Elastic scattering of ${}^3\mathrm{He}+{}^4\mathrm{He}$ with the SONIK scattering chamber

Measurements of the elastic scattering cross section of ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ are important in order to improve constraints on theoretical models of ${}^4\mathrm{He}({}^3\mathrm{He},\gamma ){}^7\mathrm{Be}$, a key reaction in Big Bang nucleosynthesis and solar neutrino production. The astrophysical $S$ factor for this reaction is a significant source of uncertainty in the standard-solar-model prediction of the ${}^7\mathrm{Be}$ and ${}^8\mathrm{B}$ solar neutrino fluxes. The elastic scattering measurements reported in the literature do not extend to low energies and lack proper uncertainty quantification. A new measurement of the ${}^4\mathrm{He}({}^3\mathrm{He},{}^3\mathrm{He}){}^4\mathrm{He}$ reaction has been made at center-of-mass energies $E_{\text{c.m.}}=0.38-3.13$ MeV using the Scattering of Nuclei in Inverse Kinematics (SONIK) scattering chamber: a windowless, extended gas target surrounded by an array of 30 collimated silicon charged particle detectors situated at TRIUMF. This is the first elastic scattering measurement of ${}^3\mathrm{He}+{}^4\mathrm{He}$ made below 500 keV and it has greater angular range and better precision than previous measurements. The elastic scattering data were analyzed using both $R$-matrix and halo effective field theory frameworks, and values of the $s$-wave scattering length and effective range were extracted. The resulting improvement in knowledge of the $s$-wave effective-range function at low energies reduces the overall uncertainty in $S_{34}$ at solar energies.

Figure 1

(a) Three-dimensional model of SONIK. (b) SONIK design details. The ${}^3\mathrm{He}$ beam traverses the ${}^4\mathrm{He}$ gas target from left to right in the figure. The stars represent the interaction regions in the gas target. (c) A detector telescope assembly. The dimensions in the figures are in millimeters.

Figure 2

Typical spectrum from the experiment. The red and blue histograms represent the spectra obtained at ${\theta }_{\text{lab}}={40}^{\circ }$ for $E\left[{}^3\mathrm{He}\right]=1.303$ and 3.608 MeV, respectively. The ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$ peaks are resolved at the higher incident energy but not at the lower beam energy.

Figure 3

Determination of stopping power for $E\left[{}^3\mathrm{He}\right]=1.767$ MeV. The error bars are entirely due to the systematic error associated with the constant $C_{\text{mag}}$ in Eq. (2).

Figure 4

A detector telescope of SONIK.

Figure 5

The $G$ factor as a function of kinetic energy of the scattered particle. The blue circles and green squares are the computed $G$ factor for each detector at each interaction point in SONIK for each incident beam energy from the geant4 simulation for ${}^3\mathrm{He}$ and ${}^4\mathrm{He}$, respectively. The red dashed line is the $G$ factor calculated using Eq. (6). The dotted line is explained in the text.

Figure 6

The $G$ factor as a function of scattering angle in laboratory frame of reference for $E\left[{}^3\mathrm{He}\right]=721$ keV. The solid (open) points represent the $G$ factor obtained from a geant4 simulation with (without) introducing the multiple scattering model in the simulation. The red dashed line is the $G$ factor calculated using Eq. (6).

Figure 7

The ratio of the experimental differential scattering cross section to the cross section calculated using $R$-matrix parameters from Ref. [10] at beam energies of 5.490, 4.347, and 3.608 MeV. The interaction region III measurements of this work, represented by the blue points, are, in general, consistent with previous determinations but are more precise. Only interaction region III data are shown for these comparisons.

Figure 8

The ratio of the experimental differential scattering cross section to the cross section calculated using $R$-matrix parameters from Ref. [10] at beam energies of 2.633 and 1.767 MeV. The discrepancies between the interaction region III measurements of this work, represented by the blue points, and previous data (Mohr et al. [23] and Chuang [20]) are discussed in the text.

Figure 9

The levels and separation energies introduced in the $R$-matrix fit were taken from Ref. [38]. The energy range covered in the present measurement is represented by the curly brace. All energies are in MeV.

Figure 10

Differential elastic scattering cross sections, relative to Rutherford's prediction, as measured with SONIK. Results are obtained from an MCMC analysis of SONIK and Barnard data. Bands encompass the 16th to 84th percentile of the inferred probability distributions. Blue bands correspond to the $R$-matrix analysis, and green bands correspond to the halo EFT analysis. Red circles with error bars indicate ${}^3\mathrm{He}$ peaks. Purple squares with error bars indicate ${}^4\mathrm{He}$ peaks. In some cases, the size of the marker representing the data point is greater than the error bar. The three panels along a row for a given $E\left[{}^3\mathrm{He}\right]$ beam energy are from interaction regions I, II, and III, respectively.

Figure 11

Differential elastic scattering cross sections, relative to Rutherford's prediction, as measured with SONIK. Colors and symbols are as described in Fig. 10. The additional yellow band in the bottom row corresponds to the second run at $E\left[{}^3\mathrm{He}\right]=2.633$ MeV discussed in Sec. 3c.

Figure 12

Differential elastic scattering cross sections, relative to Rutherford's prediction, as measured with SONIK. Colors and symbols are as described in Fig. 10.

Figure 13

Differential elastic scattering cross sections, relative to Rutherford's prediction, as reported in Ref. [17]. Results are obtained from an MCMC analysis of SONIK and Barnard data. Bands are as in Fig. 10 and grey circles represent the data from Ref. [17]. Total ${\chi }^2$ at maximum posterior probability for the $R$-matrix fit is 1098.56, resulting in ${\chi }^2/\mathrm{datum}=1.70$. The total ${\chi }^2$ for the EFT fit to these data is 1996.24.

Figure 14

Posterior distributions of the $R$-matrix parameters sampled in this analysis.

Figure 15

Scattering length and effective range posterior probability distributions from each of the seven $R$-matrix analyses and the EFT analysis.

Figure 16

Normalization factor results at each SONIK data point are shown as a product of the three different systematic effects described in Sec. 5. The results are shown in increasing energy (from left to right) as a function of data point index. Each vertical, shaded region corresponds to a different energy bin. $R$-matrix results are shown as blue circles with error bars. EFT results are shown as green squares with error bars.

Figure 17

Top: The scattering phase shifts for $\ell =0$ are shown in comparison to the analyses in Ref. [22] and Ref. [18]. The solid blue line represents the median calculated in the SB $R$-matrix analysis. The dashed green line represents the median calculated in the EFT analysis. The red shaded region indicates the energy range over which the SONIK measurements were carried out. White squares with error bars and grey x's indicate the analyses of Refs. [22] and [18], respectively. Bottom: The effective range function, $K\left(E\right)$, is plotted as a function of the center-of-mass energy.