Lawson Criterion Analysis of D-3He Fusion Reaction

The Lawson criterion constitutes a pivotal condition for the viability of sustained fusion reactions. This study employs numerical methods and data analysis in Microsoft Excel to ascertain the Lawson criterion for D-³He fusion at temperatures ranging from 75 to 150 keV. Furthermore, the investigation examines the influence of bremsstrahlung radiation and evaluates fusion reactivity based on Hively and Bosch-Hale parameters. Hively’s reactivity value falls within the range of 1.121 × 10−22 – 6.750 × 10−22 m3/s, the instability occurs at temperatures < 100 keV. The Bosch-Hale value differs significantly from the Hively reactivity, which is 1.208 × 10−22 – 2.340 × 10−22 m3/s. The bremsstrahlung radiation within the range of 4.819 × 10−20 – 6.802 × 10−20 keV exhibits minimal impact on the reaction. The Lawson criterion value within the scope of this study typically falls within the range of 0.437 × 1021 – 1.452 × 1021 s/m3. This range signifies the necessary combination of confinement time and plasma density to facilitate the generation of clean energy within a temperature range of 75 to 150 keV, thus propelling the fusion process toward the ignition phase.


Introduction
Nuclear fusion has become an alternative method for generating larger and safer sources of energy compared to fission reactions or fossil fuel combustion [1].Fusion reaction involves the merging of two light atomic nuclei to form a heavier new atomic nucleus while releasing energy.Fusion reactions occur in the form of ionized gas known as plasma.The energy required for the fusion process is exceptionally high because it must overcome the Coulomb repulsion force so that the two positively charged atomic nuclei can merge.The minimum temperature required to produce a sufficient amount of energy is around 100.000.000℃ or 8.6 keV [2].Magnetic fields are employed in fusion reactors to confine the plasma at sufficiently high temperatures.
The commonly used fusion fuels are deuterium (D) and tritium (T).Deuterium is abundantly available in nature, while tritium is a highly rare radioactive isotope on Earth.D-T fusion yields approximately 17.6 MeV of energy, with 80% of the energy carried by neutrons that can damage the reactor's walls.Deuterium (D) and helium-3 (³He) have the potential to become alternative fuels because they are safer, do not produce neutrons, and yield greater energy, approximately 18.4 MeV.However, the D-³He fusion reaction requires higher energy and temperature compared to D-T fusion [3].
Fusion reactions have specific criteria that must be met, namely the Lawson criteria, to predict whether fusion reactions are economically advantageous or not.The Lawson criteria compare the rate of energy produced by the fusion reaction to the rate of energy loss during the fusion process.J D Lawson concluded in his research that the fusion energy obtained is a function of temperature (T), plasma density (n), and plasma confinement time (τ) [4].The energy lost during the fusion process is due to bremsstrahlung radiation, which is electromagnetic radiation emitted by high-speed electrons as they lose energy and are deflected when in close proximity to atomic nuclei.
This research has the purpose to investigate the Lawson criteria for the D-³He fusion reaction with variations in temperature, the influence of bremsstrahlung radiation, and fusion reactivity using the Hively and Bosch-Hale parameters.The Hively and Bosch-Hale parameters are reaction rate models based on various experimental cross-section data for calculating reactivity in Maxwellian fusion.The ratio between ion and electron temperatures in hot plasma condition is not significantly different for both parameters.Therefore, Hively and Bosch-Hale parameters are often used in fusion reaction research.

Method
The initial stage of this research involved a literature study on plasma, fusion reaction, deuterium, helium-3, bremsstrahlung radiation, fusion reactivity and Lawson criteria.Next, the authors determined the parameters used for the Lawson criteria equation.The fusion reaction of the D-T fuel produces maximum fusion reactivity at a temperature of 64 keV in reference [5] and 70 keV in reference [6].The D-³He fusion reaction requires a higher temperature, so the temperature used in this research is 75 -150 keV.Subsequently, a numerical method is employed by inputting temperature parameters into the Hively and Bosch-Hale reactivity and bremsstrahlung radiation equation.
where T is plasma temperature in keV.The values of   and r for Hively reactivity are given in the tables in reference [7].Numerical values of C 1 and reduce mass of the particle (m r c 2 ) of Bosch-Hale reactivity also given in the tables in reference [8].Bremsstrahlung radiation rate equation is written as follows.
After the numerical method is conducted, the next step was to modify the Lawson criteria equation by adding the bremsstrahlung radiation term.The final step involved data analysis using Microsoft Excel to determine the Lawson criteria values with the Hively and Bosch-Hale parameters, both with and without bremsstrahlung radiation.The modified Lawson criterion equation is written as follows.
).Based on J D Lawson's research, the maximum value of energy recovery efficiency η is about 1/3.E fus in the equation above represents the energy produced during the fusion process.

Fusion Reactivity
Fusion reactivity, as a parameter of fusion reactions, represents the probability of a fusion reaction occurring per unit time per unit density of nuclei.The higher the value of fusion reactivity, the greater the number of fusion reactions taking place within the reactor.The results of fusion reactivity calculations using the Hively and Bosch-Hale parameters are presented in Table 1.Based on calculations with the Hively parameter, the fusion reactivity values range from of 1.121 × 10 −22 to 6.750 × 10 −22 m 3 /s, which is significantly different from the Bosch-Hale parameter, where the range is of 1.208 × 10 −22 to 2.340 × 10 −22 m 3 /s

Bremsstrahlung Radiation
The low reactivity values require D-³He fusion to be operated at extremely high plasma temperatures to achieve the ignition stage, where the fusion reaction produces more energy than is used, and external heating is not required [10].The bremsstrahlung radiation varies for each fusion fuel due to the influence of the effective charge of the nucleus (Zeff).In this study, an effective charge of 1.67 is used, assuming a 50:50 ratio of deuterium and helium-3.Table 2 shows that the fusion energy yield is greater than bremsstrahlung radiation, indicating that the D-³He fusion reaction at temperatures of 75 to 150 keV produces advantageous energy.Fusion energy production rate ranges from 5.124 × 10 −19 to 30.881 × 10 −19 keV for the Hively parameter and from 5.528 × 10 −19 to 10.705 × 10 −19 keV for the Bosch-Hale parameter.Bremsstrahlung radiation falls within the range of 4.819 × 10 −20 to 6.802 × 10 −20 keV.

Lawson Criterion Analysis
The Lawson criteria serve as the performance standard for sustained fusion reactions.The temperature, confinement time, and plasma density required in a fusion system to generate energy exceeding all losses due to what is also referred to as the triple product of the Lawson criteria.These losses are caused by energy lost due to plasma heating and plasma radiation (bremsstrahlung and synchrotron) [11].The value of the Lawson criteria is influenced by the rate of energy produced in fusion, the sigmavi parameter, and in this study, the influence of bremsstrahlung radiation is also added.Differences in parameters in fusion reactivity, whether influenced by bremsstrahlung radiation or not, result in variations in the Lawson criteria value.
Table 3 presents the calculated Lawson criteria results for each temperature.The advantageous energy obtained from the D-³He fusion reaction can be sustained at temperatures between 75 and 150 keV, with the product of plasma density and confinement time falling within the range of values from 0.437 × 10 21 to 1.347 × 10 21 s/m 3 .The Lawson criteria without bremsstrahlung radiation have lower values compared to with radiation.This indicates that the radiation increases the confinement time, and plasma density required to reach the ignition stage [12].The Lawson criterion values are around × 10 15 s/cm 3 , this aligns well with the calculations from reference [13].There are several factors that could potentially lead to a decrease in the Lawson criteria, including plasma energy loss, impurities, instability in reactor mechanisms, and other factors.Lawson's analysis assumed an ideal experiment without thermal-conduction losses [15].In this study, the Hively and Bosch-Hale parameters, for each incremental temperature increase, respectively, exhibit an average difference of approximately 6.3% and 1.4 %.Based on the difference in values and the curve shape in figure 2 and 3, the Bosch-Hale parameter had more stability than Hively parameter to calculate the Lawson criteria to achieve the ignition stage.Conditions where the fusion self-heating power is high enough to overcome the plasma cooling (energy loss) and creating a positive thermodynamic feedback loop with increasing temperature fusion ignition [16].

Conclusion
The Lawson criteria equation for the D-³He fusion reaction has been solved using the fusion energy production rate equation and the energy for plasma heating equation.The fusion reactivity with the Hively and Bosch-Hale parameters used in the calculations has significantly different values.According to the research, advantageous energy can be sustained at temperatures between 75 and 150 keV if the Lawson criteria fall within the range of 0.437 × 10 21 to 1.261 × 10 21 s/m 3 without bremsstrahlung radiation and 0.446 × 10 21 to 1.347 × 10 21 s/m 3 with radiation.Bremsstrahlung radiation does not significantly impact Lawson criteria calculations.As a result, the Lawson criteria values between with and without bresstrahlung radiation are not significantly different.

Figure 1 .
Figure 1.Fusion Reactivity with Hively and Bosch-Hale Parameter

Figures 2 and 3
Figures 2 and 3 illustrate the difference in Lawson criteria between the sharp and unstable Hively curve and the parabolic Bosch-Hale curve.The Lawson criteria values with the Hively parameter are greater than the Bosch-Hale parameter for temperatures ranging from 75 to 95 keV.Conversely, at temperatures between 100 and 150 keV, the Lawson criteria values with the Bosch-Hale parameter are higher due to the influence of the R-matrix theory for more accurate calculations.Based on Table3, the product of confinement time and plasma density with the Hively parameter falls within the range of 0.437 × 10 21 to 1.452 × 10 21 s/m 3 , and it ranges from 1.145 × 10 21 to 1.347 × 10 21 s/m 3 for the Bosch-Hale parameter at temperatures ranging from 75 to 150 keV.

Figure 2 .Figure 3 .
Figure 2. Lawson Criterion with and without Bremsstrahlung Radiation using Hively Parameter

Table 1 .
D-³He Fusion Reactivity with Hively and Bosch-Hale Parameters

Table 2 .
Bremsstrahlung Radiation and Fusion Energy Produce Rate

Table 3 .
Lawson Criterion of D-³He Fusion Reaction