Search for ²²Na in novae supported by a novel method for measuring femtosecond nuclear lifetimes

Abstract

Classical novae are thermonuclear explosions in stellar binary systems, and important sources of ²⁶Al and ²²Na. While γ rays from the decay of the former radioisotope have been observed throughout the Galaxy, ²²Na remains untraceable. Its half-life (2.6 yr) would allow the observation of its 1.275 MeV γ-ray line from a cosmic source. However, the prediction of such an observation requires good knowledge of its nucleosynthesis. The ²²Na(p, γ)²³Mg reaction remains the only source of large uncertainty about the amount of ²²Na ejected. Its rate is dominated by a single resonance on the short-lived state at 7785.0(7) keV in ²³Mg. Here, we propose a combined analysis of particle-particle correlations and velocity-difference profiles to measure femtosecond nuclear lifetimes. The application of this method to the study of the ²³Mg states, places strong limits on the amount of ²²Na produced in novae and constrains its detectability with future space-borne observatories.

Introduction

Nuclear reactions between charged particles in astrophysical environments proceed by quantum tunneling. The measurement of these reactions in the laboratory is very difficult because of the very small cross sections (σ ≲ 1 nb). To date, about 10 reactions involving radioactive nuclei and charged particles have been directly measured at low energies. Among them, the ²²Na(p, γ)²³Mg reaction has a direct impact on the amount of radioactive ²²Na produced in novae^1,2,3,4,5. The astrophysical relevance of ²²Na for the diagnosis of nova outbursts was first mentioned by Clayton and Hoyle⁶, in the context of its decay into a short-lived excited state of ²²Ne, and the subsequent emission of a γ-ray photon of 1.275 MeV released during its de-excitation. With the half-life of ²²Na being much longer than the transition time from an optically thick to an optically thin ejecta of a classical nova (about one week), the flux of the 1.275 MeV γ-ray line will remain close to its maximum value for months after the ejecta has become transparent to γ rays⁷. A precise determination of the ²²Na(p, γ)²³Mg thermonuclear rate is necessary to improve the predicted abundances of nuclei in the mass region A ≥ 20 during nova outbursts, including the ²²Na abundance in the ejecta that impacts in turn the predicted ²⁰Ne/²²Ne ratios in presolar grains⁸ of a putative nova origin, as well as, the corresponding γ-ray emission flux. The rate of this reaction is mainly dictated by a single resonance in ²³Mg at 7785.0(7) keV⁵. A direct measurement of its strength (ωγ) has been performed in three different studies with conflicting results: $\omega \gamma=5.{7}_{-0.9}^{+1.6}$⁵, 1.8(7)⁹ and <0.36 meV¹⁰. Indirect experimental methods, such as lifetime measurements, have also been employed to determine the strength of this resonance. Our simulations¹¹ show that, in the range of the debated values of the resonance strength, the mass of ²²Na ejected in novae depends on the lifetime (τ) of this key state approximately as M_ejec ∝ τ^0.7. The lifetime of this state was previously measured to be τ = 10(3) fs⁴, consistent with the recently determined upper limit of 12 fs¹². Yet, this is at odds with the predictions of the nuclear Shell Model (SM), τ_SM ≈ 1 fs.

Here we propose an experimental method for measuring short ≈fs lifetimes. This method has been applied to the key state in ²³Mg in order to obtain an independent measurement and, so, to reduce systematic uncertainties. As a result, the rate of the ²²Na(p, γ)²³Mg reaction is now well known, and realistic estimates of the maximum detectability distance of novae in 1.275 MeV γ rays are derived.

Results

The experiment was performed at GANIL, France. A ²⁴Mg beam was accelerated to 110.8(4) MeV and impinged on a ³He target of ≈2 × 10¹⁷ atoms cm⁻² uniformly implanted up to a depth of 0.1 μm below the surface of a 5.0(5) μm gold foil, producing the ²³Mg nucleus by the ³He(²⁴Mg,⁴He)²³Mg reaction. Both ²⁴Mg and ²³Mg nuclei were then stopped in a 20.0(5) μm gold foil. The two gold foils were mounted back-to-back. About 20 states were populated in ²³Mg with excitation energies between 0 and 8 MeV. The light particles produced by the reaction were identified and measured with the VAMOS++ magnetic spectrometer¹³ placed at 0° with respect to the beam direction, using two drift chambers, a plastic scintillator and two small drift chambers placed at the entrance of the spectrometer. This led to an unambiguous identification of the ⁴He particles. The ⁴He particles were detected up to an angle in the laboratory of 10.0(5)° relative to the beam axis. The excitation energy (E_x), the velocity at the time of the reaction (β_reac = v_reac/c), and the angle of the ²³Mg nuclei (θ_recoil) have been determined by measuring the momentum of the ⁴He ejectiles, with a resolution (FWHM) of ≈500 keV, ≈0.0005 and ≈ 0.05°, respectively.

The excited states in ²³Mg have been also clearly identified via their γ-ray transitions measured with the $\texttt{AGATA}$ γ-ray spectrometer^14,15. The present experimental method has taken advantage of the highest angle sensitivity of $\texttt{AGATA}$^14,15, shown in Fig. 1b. This γ-ray spectrometer is based on γ-ray tracking techniques in highly segmented HPGe detectors. It consisted of 31 crystals covering an angular range from 120° to 170°, with a total geometrical efficiency ≈ 0.6π. With $\texttt{AGATA}$, it is possible to measure accurately the energy and the emission angle of the γ rays with a resolution of 4.4(1) keV at 7 MeV and 0.7(1)°, respectively. The γ rays were observed Doppler shifted since they were emitted from a ²³Mg nucleus moving at a certain velocity β_ems = v_ems/c. The measured energy E_γ is a function of β_ems, of the center-of-mass energy E_γ,0 of the γ ray and of the angle θ between the γ ray and the ²³Mg emitting nucleus: ${E}_{\gamma }={E}_{\gamma,0}\sqrt{1-{\beta }_{{{{{{{{\rm{ems}}}}}}}}}^{2}}/(1-{\beta }_{{{{{{{{\rm{ems}}}}}}}}}\cos (\theta ))$. With $\texttt{AGATA}$, this relativistic Doppler effect can be observed continuously as a function of the angle, as shown in Fig. 1a for the case of the ${E}_{\gamma,0}=4840.{0}_{-0.4}^{+0.2}$ keV (we use energy values from the present measurement throughout this article) γ-ray transition emitted from the E_x = 5292.0(6) keV excited state in ²³Mg moving with β_ems ≈ 0.075. Conversely, if the center-of-mass energy of the γ ray is known, it is possible to determine the nucleus velocity at the time of emission β_ems from the measured E_γ and θ (see the “Methods” subsection “Determination of velocities”). Their precise measurement, enabled by the combination of both $\texttt{AGATA}$ and $\texttt{VAMOS}$++, has allowed for the accurate determination of β_ems, on an event-by-event basis.

Fig. 1: Identification of γ-ray transitions in ²³Mg.

a The energy of the measured γ rays is plotted as a function of the angle between the γ ray and the ²³Mg emitter. This matrix is conditioned with the detection of an α particle at 5.2 < E_x < 5.4 MeV in the VAMOS++ magnetic spectrometer. The γ-ray transition (${E}_{\gamma,0}=4840.{0}_{-0.4}^{+0.2}$ keV) from the E_x = 5292.0(6) keV excited state in ²³Mg is clearly observed. Its energy is Doppler shifted. The background observed here is mostly due to random coincidences between γ rays from the Compton background and α particles produced in fusion-evaporation reactions between the beam and ¹²C and ¹⁶O impurities deposited on the target. b Picture of the $\texttt{AGATA}$ γ-ray spectrometer used to detect the γ rays emitted during the reaction.

In the present work, both velocities β_ems and β_reac were measured simultaneously, on an event-by-event basis. The two velocities are not identical since (i) the ²³Mg nucleus slows down in the target before being stopped (≈500 fs) and (ii) there is a time difference between the reaction and the γ-ray emission due to the finite lifetime of the state. Thus, the profile of the velocity differences, Δβ = β_reac−β_ems, is a function of the lifetime of the state—a longer lifetime gives a larger value of Δβ. The technique proposed here, for measuring femtosecond lifetimes, is based on the analysis of velocity-difference profiles.

The results obtained in this work are shown in Fig. 2 for three different excited states of ²³Mg. These measurements were made using an average beam intensity of 2 × 10⁷ pps for a duration of ~132 h. In Fig. 2a, β_reac is shown as a function of β_ems, and in (b), the yields of the corresponding γ rays as a function of the velocity difference Δβ. The overall shape of the Δβ measured profiles can be explained in a simple way. The γ rays are emitted following the exponential decay law, i.e. N(t) = N₀ × e^−Δt/τ, with τ being the lifetime of the state and Δt the time elapsed after the reaction. For short lifetimes (τ ≲ 100 fs), the deceleration acting on the ²³Mg ions is almost constant (dE/dx ≈ constant) and consequently Δβ ∝ Δt. Therefore, the right side tail of the curves in Fig. 2b would follow the exponential decay function N(t) ∝ e^−Δβ/τ. This explains the asymmetric shape of the profile shown in blue, which corresponds to τ = 40 fs. Moreover, the experiment has a resolution in Δβ, i.e. 0.0032(1), which is measured by the width of the almost Gaussian profile observed for very short lifetimes, see the green profile for τ = 4 fs in Fig. 2b. The overall shape of the measured Δβ distributions is the convolution of the exponential decay with a Gaussian function. The sensitivity of the method, governed by the statistics, can be better than the resolution of Δβ. This makes the method sensitive to very short lifetimes, down to 0.8 fs according to simulations. From comparison of the data with Monte Carlo simulations, shown in Fig. 2b by the continuous lines (see also the Supplementary information), the best agreement is obtained with lifetimes of ${4}_{-3}^{+1}$ and $4{0}_{-7}^{+6}$ fs for the states E_x = 5292.0(6) keV (${E}_{\gamma,0}=4840.{0}_{-0.4}^{+0.2}$ keV) and E_x = 3796.8(12) keV (${E}_{\gamma,0}=3344.{8}_{-1.0}^{+0.8}$ keV), respectively, which is in excellent agreement with their known values of 5(2)¹⁶ and 41(6) fs¹⁶. This shows that the present method is a powerful tool to determine lifetimes in the femtosecond range. With the same method, it is also possible to accurately measure the center-of-mass energy of the γ-ray transitions. For the transition from the key state (E_x = 7785.0(7) keV), we measured ${E}_{\gamma,0}=7333.{0}_{-0.2}^{+0.5}$ keV, in good agreement with the referenced value of E_γ,0 = 7333.2(11) keV¹⁶.

Fig. 2: Angle-integrated velocity-difference profiles.

a The ²³Mg velocity at the time of reaction (β_reac) is shown against the velocity at the time of the γ-ray emission (β_ems), for three excited states. The line corresponds to the prompt γ-ray emission when β_ems = β_reac. The points observed on the left of the line (β_ems < β_reac) correspond to delayed γ-ray emissions. b The corresponding angle-integrated velocity-difference profiles for the three states compared with simulations (continuous lines). It shows unambiguously that the key state (in red) has a lifetime 4 < τ < 40 fs. The red-shaded area corresponds to the simulations with lifetimes within 1σ uncertainty. The horizontal error bars correspond to the width of the bins, which are larger than the real experimental uncertainty, and the vertical error bars to the statistical uncertainty.

The present method has many advantages. Since the excitation energy of the state is selected with $\texttt{VAMOS}$++, we can ignore any possible top-feeding contribution to the state, and, therefore, the measure is not affected by the lifetime of higher-lying states. Moreover, this method is independent of the reaction mechanism populating the states, i.e. direct transfer or compound-nucleus formation (²⁴Mg + ³He → ²⁷Si^* → ²³Mg + ⁴He). Furthermore, the result does not depend on the angular distribution of the emitted γ ray and of the charged particles, nor on the angle-dependent detection efficiency. A single spectrum concentrates all the statistics of the experiment, which maximizes the signal-to-noise ratio. The statistical uncertainty on the lifetime of the key state is ~50%. The main sources of systematic uncertainties are estimated to be 26% and the individual contributions are presented in Table 1.

Table 1 Systematic uncertainties (in %) on the lifetime of the key state

The result obtained for the lifetime of the astrophysical state is $\tau=1{1}_{-5}^{+7}$ fs, including statistical and systematic uncertainties (see Fig. 2), confirming the measured value of Jenkins et al.⁴ (τ = 10(3) fs). The reduced magnetic dipole transition probability B(M1) was deduced from the measured lifetime assuming a negligible E2 electric quadrupole contribution: $B(M1)=0.01{7}_{-0.008}^{+0.011}\,{\mu }_{N}^{2}$. This value is among the lowest values measured for M1 transitions. To get insight from theory, we have performed shell-model calculations in the sd shell with the well-established phenomenological USDA and USDB^17,18 interactions using the $\texttt{NushellX@MSU}$ code¹⁹. These calculations confirm that the key state is characterized by a low B(M1) value of 0.14 ${\mu }_{N}^{2}$, assuming 7/2⁺ spin and parity assignment (see the discussion below) and optimum values of the effective proton and neutron g-factors. Although this value is almost a factor of 10 larger than the experimentally determined one, the difference is not far from the typical rms error on M1 transition probabilities established for the sd shell with these interactions^11,20. Therefore, the measured values of the lifetime are compatible with shell-model calculations.

The measured lifetime was used to determine the strength of the resonance: $\omega \gamma=\frac{2{J}_{{}^{23}{{{{{{{\rm{Mg}}}}}}}}}+1}{(2{J}_{{}^{22}{{{{{{{\rm{Na}}}}}}}}}+1)(2{J}_{{\rm {p}}}+1)}\times \frac{{\Gamma }_{\gamma }{\Gamma }_{{\rm {p}}}}{{\Gamma }_{\gamma }+{\Gamma }_{{\rm {p}}}}=\omega {\rm {B{R}}}_{{\rm {p}}}(1-{\rm {B{R}}}_{{\rm {p}}})\frac{\hslash }{\tau }$, with J and Γ being the spin and the partial widths of the key state. Actually, the spin of this state has also been a subject of debate^4,21, where three values have been proposed: ${J}_{{}^{23}{{{{{{{\rm{Mg}}}}}}}}}^{\pi }=3/{2}^{+}$²¹, 5/2⁺²¹ or 7/2⁺⁴. We assume it to be 7/2⁺. This state is only 18(1.5) keV away from the known 5/2⁺ isobaric analog state of the ²³Al ground state. In the case of the 5/2⁺ assignment, the states would interfere, as predicted by the nuclear shell-model calculations¹¹, and such mixing has not been observed^5,9,22,23. Moreover, concerning the γ-ray decay pattern, shell-model calculations agree with a 7/2⁺ spin¹¹. The proton branching ratio, BR_p, was also measured in this work by detecting protons emitted from the ²³Mg unbound states with an annular silicon detector placed downstream of the target (see Supplementary information). The obtained value, BR_p = 0.68(17)%, is in excellent agreement with the latest published value, BR_p = 0.65(8)%²³.

Combining all measured values into the new recommended values: BR_p = 0.66(7)% and τ = 10.2(26) fs, results in a consolidated resonance strength of $\omega \gamma=0.2{4}_{-0.04}^{+0.11}$ meV, which is compatible with the direct measurement¹⁰ (ωγ < 0.36 meV) but not with the other two direct measurements. It should be pointed out that these direct measurements using radioactive targets are not in agreement with each other, and also that the obtained value is very low, close to the sensitivity limit of these direct experiments. The new value of the resonance strength and a Monte-Carlo approach²⁴ were used to deduce a new rate for the ²²Na(p, γ)²³Mg reaction (see Fig. 4 and Table 1 in Supplementary information). The reliably estimated experimental uncertainties of the present method allowed a more accurate determination of this rate. This new rate was found to be very reliable at maximum nova temperatures, with uncertainties reduced to 40% (10%) at T = 0.1 GK (0.5 GK) (see Supplementary information).

To quantitatively assess the impact of the new ²²Na(p, γ)²³Mg reaction rate obtained from this work on nova nucleosynthesis, a series of hydrodynamic simulations have been performed. Four physical magnitudes determine the strength of a nova outburst, and, in turn, the synthesis of ²²Na: the white dwarf mass M_WD (or radius R_WD), its initial luminosity L_WD,ini, the mass-accretion rate $\dot{M}$, and the metallicity of the accreted material. The influence of the white dwarf mass on the synthesis of ²²Na has been analyzed. Three different values for the white dwarf mass have been considered: 1.15 M_⊙, 1.25 M_⊙, and 1.35 M_⊙. In these simulations, the star hosting the nova outbursts is assumed to be an ONe white dwarf, with initial luminosity, L_WD,ini = 10⁻² L_⊙, and accreting solar composition material²⁵ from the secondary star at a constant mass-accretion rate of $\dot{M}$ = 2 × 10⁻¹⁰ M_⊙/yr. The accreted matter is assumed to mix with material from the outermost white dwarf layers²⁶. For consistency and completeness, the nova outbursts on 1.15 M_⊙ white dwarfs have been computed with two different, one-dimensional stellar evolution codes: $\texttt{MESA}$^{27,28,29,30,31,32} and $\texttt{SHIVA}$^33,34.

At the early stages of the outburst, when the temperature at the base of the envelope reaches T_base ~ 5 × 10⁷ K, the chain of reactions ²⁰Ne(p, γ)²¹Na(β⁺)²¹Ne(p, γ)²²Na powers a rapid rise in the ²²Na abundance. When the temperature reaches T_base ~ 8 × 10⁷ K, ²²Na(p, γ)²³Mg becomes the main destruction channel. At T_base ~ 10⁸ K, ²²Na(p, γ)²³Mg becomes the most important reaction involved in the synthesis and destruction of ²²Na, and therefore, its abundance begins to decrease. When T_base ~ 2 × 10⁸ K, ²¹Na(p, γ)²²Mg becomes faster than ²¹Na(β⁺)²¹Ne, such that ²⁰Ne(p, γ)²¹Na(p, γ)²²Mg(β⁺)²²Na takes over as the dominant path, favoring the synthesis of ²²Na, which achieves a second maximum after the temperature peak T_peak (see Table 2). This pattern continues up to ²²Na(p, γ)²³Mg becomes again the most relevant reaction when the temperature drops. During the subsequent and final expansion and ejection stages, as the temperature drops dramatically, the evolution of ²²Na is fully governed by ²²Na(β⁺)²²Ne. The most relevant results obtained in these simulations are summarized in Table 2. The larger peak temperatures achieved during nova outbursts on more massive white dwarfs yield larger mean, mass-averaged abundances of ²²Na in the ejecta. However, it is worth noting that the total mass of ²²Na ejected in a nova outburst decreases with the white dwarf mass (8 × 10⁻⁹ M_⊙, for Model 115b, 7 × 10⁻⁹ M_⊙, for Model 125, and 4.2 × 10⁻⁹ M_⊙, for Model 135), since more massive white dwarfs accrete and eject smaller amounts of mass, see M_ejec in Table 2.

Table 2 Abundances of (Ne, Na) isotopes in the ejecta obtained for different nova models calculated with the $\texttt{MESA}$ and $\texttt{SHIVA}$ codes (see text for the parameters)

The new and reliable reaction rate obtained in this work also opens the door to further advanced sensitivity studies on astrophysical parameters, performed here for the first time. The flux has been obtained in a series of nova simulations for a large range of mass-accretion rates $\dot{M}$ and white dwarf luminosities L_WD,ini. For example, Fig. 3 shows the expected flux of ²²Na for a nova event located 1 kpc from the Earth, with M_WD = 1.2 M_⊙ and accreting solar composition material. This figure shows that the amount of ²²Na does depend on the two parameters, varying smoothly by up to a factor of 3. The change is abrupt only on the lower right side of the figure. In this region, the ²²Na/²²Mg and ²²Na/²¹Ne ratios around the peak temperatures have the same values as in the other regions, which means that the production pathways are unchanged. On the contrary, the ²³Mg/²²Na ratio is found 4 times larger, indicating that the destruction by the ²²Na(p, γ)²³Mg reaction is more intense in this region. This graph provides a link between the unknown astrophysical parameters of the novae and the predicted ²²Na flux. In the future, precise measurements of the ²²Na γ-ray flux will constrain these astrophysical parameters.

Fig. 3: Prediction of the 1.275 MeV γ-ray flux emitted from a nova.

The ²²Na γ-ray emission flux is shown as a function of the white dwarf initial luminosity and the mass-accretion rate. This is calculated from the ²²Na mass-averaged abundance within the ejected shells. Computations were done for a 1.2 M_⊙ ONe white dwarf located 1 kpc from the Earth, using the $\texttt{MESA}$ code^{27,28,29,30,31,32}.

New instruments for γ-ray astronomy are under study or under construction: ESA’s enhanced e-ASTROGAM³⁵ and NASA’s COmpton Spectrometer and Imager, COSI³⁶. Both are, in principle, capable of detecting the 1.275 MeV γ rays released in the decay of ²²Na produced in nova outbursts, since they are being designed with a higher efficiency than the past missions, INTEGRAL-SPI³⁷ and COMPTEL-CGRO³⁸. The very high precision in the determination of the ²²Na(p, γ)²³Mg rate reported in this work permits deriving, for the first time, realistic estimates of the maximum detectability distance of novae in γ rays, through the 1.275 MeV line. With the expected sensitivities of e-ASTROGRAM (3 × 10⁻⁶ photons cm⁻² s⁻¹) and COSI (1.7 × 10⁻⁶ photons cm⁻² s⁻¹) at 1 MeV, and assuming model 125 (Table 2), the new maximum detectability distances of 2.7 and 4.0 kpc, respectively, have been derived, on which the reaction rate uncertainty obtained here leads to an uncertainty of 18%. These new estimates suggest a large chance for the possible detection of the ²²Na γ rays produced in ONe novae by the next generation of space-borne γ-ray observatories.

Methods

Determination of velocities

The velocity β_reac of the ²³Mg nuclei at the reaction time was derived from the momentum of the α particles measured with the VAMOS++ magnetic spectrometer and from the kinematics laws of energy and momentum conservation in the case of the ³He(²⁴Mg,α)²³Mg two-body reaction

$${\beta }_{{{{{{{{\rm{reac}}}}}}}}}=\sqrt{1-{\gamma }_{{{{{{{{\rm{reac}}}}}}}}}^{-2}}$$

(1)

with

$${\gamma }_{{{{{{{{\rm{reac}}}}}}}}}=\sqrt{1+\frac{\frac{{m}_{\alpha }^{2}{\beta }_{\alpha }^{2}}{1-{\beta }_{\alpha }^{2}}+\frac{{m}_{{{{{{{{\rm{beam}}}}}}}}}^{2}{\beta }_{{{{{{{{\rm{beam}}}}}}}}}^{2}}{1-{\beta }_{{{{{{{{\rm{beam}}}}}}}}}^{2}}-2\cos ({\theta }_{\alpha }){m}_{{{{{{{{\rm{beam}}}}}}}}}{m}_{\alpha }\sqrt{{\gamma }_{\alpha }^{2}-1}\sqrt{{\gamma }_{{{{{{{{\rm{beam}}}}}}}}}^{2}-1}}{{m}_{{{{{{{{\rm{recoil}}}}}}}}}^{2}}}$$

(2)

where m_α, m_beam and m_recoil are the rest-mass energies of the α, ²⁴Mg, and ²³Mg nuclei, θ_α the angle between the α particle and the beam axis. The parameter γ_α was measured with VAMOS++ and corrected for the energy losses in the target using the $\texttt{SRIM}$ code³⁹. The parameter γ_beam was measured prior to the experiment, γ_reac was then determined from the measured γ_α and a γ-ray transition was detected in coincidence.

The velocity β_ems of the ²³Mg nuclei at the γ-ray emission time was derived from the measured γ rays. Since the ²³Mg nuclei are moving at the time of the γ-ray emission, the γ-ray energy is Doppler shifted, with the measured energies E_γ shifted from the center-of-mass energy E_γ,0 according to

$${E}_{\gamma }={E}_{\gamma,0}\frac{\sqrt{1-{\beta }_{{{{{{{{\rm{ems}}}}}}}}}^{2}}}{1-{\beta }_{{{{{{{{\rm{ems}}}}}}}}}\cos (\theta )}$$

(3)

It follows that

$${\beta }_{{{{{{{{\rm{ems}}}}}}}}}=\frac{{R}^{2}\cos (\theta )+\sqrt{1+{R}^{2}{\cos }^{2}(\theta )-{R}^{2}}}{{R}^{2}{\cos }^{2}(\theta )+1}$$

(4)

with R = E_γ/E_γ,0. Here, R < 1 since the $\texttt{AGATA}$ detector was located upstream of the target. The angle θ between the γ ray and the ²³Mg recoil nucleus was derived from the measured (θ, ϕ) of the γ ray and the α particle using the formulas

$$\cos (\theta )= \sin ({\theta }_{\gamma })\sin ({\theta }_{{{{{{{{\rm{recoil}}}}}}}}})[\cos ({\phi }_{\gamma })\cos ({\phi }_{{{{{{{{\rm{recoil}}}}}}}}}) \\ +\sin ({\phi }_{\gamma })\sin ({\phi }_{{{{{{{{\rm{recoil}}}}}}}}})]+\cos ({\theta }_{\gamma })\cos ({\theta }_{{{{{{{{\rm{recoil}}}}}}}}})$$

where

$${\theta }_{{{{{{{{\rm{recoil}}}}}}}}}={{{{{{{\rm{acos}}}}}}}}\left(\frac{{m}_{{{{{{{{\rm{beam}}}}}}}}}\sqrt{{\gamma }_{{{{{{{{\rm{beam}}}}}}}}}^{2}-1}-{m}_{\alpha }\cos ({\theta }_{\alpha })\sqrt{{\gamma }_{\alpha }^{2}-1}}{{m}_{{{{{{{{\rm{recoil}}}}}}}}}\sqrt{{\gamma }_{{{{{{{{\rm{recoil}}}}}}}}}^{2}-1}}\right) \,{\rm {and}}\, {\phi }_{{{{{{{{\rm{recoil}}}}}}}}}=\pi+{\phi }_{\alpha }$$

Fit of velocity-difference profiles

Velocity-difference profiles were numerically simulated with a Monte-Carlo approach developed in the $\texttt{EVASIONS}$ C++/ROOT code⁴⁰. Simulated velocity-difference profiles were normalized to the measured ones via the profile integrals. The goodness of fit between experimental and simulated profiles was quantified with the Pearson χ² tests where the lifetime and the γ-ray center-of-mass energy were taken as free parameters.

Branching ratios

The E_x = 7785.0(7) keV astrophysical state can decay via proton or γ-ray emission. Therefore, after applying a selection on E_x in ²³Mg by using the measured α particles, the number of detected protons and γ rays allowed us to determine the proton and γ-ray branching ratios. These values were corrected for detection efficiencies. On the one hand, the geometrical efficiency of the silicon detector was estimated by numerical simulations. The angular distribution was considered isotropic for the emitted ℓ = 0 protons. On the other hand, the $\texttt{AGATA}$ efficiency was measured at low energies with a radioactive ¹⁵²Eu source, and simulated with the $\texttt{AGATA}$ $\texttt{Geant4}$ code library^41,42 to determine the efficiency at high energies after scaling the simulations to the measured efficiencies at low energies.

Determination of the ²²Na flux in novae

The amount of ²²Na ejected in a nova outburst was obtained with the simulation codes $\texttt{MESA}$^{27,28,29,30,31} and $\texttt{SHIVA}$^33,34 from its abundance in the ejected layers. $\texttt{SHIVA}$ and $\texttt{MESA}$ use two basic criteria for the ejection of a specific layer: if its velocity achieves the escape velocity (~1000 km s⁻¹), or if its luminosity becomes higher than the Eddington limit (when the force exerted by radiation exceeds the gravitational pull) and hence, the condition of hydrostatic equilibrium no longer holds.