Magnetic Fields in Inertial Fusion Plasmas

Abstract

The behaviour of magnetic fields in plasmas is explored through theory, experimental evidence and simulations. The Nernst effect, the advection of magnetic fields due to the thermal force, has been found to lead to magnetic cavitation in laser-plasmas, modifying heat transport and reducing fusion yield. Coupling between the Nernst effect and Righi-Leduc heat flow, in which the advected magnetic field distorts the heat transport, have been outlined as the cause of an instability termed the magnetothermal instability (MTI). The MTI leads to turbulence and significantly reduces fusion yield. These effects have been simulated using particle-tracing algorithms. For the Nernst effect, synthetic radiographs agree with experimental results. The MTI is yet to be tested experimentally.

Introduction

Despite their potential to enhance fusion yield, magnetic fields in ICF plasmas are subject to transport processes and instabilities that can severely degrade performance and reduce fusion yield. The Nernst effect expels magnetic fields out of the fuel core, reducing the internal magnetic flux, while the magnetothermal instability (MTI) amplifies small perturbations into turbulence through a feedback loop with heat flow. This literature review outlines the experimental and computational evidence for these phenomena. It begins by detailing the benefits of applied magnetic fields and the natural generation of fields via the Biermann battery effect, before discussing the discovery of Nernst-driven magnetic cavitation and the current theoretical understanding of the MTI, including dependence on magnetisation and possible avenues for mitigating their presence.

Magnetic fields in plasmas

Self-generated magnetic fields in inertial fusion plasmas can have profound consequences on the processes within the plasma and severely affect fusion yield. In addition, inertial fusion experiments utilising applied magnetic fields in efforts to relax ignition conditions and increase fusion yield are likewise subject to effects from the unintended behaviour of the magnetic fields present. An applied magnetic field in a plasma can increase the fusion yield through suppressing electron heat transport which lowers the required energy of the driver, and through magnetising the alpha particles in the hotspot, which decreases their Larmor radius, trapping them and increasing self-heating [1], [2], [3]. One example is MagLIF, in which a strongly magnetised Z-pinch structure compresses the fuel capsule. 2D integrated simulations performed by Sefkow et al. predict significantly lower necessary stagnation pressure, higher alpha particle confinement, lower implosion velocity and therefore driver power, and crucially, higher neutron yield [1]. Hohenberger et al. obtained the first experimental validation of such simulations using the OMEGA laser, in which a magnetic field of 8T applied to a direct-drive scheme resulted in a 30% increase in neutron yield [2]. These results have been replicated since at other facilities, such as the NIF in which Moody et al. used a 26T axial magnetic field to obtain a multiple of 3.2 increase in neutron yield for an indirect-drive experiment [3].

Even in inertial fusion plasmas without an applied magnetic field, self-generation of fields can occur through effects such as the Biermann battery or Weibel instabilities. Velocity distribution anisotropies, such as from opposing plasma flows as utilised by Huntington et al., lead to current filamentation which generate individual azimuthal magnetic fields [4]. These filaments can merge, resulting in stronger magnetic fields, a result of Weibel instability growth. The Biermann battery is a phenomenon in which perpendicular temperature and density gradients result in a magnetic field, as per Ohm’s law. In inertial plasmas the rapid and targeted laser-heating of the gas results in such gradients, to which the electrons respond faster than the ions. The resulting electron displacement results in a thermoelectric current and solenoidal electric field, generating the magnetic field [5]. In the landmark experiment attributing the magnetic field to the Biermann battery, Stamper et al. fired a neodymium-doped glass laser at a target and use magnetic probes to measure the rate of change of B [5].

Proton radiography

Another is Li et al.’s measurement of EM fields using the OMEGA laser, in which a CH foil is irradiated and monoenergetic protons are fired through the resulting plasma to measure their deflection and calculate the strength of the magnetic field [6]. Proton radiography is a popular method of measuring any present EM fields due to its high spatial and temporal resolution [7]. Further, the energy of the protons can be selected, such as choosing a monoenergetic beam of 14.7MeV protons by imploding a D 3He capsule, as developed by Li et al. [6], [8]. By firing multiple shots at various times throughout the lifetime of the plasma Li et al. obtained a time evolution of the magnetic field, and the use of a CR-39 detector stack enabled high temporal resolution [6]. This can be replicated in simulations, with a common technique being that of a PIC particle-tracing algorithm such as that implemented by PlasmaPy [7], [9].

A typical simulation workflow for a particle-tracing algorithm is demonstrated in Figure 1. Protons are generated in a random configuration to mimic experimental conditions, with the ability to set their initial energy distribution and number. They are iteratively pushed through a region with a set EM field, and the effect on the protons due to the Lorentz force is calculated. Finally, their final positions are stored as a synthetic radiograph. Each element incorporates a high degree of flexibility, allowing for custom field profiles and particle generation conditions.

Fig. 1: Typical simulation workflow for a particle-tracing algorithm [7].

The line-integrated magnetic field is obtained from the proton radiograph using the equation

\[\int_0^{l_{path}} ds \left\{ E_{x} + \frac{(\mathbf{\nu}_p \times \mathbf{B})_{x}}{c} \right\} \approx \frac{m_p v_p^2}{e} \frac{r_s + r_d}{r_s r_d} \left( x - x_0 \right)\]

where $l_{path}$ is the thickness of the path a proton has travelled along, $E_{x}$ and $B_{x}$ and the electric and magnetic fields perpendicular to the proton trajectories, $\mathbf{v}_p$ and $v_p$ are the proton velocity vector and speed, $r_s$ is the distance from the source to the plasma, $r_d$ is the distance from the plasma to the detector, $x_0$ and $x$ are the initial and final proton positions [7]. The position, $x$, is rescaled to the position within the plasma using $x=d/ \mathcal{M}$, where $d$ is the displacement on the detector and $\mathcal{M}$ is the magnification.

Nernst advection

Measuring the time evolution of the magnetic field in this way allows for an analysis of its behaviour. As well as following the plasma expansion, the magnetic field has been shown to move with electron heat flow, resulting in a magnetic cavity in the centre of the plasma as theorised by Haines in 1986 [10] and as shown by the first experimental results achieved by Arran et al. in 2023 in Figure 2. [11]. By firing the 10.6MeV protons at 0.4ns and 1.1ns, they obtained two proton radiographs demonstrating this cavitation. These experimental results compare well with both kinetic VFP (IMPACT) and fluid XMHD (CTC) code when Nernst advection is included, suggesting this as the cause of magnetic cavitation [11], [12].

In fact, assuming the frozen-in-flow MHD model is not accurate for inertial fusion as there inevitably are temperature and magnetic field gradients, a result determined by Davies et al.’s OMEGA laser modelling [13]. This Nernst advection results in a modified heat flow which distorts the temperature profile, breaking the symmetry. MagLIF simulations undertaken by Slutz et al. obtain similar results, with a significant decrease in magnetic flux in the centre – from 25% flux loss to 70% - when including the Nernst term [14]. This has substantial consequences for fusion yields; according to Sefkow et al.’s simulations, fusion yield can reduce by 10-40% [1], while simulations run by Slutz et al. found that fusion yield reduced by 70% when the Nernst term was included [14]. Therefore, higher field strengths will be required than would otherwise be expected to counteract magnetic flux losses in magnetised inertial fusion reactors [11].

Fig. 2: Proton radiography results at 0.4ns and 1.1ns from an experiment measuring the evolution of a magnetic field in a laser-plasma. The radiograph (i) shows the protons deflected outwards, and the magnetic field is reconstructed (ii) with its longitudinally averaged equivalent (iii) [11].

Magnetothermal instability

When coupled with perturbed or Righi-Leduc heat flow, the Nernst effect can lead to the magnetothermal instability (MTI), in which anisotropic heat transport leads to amplified magnetic fields, increasing the perturbed heat flow and resulting in a feedback loop that causes turbulence. First identified by Tidman and Shanny in 1974 as a thermal instability due to a Biermann battery-generated seed magnetic field [15], the MTI has since been expanded in terms of its constituent effects to better understand its causes and how one might mitigate it. For example, Bissell et al. derived the MTI can be explained by a coupling between the Righi-Leduc and Nernst effects, and that an electron temperature gradient perpendicular to the temperature gradient is not needed, thus separating the MTI from the earlier instantiation as a field-generating thermal instability [16]. They additionally performed CTC+ simulations of the effect, as shown in Figure 3, demonstrating the increased spread of heat flow and the compressed magnetic field. García-Rubio et al. discerned a difference between absolute and convective MTI in which the former grows exponentially whereas the latter is convected away and thus has a more limited impact [17]. Absolute MTI, however, can significantly affect the ICF capsule implosion. They therefore concluded that the Biermann, Nernst and Righi-Leduc terms must be included in fusion simulations, as the MTI operates on timescales relevant to ICF.

Temperature and magnetic field evolutions after 460ps of a plasma perturbed at 300ps resulting in the MTI [16].

Conclusion

The Nernst effect is magnetisation-dependent, decreasing as the Hall parameter increases. Slutz et al. found flux losses decreased from 70% to 45% when the applied magnetic field was increased from 10 T to 30 T [12], consistent with Velikovich et al.’s finding that increased magnetisation can reduce losses by an order of magnitude [16]. The MTI itself can be stabilised magnetically; a stronger horizontal field increases the Alfvén speed, and magnetic tension can fully quench the instability when the Alfvénic response is faster than the instability growth rate [17]. Due to such complex and often detrimental effects, studying magnetic field behaviour in plasmas is essential to minimise instabilities and maximise ICF yield.

References

[1] A. B. Sefkow et al., ‘Design of magnetized liner inertial fusion experiments using the Z facility’, Physics of Plasmas, vol. 21, no. 7, p. 072711, Jul. 2014, doi: 10.1063/1.4890298.

[2] M. Hohenberger et al., ‘Inertial confinement fusion implosions with imposed magnetic field compression using the OMEGA Laser’, Physics of Plasmas, vol. 19, no. 5, p. 056306, May 2012, doi: 10.1063/1.3696032.

[3] J. D. Moody et al., ‘Increased Ion Temperature and Neutron Yield Observed in Magnetized Indirectly Driven D 2 -Filled Capsule Implosions on the National Ignition Facility’, Phys. Rev. Lett., vol. 129, no. 19, p. 195002, Nov. 2022, doi: 10.1103/PhysRevLett.129.195002.

[4] C. M. Huntington et al., ‘Magnetic field production via the Weibel instability in interpenetrating plasma flows’, Physics of Plasmas, vol. 24, no. 4, p. 041410, Apr. 2017, doi: 10.1063/1.4982044.

[5] J. A. Stamper, K. Papadopoulos, R. N. Sudan, S. O. Dean, E. A. McLean, and J. M. Dawson, ‘Spontaneous Magnetic Fields in Laser-Produced Plasmas’, Phys. Rev. Lett., vol. 26, no. 17, pp. 1012–1015, Apr. 1971, doi: 10.1103/PhysRevLett.26.1012.

[6] C. K. Li et al., ‘Measuring E and B Fields in Laser-Produced Plasmas with Monoenergetic Proton Radiography’, Phys. Rev. Lett., vol. 97, no. 13, p. 135003, Sep. 2006, doi: 10.1103/PhysRevLett.97.135003.

[7] D. B. Schaeffer et al., ‘Proton imaging of high-energy-density laboratory plasmas’, Rev. Mod. Phys., vol. 95, no. 4, p. 045007, Dec. 2023, doi: 10.1103/RevModPhys.95.045007.

[8] N. L. Kugland, D. D. Ryutov, C. Plechaty, J. S. Ross, and H.-S. Park, ‘Invited Article: Relation between electric and magnetic field structures and their proton-beam images’, Review of Scientific Instruments, vol. 83, no. 10, p. 101301, Oct. 2012, doi: 10.1063/1.4750234.

[9] PlasmaPy Community et al., PlasmaPy. (Feb. 20, 2026). Zenodo. doi: 10.5281/ZENODO.18706665.

[10] M. G. Haines, ‘Heat flux effects in Ohm’s law’, Plasma Phys. Control. Fusion, vol. 28, no. 11, pp. 1705–1716, Nov. 1986, doi: 10.1088/0741-3335/28/11/007.

[11] C. Arran et al., ‘Measurement of Magnetic Cavitation Driven by Heat Flow in a Plasma’, Phys. Rev. Lett., vol. 131, no. 1, p. 015101, Jul. 2023, doi: 10.1103/PhysRevLett.131.015101.

[12] C. P. Ridgers, R. J. Kingham, and A. G. R. Thomas, ‘Magnetic Cavitation and the Reemergence of Nonlocal Transport in Laser Plasmas’, Phys. Rev. Lett., vol. 100, no. 7, p. 075003, Feb. 2008, doi: 10.1103/PhysRevLett.100.075003.

[13] J. R. Davies, R. Betti, P.-Y. Chang, and G. Fiksel, ‘The importance of electrothermal terms in Ohm’s law for magnetized spherical implosions’, Physics of Plasmas, vol. 22, no. 11, p. 112703, Nov. 2015, doi: 10.1063/1.4935286.

[14] S. A. Slutz et al., ‘Pulsed-power-driven cylindrical liner implosions of laser preheated fuel magnetized with an axial field’, Physics of Plasmas, vol. 17, no. 5, p. 056303, May 2010, doi: 10.1063/1.3333505.

[15] D. A. Tidman and R. A. Shanny, ‘Field-generating thermal instability in laser-heated plasmas’, The Physics of Fluids, vol. 17, no. 6, pp. 1207–1210, Jun. 1974, doi: 10.1063/1.1694866.

[16] J. J. Bissell, C. P. Ridgers, and R. J. Kingham, ‘Field Compressing Magnetothermal Instability in Laser Plasmas’, Phys. Rev. Lett., vol. 105, no. 17, p. 175001, Oct. 2010, doi: 10.1103/PhysRevLett.105.175001.

[17] F. García-Rubio, R. Betti, J. Sanz, and H. Aluie, ‘Theory of the magnetothermal instability in coronal plasma flows’, Physics of Plasmas, vol. 29, no. 9, p. 092106, Sep. 2022, doi: 10.1063/5.0109877.