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Carbon-14 has six protons and eight neutrons. Atmospheric production is dominated by ${}^{14}\mathrm{N}(n,p){}^{14}\mathrm{C}$ using secondary cosmic ray neutrons. It undergoes beta-minus decay to nitrogen-14, with a physical half-life of approximately $5730$ years.

A hotter stellar core can favor a higher-barrier route despite fewer catalyst nuclei. To isolate this competition, prescribe a local hydrogen burning toy model at fixed mass density $\rho$, with reference $T_0=15\mathrm{MK}$ andChoose $E_{G,pp}=493\mathrm{keV}$ and $E_{G,\mathrm{CNO}}=45000\mathrm{keV}$, rounded barrier parameters for a proton pair and proton capture on nitrogen-14. These prescribe the temperature dependence, while the following freely chosen heating normalization incorporates effective nuclear factors and deposited energy:Here $X$ is hydrogen mass fraction, $X_{\mathrm{CNO}}$ is CNO mass fraction, and ${ϵ}_{\ast }$ is an arbitrary positive unit of stellar energy-generation rate. For $X=0.70$ and $X_{\mathrm{CNO}}=0.01$, the reference ratio is $0.01$ by construction. It is not a measured solar ratio. The CNO cycle bottleneck is represented by one effective rate, without evolving the existing proton-proton chain branches or catalyst network.

At $T_0$, the logarithmic temperature slopes are $({\tau }_{pp,0}-2)/3=3.90$ and $({\tau }_{\mathrm{CNO},0}-2)/3=19.90$. The larger Coulomb barrier gives a steeper fractional response. Because these slopes change with temperature, this example retains the exponential rather than extending two fixed power laws. Writing $R_0=0.01(X_{\mathrm{CNO}}/0.01)(0.70/X)$ and $D={\tau }_{\mathrm{CNO},0}-{\tau }_{pp,0}$ givesprovided the bracket is positive. With the chosen composition, $T_{\ast }=20.30\mathrm{MK}$. Reducing $X_{\mathrm{CNO}}$ to $0.001$ at unchanged $X$ shifts it to $23.91\mathrm{MK}$. For $X_{\mathrm{CNO}}=0$, this catalytic route supplies no heat in this model. These are local toy crossovers: a whole star's luminosity requires integrating its stellar energy-generation rate through its stellar composition profile and stellar structure. Nuclear features, plasma screening and changes of the dominant bottleneck limit the model's extrapolation.

In 1953, David Noel Ferguson Dunbar and collaborators used accelerated deuterons on nitrogen-14, populating carbon-12 through ${}^{14}\mathrm{N}(d,\alpha ){}^{12}\mathrm{C}$. Magnetic analysis resolved outgoing alpha particle groups. At $620\mathrm{keV}$ incident energy and laboratory angle $90$ degrees, they reported excitation $7.68\pm 0.03\mathrm{MeV}$, with yield $6\%$ of the $4.43\mathrm{MeV}$ group's yield. The stronger group supplied an internal comparison. The paper's opening explicitly relates the search to Fred Hoyle's abundance argument. This is evidence for a nuclear level from a different entrance reaction than stellar fusion.^[ref]

The inference uses energy and momentum conservation: at fixed incident energy and observation angle, more energy retained as carbon-12 excitation leaves less outgoing kinetic energy. A resolved particle group therefore identifies a nuclear level. This laboratory nuclear line supports an input required by a stellar synthesis model. It does not reproduce core helium burning, measure a complete stellar carbon yield, or by itself determine the stabilizing decay probability.

Dating principle. Atmospheric cosmic rays produce secondary neutrons, which can generate carbon-14 through ${}^{14}\mathrm{N}(n,p){}^{14}\mathrm{C}$. Carbon then enters atmospheric carbon dioxide and living organisms. While an organism exchanges carbon with its environment, its carbon-14 to carbon-12 ratio approximately tracks that reservoir. After exchange stops, surviving carbon-14 undergoes beta-minus decay, whereas the stable carbon remains. On the closed-sample, constant-initial-ratio model, radioactive decay givesThus radiocarbon dating infers elapsed time from the loss of an isotope ratio or of radioactive activity per fixed carbon mass. It dates cessation of carbon exchange, which need not equal the date an object was manufactured or buried. Actual calendar dating also requires radiocarbon calibration and control of contamination and reservoir differences.

Mean lifetime. For exponential decay, the half-life obeys $e^{-\lambda T_{1/2}}=1/2$, while the mean lifetime is $\tau =1/\lambda$. HenceUsing $1\mathrm{yr}=365.25\times 86400\mathrm{s}$ gives $\lambda =3.83325\times 10^{-12}{\mathrm{s}}^{-1}$ and $\tau =2.60875\times 10^{11}\mathrm{s}$.

The mass minimum and beta stability. Treat the supplied masses as nuclear rest energies in $\mathrm{MeV}$, and omit nuclear pairing as instructed. At fixed mass number $A$, the volume and surface terms of the semi-empirical mass formula are constant. Expanding the remaining terms gives a convex nuclear mass parabola,The continuous minimum in atomic number is thereforeAt $A=14$, the quadratic coefficient is $7.01302\mathrm{MeV}$ and $Z_{\ast }=6.79450$. The nearest integer is $Z=7$; the nuclear isobar of minimum mass is nitrogen-14. Maximizing nuclear binding energy alone would omit the neutron-proton rest-mass difference; its continuous optimum is slightly different, but also selects integer $Z=7$ here.

The values most useful for checking neighboring decay thresholds areConsequently carbon-14 can lower its nuclear mass by changing a neutron into a proton:For bare nuclear masses, the Q value must include the emitted electron mass. Taking $m_e{c}^2=0.511\mathrm{MeV}$,The opposite proton-to-neutron conversion would lead to $Z=5$, whose nuclear mass is $18.1567\mathrm{MeV}$ higher than that of carbon-14. Neither beta-plus decay nor electron capture can bridge that gap by supplying an electron-mass correction. For the nitrogen-14 daughter, both neighboring nuclear masses are higher: $Z=6$ by $4.13070\mathrm{MeV}$ and $Z=8$ by $9.89535\mathrm{MeV}$. Even electron capture to $Z=6$ has $Q=0.511-4.13070<0$. Thus the daughter has no further energetically allowed single beta decay in this model; physical nitrogen-14 is stable.

These threshold numbers belong to the supplied smooth semi-empirical mass formula, not to an accurate prediction of the carbon-14 electron spectrum or its long half-life. Semi-empirical mass errors can change beta-decay thresholds. Also, the instruction to omit nuclear pairing is a model simplification: even $A=14$ does not make physical pairing vanish, and even-even carbon and odd-odd nitrogen need not have the same pairing correction. No omitted pairing coefficient is guessed here.

Activity of modern carbon. Neglect the small contribution of other stable carbon isotopes when counting atoms in a gram. The Avogadro constant givesThe expected radioactive activity of that one gram isor approximately $15.0$ decays per minute per gram of carbon. A becquerel is one decay per second; the normalization is per gram of carbon, rather than per gram of the entire organic sample.

Artefact age. Its remaining activity fraction is $0.13/0.25008=0.51983$. Inverting the exponential decay law,This is the radiocarbon age on the supplied constant-initial-activity, $5730$-year half-life model. It is not a calibrated calendar date; the sample's historical initial ratio and any later carbon exchange are not supplied.