The Effect of Plasma Profiles on the Critical Value of $n\uptau_{E}$ for Ignition

Abstract

The effect of plasma profiles for ignition condition in a stationary D–T plasma is investigated using the energy conservation equations for ions and electrons, assuming that steady state fusion power is produced with no external power. The alpha power heating is sufficiently large to sustain the plasma and to balance the combined Bremsstarhlung and thermal conduction losses. The space dependent Lawson criteria is derived and critical condition is identified. As a result of this analysis we have shown that the optimum temperature might be $\bar{T} \approx 26\,{\text{keV}}$ and that the peaked profiles with $n\sim\left( {1 - \frac{{r^{2} }}{{a^{2} }}} \right)^{{v_{n} }}$, ν _n  = 1, and $T\sim\left( {1 - \frac{{r^{2} }}{{a^{2} }}} \right)^{{v_{T} }} ,\,v_{T} = 2$ are good to minimizing $\bar{n}\uptau_{E}$ for ignition. The results for these profiles show the critical value of $(\bar{n}\uptau_{E} )_{min} = 0.08 \times 10^{20 } \,{\text{m}}^{ - 3} \,{\text{s}}$ showing the reduction by 1/3 from the reference value limit ν _n  = ν _T  = 0. For a 26 keV plasma with an energy confinement time of 1 s, a pressure of about 6.24 atm is required for the plasma to be ignited; that is, it is sustained purely by the self-heating of the fusion alpha particles.