Stellarator Fusion Reactors

Introduction

Fig. 1: Wendelstein 7-X stellarator. (Source: Wikimedia Commons

Nuclear fusion uses the same process as the sun, whereby two light atomic nuclei combine to form a heavier nucleus and release large amounts of energy. There are two primary approaches to fusion energy: inertial and magnetic confinement. This article will discuss the latter, since stellarators are one approach within magnetic confinement. The center of a fusion device must be very hot and kept far from the vessel wall to avoid damage. The interaction of plasma with electromagnetic fields can be used for plasma confinement: this is known as magnetic confinement. The dominant design in this domain has been the tokamak, which using currents driven through the plasma for confinement. [1] Stellarators are a different approach invented by Lyman Spitzer Jr. in the 1950s while at Princeton University. [2] Rather than utilizing internal currents to to create the necessary magnetic fields for the reaction, stellarators use external coils. Overall, they are more stable compared to tokamaks but the intricate coil design process increases engineering difficulties. Currently, the German Wendelstein-7X stellarator reactor has reached the highest triple-product to date, but is still an order of magnitude away from reaching the Lawson Criterion.

The Lawson Criterion

All fusion reactors, regardless of approach, must reach the Lawson Criterion, which defines the necessary condition for a fusion reaction to produce more energy than it loses. In 1957, John Lawson introduced the double-product: a measure combining the plasma density (n) and confinement time (τ_E), the duration for which a plasma retains its thermal energy before it dissipates. For a Deuterium-Tritium (D-T) fusion reaction that produces a Helium-4 nucleus and a high-energy neutron, the double-product nτ_E must be at least 10²⁰m^-3 s. [3,4] The optimal operating regime for a fusion plasma lies between 10-20 keV, where fusion power output is maximized and the confinement requirement is weakest. For a stellarator reactor, Beidler et al. find that this implies an energy confinement requirement of nτ_E ≥ 2 × 10²⁰ m^-3 s. [5]

The triple-product nTτ_E is an expanded version of Lawson's original double-product which includes dependence on the temperature of the fuel (T). [4] Stellarator projects have had varying results in nearing the Lawson Criterion (see Table 1), but still have yet to achieve the ignition threshold.

The W7-X stellarator in Germany (see Fig. 1) serves as the global proof-of-concept for stellarators. [6] It was reported that the W7-X reached a stellarator record triple-product of nTτ_E = 6.2 × 10¹⁹ keV m^-3 s in 2017. [4] While this is significant progress since Model C's achievement of 1.10 × 10¹⁴ keV m^-3 s, it is still an order of magnitude from the established Lawson Criterion. Additionally, the W-7X took over 10 years to construct, reportedly due to delays from quality assurance and engineering limitations; construction of the 50 coils took 10⁶ hours. [1]

Stellarator Basics, Advantages, and Challenges

In magnetic confinement, magnetic field lines must twist as they travel around the torus to allow particles to remain close to a magnetic surface; it is quantified by the rotational transform, which is the number of poloidal turns per toroidal turn of a field line. [1] Toroidal refers to the direction the long way around the reactor, following the main ring of the torus. Poloidal refers to the direction the short way around the cross-section of the plasma, providing the essential twist needed to keep particles confined. The rotational transform can be defined in terms of the change in poloidal angle, (θ_k) after the kth toroidal turn along a field line: [1]

While tokamaks produce this essential magnetic twist by combining external fields with a powerful internal plasma current, stellarators generate the rotational transform entirely through the use of complex, 3D-shaped external coils. [6] By eliminating the need for an induced internal current, stellarators gain the primary advantage of being inherently steady-state devices that lack the hard density limits found in other designs. In comparison, tokamaks are restricted by the Greenwald limit: the maximum density threshold beyond which the internal plasma current becomes unstable and confinement suddenly collapses. [5] However, the trade-off for this operational stability is the immense engineering challenge posed by the intricate 3D geometry required for the stellarator's modular coils.

Optimized stellarator designs utilize intricate modular coils to create precise magnetic twists that minimize radial drift and neoclassical transport losses. [7] Neoclassical theory accounts for the complex, curved geometry of toroidal devices like tokamaks and stellarators in comparison to classical theory which examines particle collision and drift in simple, straight magnetic fields. Then, neoclassical losses occur when Coulomb collisions and the drift of trapped or circulating particles push them out of the magnetic surface. This transport is a significant concern for reactor performance because particles can become trapped in local magnetic mirrors, which are regions of high magnetic field strength created by the complex coil geometry. These trapped particles are highly susceptible to radial drift across magnetic field lines: instead of following their intended path, they move sideways and outward toward the reactor walls. If excessive particles and heat escape, the plasma cools too quickly to maintain the confinement time required for ignition. [7]

Conclusion

Stellarators stand as a more stable alternative to tokamaks, but the design has yet to achieve the Lawson Criterion and challenges remain in the intricacy of coil design. The German W7-X has demonstrated that stellarators can sustain plasma with manageable losses, establishing potential for a future steady-state fusion power plant. Various optimization methods have been introduced to focus on further refining modular coil geometries for purposes of suppressing neoclassical transport and radial drift. [1] Achieving this level of precision in magnetic confinement will be significant in closing the gap toward the Lawson Criterion and transitioning stellarators from experimental models to viable energy sources.

© Chuer Yang. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] L.-M. Imbert-Gérard, E. J. Paul, and A. M. Wright, An Introduction to Stellarators: From Magnetic Fields to Symmetries and Optimization" (Society for Industrial and Applied Mathematics, 2024).

[2] L. Spitzer Jr., "The Stellarator Concept," Phys. Fluids 1, 253,(1958).

[3] J. D. Lawson, "Some Criteria for a Power Producing Thermonuclear Reactor," Proc. Phys. Soc. B 70, 6 (1957).

[4] S. E. Wurzel and S. C. Hsu, "Progress Toward Fusion Energy Breakeven and Gain as Measured Against the Lawson Criterion," Phys. Plasmas 29, 062103 (2022).

[5] C. D. Beidler et al., "Stellarator Fusion Reactors - An Overview," Max Plank Institute für Plasmaphysik, December 2001.

[6] D. A. Gates et al., "Stellarator Research Opportunities: A Report of the National Stellarator Coordinating Committee," J. Fusion Energy 37, 51 (2018).

[7] H. Wobig, "The Theoretical Basis of a Drift-Optimized Stellarator Reactor," Plasma Phys. Control. Fusion 35 903 (1993).