than ${T}_{\mathrm{e}}$ , since the heating of the DT ion is through relaxation with bulk electrons. A temperature difference between the EB and non-EB cases emerges after $1\;\mathrm{ps}$ , when the field has grown beyond $100\;\mathrm{T}$ and begins to concentrate injected electrons. As the pinching effect intensifies, the temperature gap finally increases to $1\;\mathrm{keV}$ at $t=10\;\mathrm{ps}$ , when ${T}_{\mathrm{i}}$ in the EB case has reached $4.3\;\mathrm{keV}$ . To save computational resources, the nuclear reaction module is not included in simulations. If considering additional fusion α-particle heating from burning of the DT fuel, the sensitively $T$ -dependent fusion reactivity in the EB case is nearly two orders of magnitude that in the non-EB case^[ Reference Bosch and Hale^50], and the gap of ${T}_{\mathrm{i}}$ between EB and non-EB cases will be far more remarkable. It is expected that the entire heated region in the EB case would exceed $5\;\mathrm{keV}$ , forming a hot spot for ignition. Using the same estimation of the heating region as described in the last paragraph, and defining the heating efficiency as the increase of DT ion internal energy in the heated region divided by the total energy injected by the REB, the heating efficiency is $11.69\%$ in the non-EB case and rises to $19.75\%$ in the EB case, which comes out as nearly a two-fold increase. This result indicates that the effect of the pinching and self-generated EB field in the high-density plasma has been significantly underestimated in previous studies.