Enhancement of fusion reactivities using non-Maxwellian energy distributions

We discuss conditions for the enhancement of fusion reactivities arising from different choices of energy distribution functions for the reactants. The key element for potential gains in fusion reactivity is identified in the functional dependence of the tunneling coefficient on the energy, ensuring the existence of a finite range of temperatures for which reactivity of fusion processes is boosted with respect to the Maxwellian case. This is shown using a convenient parametrization of the tunneling coefficient dependence on the energy, analytically in the simplified case of a bimodal Maxwell-Boltzmann distribution, and numerically for kappa distributions. We then consider tunneling potentials progressively better approximating fusion processes and evaluate in each case the average reactivity in the case of kappa distributions.

Figure 1

Probability densities for non-Maxwell-Boltzmann energy distributions and for the corresponding MB energy distributions with the same average energy versus $E$ (in arbitrary units). Three cases of bMB distributions are shown in panel (a), with inverse temperatures ${\beta }_1=1$ and ${\beta }_2={10}^{-1}$ (in arbitrary units) for different weights of $x_1=0.25$, with long dashed red (light gray) line, $x_1=0.50$ (dotted black line), $x_1=0.75$ (double-dot-dashed blue line). The corresponding MB distributions with the same average energy and therefore inverse temperature ${\beta }_0$ [derived from Eq. (3)] are also shown $x_1=0.25$ with continuous red (light gray) line, $x_1=0.5$ (short dashed black line), $x_1=0.75$ (blue dot-dashed line). The case of $\kappa$ distributions is shown in panel (b) for different values of $\kappa =0.6,1,2,20$ and constant $\eta =-1/2$. For sufficiently large $\kappa$, the distribution approaches the MB distribution. A common feature of the two non-Maxwellian distributions with respect to the corresponding MB distribution is the presence of regions with higher probability densities at both low and high energy, with a region in between for which MB instead has higher probability densities.

Figure 2

Parameter $\delta R_{\mathrm{bMB}}/R_{\mathrm{MB}}$ versus the $\alpha$ exponent of the tunneling coefficient assumed in Eq. (15). Four cases of bMB with $x_1=0.2,0.4,0.6,0.8$, all with ${\beta }_1=1, {\beta }_2={10}^{-1}$, are depicted and (a) ${\beta }_0{E}_0={10}^{-2}$, (b) ${\beta }_0{E}_0={10}^2$.

Figure 3

Parameter $\delta R_{\kappa }/R_{\mathrm{MB}}$ versus the $\alpha$ exponent of the tunneling coefficient. Four cases are shown with $\kappa =1,2,5,10$ and (a) $\beta E_0={10}^{-2}$, (b) $\beta E_0={10}^2$. This plot differs from the one of bMB distributions because there is a gain in using $\kappa$ distributions also at very low $\alpha$ and because $\delta R_{\kappa }=0$ at two values of $\alpha$: at $\alpha =0.5$ independently of $\kappa$, and at a value of $\alpha$ progressively increasing with decreasing $\kappa$.

Figure 4

(a) Tunneling coefficient versus the energy of the incident particle for a stepwise double barrier potential, with different cases of positional spreading of a Gaussian state, $\xi =5$ (green dashed line), $\xi =20$ (red dot-dashed line), and the case of a plane wave (black continuous line). The parameters of the potential are, with the notation used in Eq. (30), $c=0.734, b=-c, d=c+5, a=-d, V_0=5.4, V_1=-40.33$. (b) Same for the case of a GWS potential, with $\lambda =1.67, L=0.5, U_0=25, W_0=56$, all lengths being expressed in fm and all energies in MeV. The reduced mass is $m=1u$.

Figure 5

Reactivity dependence on temperature for different states and energy distributions. In the top panels we show the cases of a stepwise double barrier potential, with (a) $\xi =20$ fm and (b) $\xi =5$ fm. In the bottom panels the corresponding cases of the GWS potential are shown for (c) $\xi =20$ fm and (d) $\xi =5$ fm, respectively. The various cases in each plot are relative to the choice of $\kappa$-distributed energies, with $\kappa =0.6,1,2,20$, and $\eta =-1/2$, and the corresponding MB distribution.

Figure 6

Reactivity gain in using $\kappa$-distributed energies. The ratio of reactivities with the most favorable case of $\kappa =0.6$ and the case of an MB distribution is plotted for the two positional spreadings $\xi =5$ fm and $\xi =20$ fm, for (a) the stepwise potential and (b) the GWS potential, the same parameters as in Fig. 4.

Figure 7

Reactivity gain in using $\kappa$-distributed energies for deuterium-deuterium (continuous lines) and deuterium-tritium (dashed lines) plasmas. The ratio between reactivities with various $\kappa$ distributions ($\kappa =1.6,2,10$, with the $\kappa =1.6$ case very close to the threshold value having $\eta =-3/2$ in the three-dimensional case) and the case of an MB distribution is plotted (a) versus temperature in a range of interest for the current experiments with Tokamak machines, and (b) versus the $\kappa$ parameter for the three cases of temperatures of 5, 10, and 20 keV. The reactivities have been computed by using the experimental cross sections parametrized as in Ref. [19] by integrating the reactivity over energy intervals in which the interpolation is validated, 0.5–550 keV for the deuterium-tritium reaction, 0.5–5000 keV for $\mathrm{D}(d,p)\mathrm{T}$, 0.5–4900 keV for $\mathrm{D}(d,n){}^3\mathrm{He}$ (see Table IV in Ref. [19]).