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¹²C(α,γ)¹⁶O

The “holy grail of nuclear astrophysics”. A helium-burning core makes carbon through the Hoyle state of ¹²C; part of that carbon then captures one more α and becomes oxygen. How much of each is left sets the rest of a massive star's life, the composition of white dwarfs, supernova yields and even where the black-hole mass gap begins. At the stellar energy the capture cross section is about 10⁻¹⁷ b, far below anything measured, so the rate is extrapolated from data above about 1 MeV through the levels of ¹⁶O.

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Helium-burning temperature

T₉

Star (MESA models of deBoer et al.)

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Why it cannot be measured

An α particle at E₀ ≈ 0.3 MeV meets a Coulomb barrier of a few MeV and turns around about 60 fm out, more than ten times the nuclear radius; the tunnelling factor e^−2πη is about 10⁻¹⁶. The review of deBoer et al. (2017) estimates σ(300 keV) ≈ 2×10⁻¹⁷ b, about five orders of magnitude below the few picobarns reached by the best measurements, whose lowest points lie near E_c.m. ≈ 1 MeV.

The extrapolation is hard because the cross section there is made of tails. The ground-state transition has an E1 part, carried by the subthreshold 1⁻ state at 7.12 MeV (45 keV below the α threshold) interfering with the broad 1⁻ at 9.585 MeV, and an E2 part, carried by the subthreshold 2⁺ at 6.92 MeV and by direct capture. Isospin suppresses E1 in this N = Z system, so the two parts are of similar size; cascades through the bound excited states add about 5% at 300 keV (deBoer et al., Sects. I, V.C).

Indirect routes and underground labs

α transfer. Sub-Coulomb ¹²C(⁶Li,d) and (⁷Li,t) transfer fixes the asymptotic normalization coefficients of the subthreshold 2⁺ and 1⁻ states, which set the size of their tails (Brune et al., PRL 83, 4025 (1999); Avila et al., PRL 114, 071101 (2015), who also measured the 6.05 and 6.13 MeV states).

β-delayed α decay of ¹⁶N. ¹⁶N β-decays into 1⁻ strength of ¹⁶O above the α threshold, which breaks up into α + ¹²C; the low-energy shape of the α spectrum constrains the subthreshold 1⁻ contribution (Azuma et al., PRC 50, 1194 (1994); Tang et al., PRC 81, 045809 (2010)).

Trojan horse. In the review the Trojan-horse method appears only for the related ¹⁵N(p,α)¹²C reaction. ¹⁶O = α + ¹²C has itself served as a Trojan-horse nucleus for carbon fusion (Li et al., PLB 879, 140675 (2026), arXiv:2609.08819); I found no Trojan-horse determination of ¹²C(α,γ)¹⁶O.

Underground. In 2017 the review lists LUNA (waiting for the 3.5 MV LUNA MV machine, since the old one gives α beams only up to 400 keV), CASPAR and Jinping. JUNA in Jinping ran ¹²C(α,γ)¹⁶O in 2021 with a beam near 1 emA, reaching an upper-limit sensitivity of 10⁻¹³ b at E_c.m. = 538 keV, with the analysis then still in progress (Liu et al., EPJ Web Conf. 260, 08001 (2022)).

Open question

Extrapolated S(300 keV) values have scattered by factors of several over fifty years (deBoer et al., Table IV; Kunz et al. 2002, Table 1). The 2017 global R-matrix analysis gives 140 ± 21 (MC) ⁺¹⁸₋₁₁ (model) keV b, while other recent analyses sit near 160 keV b (side table); the stated goal is about 10%. From the edge of the black-hole mass gap seen in the first ten gravitational-wave black holes, Farmer et al. (ApJL 902, L36 (2020)) infer S₃₀₀ > 175 keV b at 68% confidence, a stellar-model inference whose degeneracies, they write, need further study. This page does not take a side.

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