|
|
| Fig. 1: Fusion Reactivities. [4] (Source: Wikimedia Commons) |
Nuclear fusion has the potential to produce energy at a global scale and replace traditional energy sources like fossil fuels, but these lofty goals require a fusion reactor system that produces net energy. While this requires the system to have a rate of energy production higher than its rate of energy loss, an additional requirement for scaling nuclear fusion is sustained burn, which occurs when enough excess energy is captured by the fuel of the reactor to allow for self-sustained fusion. One figure physicists prominently use to identify the threshold for sustained burn is the Lawson Criterion. Originally formulated by John Lawson in the 1950s, it defines the minimum temperature needed for successful sustained burn in fusion reactors based on plasma density and confinement time. [1]
Modern-day magnetic confinement reactors like ITER rely on deuterium-tritium (D-T) fusion, so a sample derivation of the Lawson criterion for D-T fusion will be used to demonstrate the principles behind Lawson function. [2] The minimum "temperature" (given by kBT where T is electron temperature in Kelvin) needed for sustained burn is found by setting the energy generated by the nuclear fusion reaction, which primarily comes from the released α particle, equal to the energy lost. [3] Assuming a 1:1 ratio of deuterium and tritium, such that the number of each isotope is equal to half the number of electrons, the power generated per unit volume by the fusion reaction is one-fourth times the product of the number density of electrons (n) squared times the value for D-T fusion reactivity at a given ion temperature T (<συ>, see Fig. 1) times kinetic energy of the α particle generated by D-T fusion (εα). [3,4] The D-T fusion reactivity refers to the rate at which deuterium and tritium nuclei go through fusion reactions in a plasma under specific temperature conditions, and it is modeled by the parameters <συ>, where σ is the fusion cross-section (the experimentally determined probability of a fusion event occurring under specific conditions) and υ is the is the relative velocity of the reacting particles. [3] The product of these term is averaged over the Maxwellian distribution of the velocities of the reacting particles. [3]
 |
| Fig. 2: The Lawson Function for D-T Fusion. [3] (Image Source: T. Schouten) |
To calculate energy losses, the thermal energy density is used and can be calculated via the average translational kinetic energy formula, combined with the assumption that all ions and isotopes are at the same temperature, to get W = 3n kBT, where kB is Boltzmann's constant. [3] The energy loss rate per unit volume (P) can then be described by P = W/τe, where τe, known as the confinement time, is the time a plasma's thermal energy stays within the plasma before loss. Given the total energy contained in a plasma E, τe is described by dE/dt = - E/τe in the absence of fusion reactions. Notably, the energy loss here is non-specific as modeled, but specific types of energy losses include losses due to radiation and losses due to thermal conductivity. [1]
Taking the minimum criteria that the power generated per unit volume by the fusion reaction be greater than or equal to the sum of the energy losses, substitution and algebraic rearrangement gives the following result:
| nτe | ≥ | 12 kBT <συ> εα |
= | f(T) |
The right hand-side is known as the Lawson Function, or f(T). Finding the minimum of the Lawson Function above 10 keV yields the product of the electron density and the confinement time. This must be greater than or equal to about 1.5 × 1020 s m-3, which is the standard Lawson criterion. This minimum occurs at a kBT value of about 25.67 keV (see Fig. 2).
 |
| Fig. 3: Triple Product for D-T Fusion. [3] (Image Source: T. Schouten) |
While the original formulation of the Lawson criterion only relied on the plasma density and confinement time, physicists today often use the triple product, which includes the fuel temperature of the reactor fuel as a parameter for determining minimum sustained burn criteria in order to model the constraints of achievable pressure. [5] The resulting equation is easily derived by multiplying both sides of the Lawson function by the fuel temperature of the reactor T:
| nTτe | ≥ | 12 kBT2 <συ> εα |
As before, taking the minimum of the triple product above 10 keV yields that the product of the electron density, the confinement time, and the fuel temperature of the reactor. This must be greater than or equal to about 2.76 × 1021 keV s m-3. This triple product formulation of the Lawson criterion occurs at a kBT value of about 13.54 keV (see Fig. 3).
Nuclear fusion has famously been achieved in a home garage by a high schooler, so why is ITER, the multi-decade international fusion reactor project still under development, so large? [6] Indeed, ITER weighs about 23 kilotons and has a height of almost 30 meters. [7] The purpose of ITER's large size is to increase the confinement time τe. Using the parameters reported by the ITER Physics Expert Group, the average number density of electrons is 9.8 × 1019 m-3 and the average kBT value is 12.9 keV. [2] Substituting these values into the derived Lawson criterion and triple product value, the minimum confinement time required for successful fusion in ITER is between 1.59 seconds (via the Lawson function) and 2.33 seconds (via the triple product formulation).
Using the values from the Lawson function and the triple product formulation, we can estimate that the confinement time of ITER should be greater than 2.33 seconds for sustained burn. The expected confinement time of ITER, based on its design, is 3.7 seconds, which meets the minimum value derived via the Lawson criterion. [2] However, since the large size of a fusion reactor is intended to produce a large τ, exactly how large it should be depends on the specific details of energy loss for a given reactor. Additional design considerations have led to other slightly varying estimates. [8]
© Troy Schouten. 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.
[1] J. D. Lawson, "Some Criteria For a Power Producing Thermonuclear Reactor," Proc. Phys. Soc. B 70, 6 (1957).
[2] M. Shimada et al., "Overview and Summary: Progress in the ITER Physics Basis," Nucl. Fusion 47, S1 (2007).
[3] R. Fitzpatrick., Tearing Mode Dynamics in Tokamak Plasmas (IOP Publishing, 2023).
[4] H. S. Bosch and G. M. Hale, "Improved Formulas For Fusion Cross-Sections and Thermal Reactivities," Nucl. Fusion 32, 611 (1992).
[5] 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).
[6] M. Moynihan and A. B. Bortz, Fusion's Promise (Springer, 2023), p.225.
[7] M. D. Mathew, "Nuclear Energy: A Pathway Towards Mitigation of Global Warming", Prog. Nucl. Energy 143, 104080 (2022).
[8] E. J. Doyle et al., "Plasma Confinement and Transport: Progress in the ITER Physics Basis," Nucl. Fusion 47, S18 (2007).
