Evaluation of the Lawson criterion for aneutronic proton-boron-11 fusion: effects of ion temperature and bremsstrahlung losses

Nuclear fusion, the process of combining light nuclei to form heavier nuclei, offers a promising pathway to sustainable clean energy with minimal radioactive waste. The Lawson criterion, expressed as the product of plasma density, confinement time, and temperature, establishes the conditions required for ignition and net energy gain. This study investigates the Lawson criterion for proton-boron-11 (p- 11 B) fusion across ion temperatures of 75–500 keV, incorporating fusion reactivity data from Tentori-Belloni and Nevins-Swain, as well as energy losses from bremsstrahlung radiation under different electron-to-ion temperature ratios ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> = 1, 0.5, 0.25). The Tentori-Belloni dataset yields higher fusion reactivity than Nevins-Swain, resulting in more favorable Lawson values. Net energy production is achieved only when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> </mml:math> &amp;lt; <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> , with optimal operating windows identified at 190–330 keV for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> </mml:math> = 0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> and 125–500 keV for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m6"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> </mml:math> = 0.25 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m7"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> . At <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m8"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> &amp;lt; 230 keV , the Lawson criterion decreases due to plasma instabilities and confinement limitations; in this work, radiative losses are evaluated using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m9"> <mml:mrow> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow> <mml:mi>e</mml:mi> <mml:mi>f</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>2.4</mml:mn> </mml:mrow> </mml:math> derived from the p- 11 B fuel mixture ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m10"> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>B</mml:mi> </mml:msub> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mn>90</mml:mn> <mml:mo>:</mml:mo> <mml:mn>10</mml:mn> </mml:mrow> </mml:math> ) only, while external impurity contributions are not explicitly modeled. For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m11"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> &amp;gt; 230 keV , the Lawson criterion increases, reaching characteristic minima around 330 keV and 500 keV. These thresholds represent the minimum conditions required to achieve ignition and sustain a self-sufficient fusion reaction. The minimum Lawson values obtained were 1.3 × 10 22 m −3 s (no radiation), 1.2 × 10 23 m −3 s ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m12"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> </mml:math> = 0.5 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m13"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mrow> </mml:math> ), and 1.5 × 10 22 m −3 s ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m14"> <mml:mrow> <mml:msub> <mml:mi>T</mml:mi>