Decay branching ratios of excited ${}^{24}\mathrm{Mg}$

The nuclear reactions ${}^{12}\mathrm{C}({}^{12}\mathrm{C},\alpha ){}^{20}\mathrm{Ne}$, ${}^{12}\mathrm{C}({}^{12}\mathrm{C},p){}^{23}\mathrm{Na}$, and ${}^{12}\mathrm{C}({}^{12}\mathrm{C},n){}^{23}\mathrm{Mg}$ are the primary reactions in carbon burning, which occurs as part of several stellar processes. The Gamow window, which describes the energy range where most of these reactions take place, is typically around 1.5 MeV in the center-of-mass frame. Direct measurements of the cross sections at this energy are difficult due to the large Coulomb barrier present between the carbon nuclei; however, a successful surrogate measurement can provide the branching ratios between these reactions while avoiding the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ Coulomb barrier. An experiment was performed using inelastic scattering of 40 MeV $\alpha$ particles on ${}^{24}\mathrm{Mg}$ as a possible surrogate for the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ compound nucleus.

Figure 1

Experimental setup showing the $\mathrm{\Delta }E$ and $E$ silicon detectors and target within the STARS chamber and six “Clover” HPGe detectors surrounding the chamber.

Figure 2

$\mathrm{\Delta }E$ Energy loss rate vs particle energy for particle identification. Here the energy deposited in the $\mathrm{\Delta }E$ silicon detector is divided by the path length of the particle through the detector. Since the $\mathrm{\Delta }E$ is a thin detector, this places a particular particle on the same curve for all measured angles. The labels $p$ and $d$ refer to protons and deuterons. A faint band from tritons is present just above the band for deuterons. The hook in the proton band at 15 MeV is from higher-energy protons passing through both the $\mathrm{\Delta }E$ and $E$ detectors, thus not depositing their full energy.

Figure 3

The spectrum of excitation in ${}^{24}\mathrm{Mg}$ corresponding to detected $\alpha$ particles, integrated over all measured scattering angles. The contribution from ${}^{12}\mathrm{C}$ present in the target is also shown.

Figure 4

The excitation of the ${}^{23}\mathrm{Na}$ daughter as a function of the ${}^{24}\mathrm{Mg}$ excitation found with the scattered $\alpha$ particle. A proton $\mathrm{\Delta }E\text{−}E$ particle identification was required for the ejected proton.

Figure 5

The excitation of the ${}^{20}\mathrm{Ne}$ daughter as a function of the ${}^{24}\mathrm{Mg}$ excitation found with the scattered $\alpha$ particle, assuming the ejected particle is an $\alpha$ particle. States in ${}^{20}\mathrm{Ne}$ can be seen as horizontal bands. Events producing ${}^{20}\mathrm{Ne}$ in the ground and first-excited states were identified in this way.

Figure 6

The excitation of the ${}^{23}\mathrm{Na}$ daughter as a function of the ${}^{24}\mathrm{Mg}$ excitation found with the scattered $\alpha$ particle. No $\mathrm{\Delta }E\text{−}E$ particle ID was used for the ejected proton. This plot was only used for the range of ${}^{24}\mathrm{Mg}$ excitation that would lead to a proton with too little energy to pass through the $\mathrm{\Delta }E$ detector. The $\alpha$ channels leading to ${}^{20}\mathrm{Ne}$ are visible as slightly sloped horizontal bands.

Figure 7

The $\gamma$ spectrum produced in coincidence with a scattered $\alpha$ particle. The first spectrum is for events with a ${}^{24}\mathrm{Mg}$ excitation energy over 12 MeV as indicated by the scattered $\alpha$ particle. The peak at 440 keV has been expanded to show the separation of the 440 keV $\gamma$ ray belonging to ${}^{23}\mathrm{Na}$ and the 451 keV $\gamma$ ray belonging to ${}^{23}\mathrm{Mg}$. The second spectrum is for events corresponding to a ${}^{24}\mathrm{Mg}$ excitation energy below 12 MeV. This division gives an approximate split for $\gamma$ rays originating from the excited ${}^{24}\mathrm{Mg}$ nucleus and those for the daughter nuclei ${}^{20}\mathrm{Ne}$, ${}^{23}\mathrm{Na}$, and ${}^{23}\mathrm{Mg}$. The primary contaminants on the target were carbon and oxygen, of which the 4438 keV $\gamma$ ray from carbon can be seen. A Doppler correction based on the recoiling ${}^{24}\mathrm{Mg}$ nucleus has been applied. This is also the approximate correction needed for the daughter nuclei. Due to the difference in mass, the 4438 keV peak from ${}^{12}\mathrm{C}$ retains some Doppler broadening. Mg, $\alpha$, $p$, and $n$ refer to ${}^{24}\mathrm{Mg}$, ${}^{20}\mathrm{Ne}$, ${}^{23}\mathrm{Na}$, and ${}^{23}\mathrm{Mg}$, with numbers indicating the excited state.

Figure 8

The $\gamma$-ray energies vs the excitation of ${}^{24}\mathrm{Mg}$ as calculated from the coincident scattered $\alpha$ particle. The 1369 keV $\gamma$ ray from the first-excited state of ${}^{24}\mathrm{Mg}$ can bee seen strongly at lower excitation energies as it is fed by many of the higher excited states. At around 13 to 14 MeV excitation it can be seen to diminish in favor of the 440 keV $\gamma$ ray from ${}^{23}\mathrm{Na}$ and the 1634 keV $\gamma$ ray from ${}^{20}\mathrm{Ne}$, showing the opening of the particle decay channels at higher excitation energies. Counts shown in coincidence with elastic scattering (zero on the vertical axis) give a visual estimate of the accidental coincidence $\gamma$-ray spectrum. This spectrum would be scaled according to Fig. 3 to estimate the contribution at higher excitation energies.

Figure 9

Branching ratios for $\alpha$-particle-emitting channels. The bottom plot is the combined branching ratio to excited states of ${}^{20}\mathrm{Ne}$ that decay by $\alpha$-particle emission, resulting in ${}^{16}\mathrm{O}$ as the final product.

Figure 10

Total $\alpha$ branch (ground plus all excited states of ${}^{20}\mathrm{Ne}$). This includes excited states of ${}^{20}\mathrm{Ne}$ which decay by $\alpha$-particle emission. The reference data is from Refs. [7, 9, 11].

Figure 11

Branching ratios for proton-emitting channels. The bottom plot is for the proton branch to the fifth excited state of ${}^{23}\mathrm{Na}$. $\gamma$ rays from the fourth excited state were not seen. Comparison data for this plot is for the fourth and fifth excited states combined, which were not resolved.

Figure 12

Total proton branching ratio (ground plus all excited states of ${}^{23}\mathrm{Na}$).

Figure 13

Neutron branching ratio populating the first-excited state of ${}^{23}\mathrm{Mg}$. Higher excitation states were not measured, and $\gamma$ cascading from these levels may be included at higher excitation energies. The total neutron branch found by using the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}\to n+{}^{23}\mathrm{Mg}$ cross section measurements of Bucher et al. [20] divided by the ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ fusion cross section adopted by Caughlan and Fowler [21] is shown for comparison. This is not a direct comparison, since the current measurement does not include the branch to the ground state and only includes the portion of higher excited states which populate the first-excited state in ${}^{23}\mathrm{Mg}$. The reference data thus serves as an upper limit for the current measurement.

Figure 14

The sum of the measured $\gamma$, $\alpha$, proton, and neutron channels. Note that, in the energy range applicable to carbon burning (14 to 17 MeV), the sum is nearly one. The dip at 3 MeV is due to ${}^{24}\mathrm{Mg}$ not having an excited state between 2.5 and 3.5 MeV. Additionally, many states in ${}^{24}\mathrm{Mg}$ above 5 MeV have $\gamma$-decay branches which do not emit either a 1369 keV or a 4238 keV $\gamma$ ray, resulting in an underestimate of the total $\gamma$-decay branch between 5 MeV and the onset of the particle decay branches.

Figure 15

Total branching ratios for the $\alpha$, proton, and neutron channels. The ratios are normalized by setting the total of all branches shown in Fig. 14 to one for all energies. No ground-state contribution was assumed for the neutron channel. For equivalent ${}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ center-of-mass energies, subtract 13.934 MeV from the ${}^{24}\mathrm{Mg}$ excitation energy.