Capture reactions into Borromean two-proton systems at $rp$ waiting points

We investigate even-even two-proton Borromean systems at prominent intermediate heavy waiting points for the rapid proton capture process. The most likely single-particle levels are used to calculate three-body energy and structure as a function of proton-core resonance energy. We establish a linear dependence between two- and three-body energies with the same slope, but the absolute value slightly dependent on partial wave structure. Using these relations we estimate low-lying excited states in the isotones following the critical waiting points. The capture rate for producing a Borromean bound state is described based on a full three-body calculation for temperatures about $0.1–10$ GK. In addition, a simple rate expression, depending only on a single resonance state, is found to comply with the three-body calculation for temperatures between $0.1$ and 4 GK. The rate calculations are valid for both direct and sequential capture paths. As a result the relevant path of the radiative capture reactions can be determined. We present results for $E1$ and $E2$ photon emission, and discuss occurrence preferences in general as well as relative sizes of these most likely processes. Finally, we present a method for estimating proton capture rates in the region around the critical waiting points.

Figure 1

The potentials based on the spectra of the lowest three $\lambda$'s for $p_{3/2}$ (dashed red line) and $f_{5/2}$ (dotted blue line) in isolation, along with the combination of both $p_{3/2}$ and $f_{5/2}$ (solid black line). All two-body potentials have been adjusted to produce the energy $E_{2b}=0.1$ MeV. It is seen how the spectrum of the combined case always follows the lowest available potential.

Figure 2

The three-body energies as a function of two-body energies for $p_{1/2}, p_{3/2}$, and $f_{5/2}$ in isolation, and for $p_{1/2}$ and $f_{5/2}$ as well as $p_{3/2}$ and $f_{5/2}$ in combination, the latter both with and without $p-p$ interaction, and with and without a three-body potential. The linear fits are in accordance with Eq. (29). All two-body potentials were adjusted to produce the same energy, when more that one wave is allowed. Only a $0^{+}$ state is considered here. The horizontal and vertical line indicates our Borromean region of interest.

Figure 3

The probability distribution for the $0^{+}$ ground state with equal two-body binding of 0.641 MeV for both $f_{5/2}$ and $p_{3/2}$ partial waves. Projected contour curves are shown at the bottom of each figure. The distance variables correspond to the two different Jacobi sets, where panels (a) and (b) are for the first and second Jacobi sets, respectively.

Figure 4

(a) The potentials based on the spectra of the lowest five $\lambda$'s for the $2^{+}$ state allowing both $p_{3/2}$ and $f_{5/2}$ waves. The dashed, horizontal line is at the lowest three-body resonance energy. The dotted curve is the sum of Coulomb potential and two-body energy. (b) The same for the $1^{-}$ state with $p_{3/2}$ and $d_{5/2}$ waves. (c) The same as for (b) with the $d_{5/2}$ two-body energy equal to $1.5$ MeV.

Figure 5

The three-body energy as a function of two-body energy for the $0^{+}$ and $2^{+}$ state with $p_{3/2}$ and $f_{5/2}$ waves, as well as for the $0^{+}$ and $1^{-}$ state with $p_{3/2}$ and $d_{5/2}$ waves. The horizontal line is included to guide the eye.

Figure 6

The cross sections for $2^{+}\to 0^{+}$ with both $p_{3/2}$ and $f_{5/2}$ waves (solid, black curve), and only with $p_{3/2}$ waves (dashed, blue curve). Also included are the cross section for $1^{-}\to 0^{+}$ with $p_{3/2}$ and $d_{5/2}$. For the red dotted curve $E_{d_{5/2}}=0.641$ Mev, and $E_{d_{5/2}}=1.50$ MeV for the green dash-dotted curve. In both cases the $\mathcal{E}1$ transition has been scaled down by ${10}^3$ to make the figure readable. The photodissociation energy is related to the total energy $E$ and the binding energy $B$ by $E_{\gamma }=E+\left|B\right|$.

Figure 7

The reaction rates corresponding to the cross sections from Fig. 6. The result of the full calculations is given by the solid curves. The dashed curves in the same color are the result of applying Eq. (24) to the lowest peak in Fig. 6. The dotted curves are the sum of contributions for the three (two) lowest resonances for $p_{3/2}+f_{5/2}\left(p_{3/2}\right)$ using Eq. (24). The $\mathcal{E}1$ rates have been scaled down by ${10}^4$ to make the figure readable. The various resonance widths for the lowest resonances can be found in Table 3.

Figure 8

The square of the angular wave function $\mathrm{\Phi }$ from Eq. (12), multiplied by the phase factor ${cos}^2\left(\alpha \right){sin}^2\left(\alpha \right)$ and integrated over ${\mathrm{\Omega }}_x$ and ${\mathrm{\Omega }}_y$, as a function of $\rho$ and $\alpha$ for the lowest $\lambda$ of the $2^{+}$ state in the second set of Jacobi coordinates.

Figure 9

Estimates of the reaction rate for $2^{+}\to 0^{+}$ with $p_{3/2}$ and $f_{5/2}$ based only on two-body energies. The blue curve at $E_{2b}=0.74$ MeV corresponds to the upper limit established in Sec. 5b. The low temperature region has been scaled up to demonstrate the similarities.