---
title: "Advanced Burning, the Iron Peak, and the s/r Processes"
draft: false
module: Nuclear Astrophysics
moduleNumber: 5
lessonNumber: 4
order: 504
summary: >
  Massive stars burn carbon, neon, oxygen, and silicon in ever-shorter stages,
  building an onion-shell interior and reaching nuclear statistical equilibrium at
  the iron peak, where the binding-energy-per-nucleon curve turns over and fusion can
  release no more energy. Elements beyond iron form by neutron capture: the slow
  s-process in AGB stars tracks the valley of stability, while the rapid r-process in
  supernovae and neutron-star mergers builds the heaviest nuclei far from it.
topics: [Nuclear Astrophysics]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 15 — The Fate of Massive Stars; §15.3 Nucleosynthesis and the r- and s-Processes; Ch. 13"
  - book: Maoz
    ref: "Ch. 4 — Stellar Evolution and Stellar Remnants"
  - book: PDG
    ref: "Review of Particle Physics — atomic masses and binding energies"
---

A star of more than about $8\,M_\odot$ does not stop at carbon and oxygen. Its core
contracts and heats through a succession of burning stages, each fusing the ashes of
the last, until it assembles an iron core that can release no further energy. The
sequence is governed by two curves met earlier: the Coulomb barrier, which raises the
ignition temperature of each heavier fuel, and the binding energy per nucleon, which
peaks at iron and marks the end of energy-releasing fusion. Everything heavier than
the iron peak is built not by fusion but by capturing neutrons.

## Advanced burning stages

After core helium burning leaves a carbon–oxygen core, contraction raises the central
temperature through the ignition points of successively heavier fuels. Each stage has
a higher Coulomb barrier and therefore a higher ignition temperature, following the
Gamow scaling of [reaction
rates](/astrophysics-cosmology/nuclear-astrophysics/thermonuclear-reaction-rates-and-the-gamow-peak).

- **Carbon burning** ($\sim 8\times 10^8\ \text{K}$):
  ${}^{12}\text{C} + {}^{12}\text{C}$ yields ${}^{20}\text{Ne} + \alpha$,
  ${}^{23}\text{Na} + p$, and other channels.
- **Neon burning** ($\sim 1.5\times 10^9\ \text{K}$): photodisintegration
  ${}^{20}\text{Ne} + \gamma \to {}^{16}\text{O} + \alpha$ precedes
  ${}^{20}\text{Ne} + \alpha \to {}^{24}\text{Mg} + \gamma$. Neon burns before oxygen
  because its photodisintegration sets in at a lower temperature than oxygen fusion.
- **Oxygen burning** ($\sim 2\times 10^9\ \text{K}$):
  ${}^{16}\text{O} + {}^{16}\text{O}$ produces ${}^{28}\text{Si}$, ${}^{31}\text{P}$,
  ${}^{31}\text{S}$, and ${}^{32}\text{S}$.
- **Silicon burning** ($\sim 3\times 10^9\ \text{K}$): the Coulomb barrier for
  ${}^{28}\text{Si} + {}^{28}\text{Si}$ is too high to fuse directly. Instead,
  energetic photons photodisintegrate some silicon into alpha particles, protons, and
  neutrons, which are recaptured onto other nuclei, gradually rearranging the
  composition toward the iron peak.

Each stage releases less energy per unit mass than the last, and neutrino losses grow
steeply with temperature, so the burning lifetimes collapse. For a $20\,M_\odot$ star
the durations run from millions of years for hydrogen down to a few days for silicon.

| Stage | ignition $T$ | representative products | duration |
| --- | --- | --- | --- |
| hydrogen | $\sim 4\times 10^7\ \text{K}$ | ${}^{4}\text{He}$ | $\sim 10^7\ \text{yr}$ |
| helium | $\sim 2\times 10^8\ \text{K}$ | ${}^{12}\text{C}, {}^{16}\text{O}$ | $\sim 10^6\ \text{yr}$ |
| carbon | $\sim 8\times 10^8\ \text{K}$ | ${}^{20}\text{Ne}, {}^{23}\text{Na}$ | $\sim 300\ \text{yr}$ |
| neon | $\sim 1.5\times 10^9\ \text{K}$ | ${}^{16}\text{O}, {}^{24}\text{Mg}$ | $\sim 1\ \text{yr}$ |
| oxygen | $\sim 2\times 10^9\ \text{K}$ | ${}^{28}\text{Si}, {}^{32}\text{S}$ | $\sim 0.5\ \text{yr}$ |
| silicon | $\sim 3\times 10^9\ \text{K}$ | iron-peak nuclei | $\sim$ days |

Because each fuel ignites in the core and then continues burning in a shell around the
newly formed heavier core, the star develops a layered **onion-shell** structure, with
the heaviest ash at the center and hydrogen still burning at the outer edge.[^co-adv]

$$
% caption: The onion-shell interior of a pre-supernova massive star: each burning
% stage leaves a shell of its ash, from the hydrogen envelope through helium, carbon
% and oxygen, oxygen-neon-magnesium, and silicon, down to an inert iron core at the
% center.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% concentric shells
\draw[very thick] (0,0) circle (3.6);
\draw[very thick] (0,0) circle (2.9);
\draw[very thick] (0,0) circle (2.25);
\draw[very thick] (0,0) circle (1.65);
\draw[very thick] (0,0) circle (1.1);
\draw[acc, very thick, fill=acc!18] (0,0) circle (0.6);
% labels along a radius
\node[black!70, anchor=west] at (0.05,3.25) {H envelope};
\node[black!70, anchor=west] at (0.05,2.55) {He};
\node[black!70, anchor=west] at (0.05,1.9) {C, O};
\node[black!70, anchor=west] at (0.05,1.35) {O, Ne, Mg};
\node[black!70, anchor=west] at (0.05,0.82) {Si};
\node[acc, anchor=center] at (0,0) {Fe};
\end{tikzpicture}
$$

## The iron peak and the end of fusion

At temperatures above $\sim 4\times 10^9\ \text{K}$ the forward fusion and reverse
photodisintegration reactions come into balance, and the composition relaxes to
**nuclear statistical equilibrium** (NSE). Under equilibrium the abundances are fixed
by a Saha-like relation that favors the most tightly bound nuclei, which are the
iron-peak species around ${}^{56}\text{Fe}$, ${}^{54}\text{Fe}$, and
${}^{56}\text{Ni}$. Silicon burning drives the core into this state, and the core
becomes iron.

The reason iron is the endpoint is the **binding energy per nucleon**,

$$
\frac{B}{A} = \frac{\bigl[Z m_p + (A - Z)m_n - m(Z,A)\bigr]c^2}{A},
$$

which measures how much energy is released when $A$ free nucleons assemble into a
nucleus. This curve rises steeply from hydrogen, reaches a broad maximum of about
$8.8\ \text{MeV}$ per nucleon near $A \approx 56$–$62$, then declines gently toward
uranium. Fusion releases energy only when it moves nuclei toward the peak, that is,
when $B/A$ increases. Fusing iron-peak nuclei into anything heavier would decrease
$B/A$ and therefore absorb energy. The core can extract no more nuclear energy, loses
its pressure support, and is left to collapse, the trigger for a [core-collapse
supernova](/astrophysics-cosmology/stellar-death-and-compact-remnants/core-collapse-supernovae).

$$
% caption: Binding energy per nucleon versus mass number. The curve climbs from
% hydrogen (with a sharp local maximum at tightly bound helium-4) to a peak near iron
% at about 8.8 MeV, then declines. Fusion of light nuclei and fission of heavy nuclei
% both move toward the peak and release energy; nothing can be gained by fusing iron.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {mass number A};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {binding energy per nucleon};
% rising curve to iron peak then gentle decline
\draw[acc, very thick]
  (0.3,0.15)
  .. controls (0.55,0.9) and (0.75,1.6) .. (0.95,3.55) % He-4 bump
  .. controls (1.1,2.7) and (1.2,3.0) .. (1.5,3.4)
  .. controls (2.2,3.95) and (3.0,4.25) .. (3.6,4.3)  % iron peak
  .. controls (5.4,4.25) and (7.4,3.85) .. (9.0,3.5); % decline to U
% He-4 marker
\fill[black] (0.95,3.55) circle (1.6pt);
\node[black, anchor=south] at (0.95,3.6) {He-4};
% iron peak marker
\fill[acc] (3.6,4.3) circle (2pt);
\node[acc, anchor=south] at (3.6,4.35) {iron peak};
% direction arrows
\draw[->, black] (1.6,1.9) -- (2.9,1.9);
\node[black, anchor=north] at (2.2,1.85) {fusion};
\draw[->, black] (7.4,2.0) -- (6.1,2.0);
\node[black, anchor=north] at (6.8,1.95) {splitting};
% mass ticks
\foreach \x/\lab in {0.95/4, 3.6/56, 9.0/238}
  \node[black, anchor=north] at (\x,-0.05) {\lab};
\end{tikzpicture}
$$

## Neutron capture beyond iron

Elements heavier than the iron peak cannot be built by charged-particle fusion: the
Coulomb barrier is prohibitive and the reactions are endothermic. They are assembled
instead by **neutron capture**, which has no Coulomb barrier. A nucleus absorbs a
neutron, $(Z,A) + n \to (Z,A+1)$; if the product is stable it waits for another
neutron, and if it is unstable it $\beta^{-}$ decays, $(Z,A) \to (Z+1,A) + e^- +
\bar\nu_e$, raising the atomic number. The path a nucleus takes through the chart of
nuclides depends entirely on the ratio of the neutron-capture rate to the beta-decay
rate, which splits nucleosynthesis into two regimes.

The **slow (s) process** operates when neutron capture is slow compared with beta
decay. After each capture, an unstable nucleus decays back to stability before the
next neutron arrives, so the path hugs the valley of beta stability, stepping from one
stable isotope to the next. It occurs in thermally pulsing asymptotic-giant-branch
stars, where the reactions ${}^{13}\text{C}(\alpha,n){}^{16}\text{O}$ and
${}^{22}\text{Ne}(\alpha,n){}^{25}\text{Mg}$ supply modest neutron densities of order
$10^{7}$–$10^{11}\ \text{cm}^{-3}$. The s-process builds nuclei up to bismuth, where
an alpha-decay cycle terminates the chain.

The **rapid (r) process** operates when neutron capture is fast compared with beta
decay. A nucleus captures many neutrons in quick succession, driving it far to the
neutron-rich side of stability toward the neutron drip line, and only when the neutron
flux ends does the highly unstable product beta-decay back to stability. It requires
enormous neutron densities, of order $10^{22}$–$10^{24}\ \text{cm}^{-3}$, reached only
in explosive sites: the neutrino-driven winds of core-collapse supernovae and,
decisively, the ejecta of [neutron-star
mergers](/astrophysics-cosmology/binaries-and-gravitational-waves/multimessenger-astronomy-and-gamma-ray-bursts).
The r-process makes the heaviest nuclei, including thorium and uranium.

$$
% caption: The s-process (solid staircase) advances by single neutron captures with
% beta decays between them, tracking the valley of stability; the r-process (dashed)
% captures many neutrons at fixed proton number, running far to the neutron-rich side,
% then beta-decays back up to stability after the neutron flux ends. Vertical lines
% mark the neutron magic numbers where both processes pause.
\begin{tikzpicture}[scale=1.0, font=\footnotesize, >={Stealth[length=1.8mm]}]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {neutron number N};
\draw[->, black] (0,0) -- (0,4.6) node[above, black!70] {proton number Z};
% valley of stability band
\draw[black, line width=6pt, opacity=0.5] (0.4,0.4) -- (8.6,4.0);
\node[black, anchor=north west, rotate=25] at (5.2,2.5) {valley of stability};
% s-process staircase along stability
\draw[acc, very thick]
  (1.2,0.9) -- (1.9,0.9) -- (1.9,1.25) -- (2.6,1.25) -- (2.6,1.6)
  -- (3.4,1.6) -- (3.4,1.95) -- (4.2,1.95) -- (4.2,2.3) -- (5.0,2.3);
\node[acc, anchor=south east] at (3.9,2.15) {s-process};
% r-process: horizontal then diagonal decay back
\draw[black, very thick, densely dashed] (1.2,0.7) -- (6.6,0.7);
\draw[black, very thick, densely dashed, ->] (6.6,0.7) -- (8.2,3.3);
\node[black, anchor=north] at (4.0,0.62) {r-process (many captures)};
\node[black, anchor=west] at (7.4,2.4) {beta decays back};
% magic number lines
\draw[black, densely dotted] (3.9,0) -- (3.9,4.3);
\draw[black, densely dotted] (6.6,0) -- (6.6,4.3);
\node[black, anchor=south] at (3.9,4.3) {magic N};
\node[black, anchor=south] at (6.6,4.3) {magic N};
\end{tikzpicture}
$$

Because the two processes reach the neutron magic numbers $N = 50, 82, 126$ at
different proton numbers, they leave distinct fingerprints in the abundance pattern:
the s-process abundance peaks (at $A \approx 88, 138, 208$) sit at slightly higher
mass than the r-process peaks (at $A \approx 80, 130, 195$), because the r-process
reaches a magic neutron number while still neutron-rich and the nuclei only later
decay to lower mass along an isobar. The double-peaked abundance curve of the heavy
elements is direct evidence that both processes contribute.

## The origin of the elements

Assembling these mechanisms accounts for the cosmic abundance pattern across the
periodic table:

- **Hydrogen, helium, and a trace of lithium** were made in [Big Bang
  nucleosynthesis](/astrophysics-cosmology/the-hot-big-bang/big-bang-nucleosynthesis)
  in the first minutes.
- **Carbon through the iron peak** are forged by fusion in stellar cores, the
  lighter of these also by low-mass stars and the iron-peak nuclei predominantly in
  the explosive burning of supernovae.
- **Most nuclei heavier than iron** are built by neutron capture, split between the
  s-process in AGB stars and the r-process in supernovae and neutron-star mergers.
- **Lithium, beryllium, and boron** are largely produced by cosmic-ray spallation of
  heavier nuclei in the interstellar medium, filling a gap the stellar processes skip.

$$
% caption: Dominant nucleosynthetic origin as a function of atomic number. The
% lightest elements are primordial; fusion in stars builds up to the iron peak; the
% slow and rapid neutron-capture processes make the elements beyond iron; a narrow
% light-element window is filled by cosmic-ray spallation.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (10.0,0) node[right, black!70] {atomic number Z};
% origin bands as segments along Z
\draw[black] (0.3,0.4) rectangle (1.3,1.2);
\node[black!70, anchor=south, align=center] at (0.8,1.2) {Big Bang};
\node[black, anchor=north] at (0.8,0.35) {H, He};
\draw[black] (1.3,0.4) rectangle (2.1,1.2);
\draw[black, thin] (1.7,1.2) -- (1.7,1.7);
\node[black, anchor=south, align=center, font=\scriptsize] at (1.7,1.72) {spallation};
\draw[black] (2.1,0.4) rectangle (5.2,1.2);
\node[black!70, anchor=south, align=center] at (3.65,1.2) {stellar fusion};
\node[black, anchor=north] at (3.65,0.35) {C to Si};
\fill[acc!18, draw=acc] (5.2,0.4) rectangle (6.2,1.2);
\node[acc, anchor=south, align=center] at (5.7,1.55) {iron peak};
\draw[black] (6.2,0.4) rectangle (10.0,1.2);
\node[black!70, anchor=south, align=center] at (8.1,1.2) {neutron capture};
\node[black, anchor=north] at (8.1,0.35) {s- and r-process};
% Z ticks
\foreach \x/\lab in {0.3/1, 5.7/26, 10.0/92}
  \node[black, anchor=north] at (\x,-0.25) {\lab};
\end{tikzpicture}
$$

The nuclear physics of this module fixes where each element comes from and when. The
Gamow peak set the ignition temperatures that order the burning stages; the
binding-energy curve fixed the iron endpoint that ends fusion and collapses the core;
and neutron capture, in its slow and rapid forms, builds everything heavier. The
collapse of the iron core and the explosive nucleosynthesis that accompanies it are
taken up in [core-collapse
supernovae](/astrophysics-cosmology/stellar-death-and-compact-remnants/core-collapse-supernovae),
and the r-process ejecta of merging neutron stars are confirmed observationally in
[multimessenger
astronomy](/astrophysics-cosmology/binaries-and-gravitational-waves/multimessenger-astronomy-and-gamma-ray-bursts).

[^co-adv]: Carroll & Ostlie, §15.3 and Ch. 13 — advanced nuclear burning stages, the onion-shell model, nuclear statistical equilibrium and the iron peak, the binding-energy-per-nucleon curve, and the s- and r-processes of neutron-capture nucleosynthesis.