Model-independent determination of the astrophysical $S$ factor in laser-induced fusion plasmas

In this work, we present a new and general method for measuring the astrophysical $S$ factor of nuclear reactions in laser-induced plasmas and we apply it to ${}^2\mathrm{H}(d,n){}^3\mathrm{He}$. The experiment was performed with the Texas Petawatt Laser, which delivered 150–270 fs pulses of energy ranging from 90 to 180 J to ${\mathrm{D}}_2$ or ${\mathrm{CD}}_4$ molecular clusters (where D denotes ${}^2\mathrm{H})$. After removing the background noise, we used the measured time-of-flight data of energetic deuterium ions to obtain their energy distribution. We derive the $S$ factor using the measured energy distribution of the ions, the measured volume of the fusion plasma, and the measured fusion yields. This method is model independent in the sense that no assumption on the state of the system is required, but it requires an accurate measurement of the ion energy distribution, especially at high energies, and of the relevant fusion yields. In the ${}^2\mathrm{H}(d,n){}^3\mathrm{He}$ and ${}^3\mathrm{He}(d,p){}^4\mathrm{He}$ cases discussed here, it is very important to apply the background subtraction for the energetic ions and to measure the fusion yields with high precision. While the available data on both ion distribution and fusion yields allow us to determine with good precision the $S$ factor in the $d+d$ case (lower Gamow energies), for the $d+{}^3\mathrm{He}$ case the data are not precise enough to obtain the $S$ factor using this method. Our results agree with other experiments within the experimental error, even though smaller values of the $S$ factor were obtained. This might be due to the plasma environment differing from the beam target conditions in a conventional accelerator experiment.

Figure 1

The Faraday cup signal ($\mathrm{\Delta }V$) versus the time of flight of the deuterium ions recorded by the oscilloscope for one experiment of the campaign. The first steep peak whose tail extends to hundreds of ns overlapping with the second peak is due to the x rays produced by the interaction of the laser inside the vacuum target chamber. The second small peak ($\lesssim {10}^{-6}$ s) is associated with energetic deuterium ions produced in the Coulomb explosion. The big wide double-featured peak at 3–20 $\mu \mathrm{s}$ is due to slower sub-keV ions resulting from the blast wave propagation in the surrounding and cold cluster gas [20, 27]

Figure 2

Deuterium ion energy distribution as recorded in the FC (in red) and after background subtraction (black). The high energy tail is due to the x-ray noise from laser-cluster interaction.

Figure 3

The energy moments for $n=0$ (black), $n=1$ (red), $n=2$ (green), and $n=3$ (blue line) of the deuterium ion distribution extracted from the FC signal recorded in one experiment. Moments analysis proves to be a tool to separate the electromagnetic noise from detectable signal, approximately showing the location where the slope of the curve changes.

Figure 4

$\mathrm{\Sigma }\left(E\right)$ for the nuclear reactions (1) and (3) is plotted versus the center-of-mass energy of the fusion nuclei. The area under the curves gives the inverse of the $S$ factor. The $d+d$ BB (in red) and $d+{}^2\mathrm{H}$ BT (in blue) contributions are plotted together with their sum (binned, black) and the $d+{}^3\mathrm{He}$ one (green). The latter are also plotted without applying any background cut (thick dashed grey and cyan lines). The maximum of this quantity locates the effective Gamow peak energy. The solid (red) up and (blue) down triangles respectively represent the BB and BT contributions of (1), the solid (black) circle is their sum, and the solid (green) square is used for the reaction (3).

Figure 5

The variation of the yields $\mathrm{\Delta }Y$ as defined in Eq. (20) is plotted for each shot. The proton yields (red solid triangles) are always more affected by the choice of the energy cut than the neutron ones (black solid circles).

Figure 6

The astrophysical factor $S\left(E\right)$ for the nuclear reaction (3), obtained using the method proposed in this work (solid black circles) [23]. Other “conventional” experiments [7, 8, 9, 10] are shown for comparison.

Figure 7

The astrophysical factor $S\left(E\right)$ for the nuclear reaction (1) after the background subtraction (solid black circles), where the effective Gamow peak energy is used as $E_{c.m.}$. “Conventional” experiments [1, 2, 3, 4, 5, 6, 7] are included for comparison.

Figure 8

The astrophysical factor $S\left(E\right)$ averaged over different shots for the nuclear reaction (3), obtained using the method proposed in this work (solid black circles) Other “conventional” experiments [7, 8, 9, 10] are shown for comparison.

Figure 9

The astrophysical factor $S\left(E\right)$ averaged over different shots for the nuclear reaction (1) after the background subtraction (solid black circles), where the effective Gamow peak energy is used in place of $E_{c.m.}$. “Conventional” experiments [1, 2, 3, 4, 5, 6, 7] are included for comparison.