---
title: Thermonuclear Reaction Rates and the Gamow Peak
draft: false
module: Nuclear Astrophysics
moduleNumber: 5
lessonNumber: 1
order: 501
summary: >
  Stellar fusion proceeds only by quantum tunneling through the Coulomb barrier,
  because thermal energies are a thousand times smaller than the barrier height.
  The reaction rate is an integral over the Maxwell–Boltzmann distribution and the
  tunneling probability, whose product is sharply peaked at the Gamow energy. The
  astrophysical S-factor isolates the nuclear physics from the barrier penetration,
  and the steep temperature dependence follows from the width and position of the
  Gamow peak.
topics: [Nuclear Astrophysics]
sources:
  - book: Carroll & Ostlie
    ref: "Ch. 10 — The Interiors of Stars; §10.3 Energy Transport and Thermodynamics, Nuclear Reaction Rates"
  - book: Maoz
    ref: "Ch. 3 — Stellar Physics"
  - book: PDG
    ref: "Review of Particle Physics — constants and nuclear data"
---

The energy that supports a star against its own gravity comes from fusing light
nuclei into heavier ones. Two positively charged nuclei must approach within a
nuclear radius, about $1\ \text{fm}$, against a Coulomb repulsion that at that
separation exceeds their thermal energy by three orders of magnitude. Classically
the reaction cannot happen at stellar temperatures. Fusion proceeds only because
the wavefunction of the relative motion leaks through the barrier, and the rate is
governed by the competition between the exponentially small tunneling probability,
which favors fast particles, and the exponentially rare fast particles in the
thermal tail. Their product defines a narrow band of energies, the Gamow peak, that
supplies essentially all of a star's fusion.

## The Coulomb barrier

Two nuclei of charges $Z_1 e$ and $Z_2 e$ separated by $r$ interact through the
Coulomb potential

$$
V(r) = \frac{1}{4\pi\epsilon_0}\frac{Z_1 Z_2 e^2}{r},
$$

which rises until the nuclei touch at the contact radius
$r_n \approx 1.4\,(A_1^{1/3} + A_2^{1/3})\ \text{fm}$, where the short-range
attractive nuclear force takes over and the potential drops into a deep well. The
top of the barrier for two protons sits at

$$
E_{\rm C} = \frac{1}{4\pi\epsilon_0}\frac{Z_1 Z_2 e^2}{r_n}
\approx \frac{1.44\ \text{MeV}\,\text{fm}}{r_n}\,Z_1 Z_2
\approx 0.55\ \text{MeV}
$$

for $r_n \approx 2.6\ \text{fm}$. Compare this with the thermal energy at the center
of the Sun, $T_c \approx 1.5\times 10^7\ \text{K}$:

$$
kT_c = 1.38\times 10^{-23}\ \text{J K}^{-1}\times 1.5\times 10^7\ \text{K}
= 2.1\times 10^{-16}\ \text{J} \approx 1.3\ \text{keV}.
$$

The ratio $E_{\rm C}/kT_c \approx 430$ means a classical particle reaches the top of
the barrier with a Boltzmann probability of order $e^{-430}$, far too small to power
any star. The resolution is that the particles need not clear the barrier; they
tunnel through it.[^co-rates]

$$
% caption: The Coulomb barrier V(r) rises as 1/r until the nuclear well at r_n; a
% pair with energy E far below the barrier top E_C tunnels from the classical turning
% point r_c inward, its wavefunction decaying exponentially across the barrier.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% axes
\draw[->, black] (0,0) -- (9.0,0) node[right, black!70] {separation r};
\draw[->, black] (0,-2.2) -- (0,4.0) node[above, black!70] {potential V};
% Coulomb 1/r barrier from r_n outward
\draw[acc, very thick] (1.4,3.5) .. controls (2.2,2.1) and (3.4,1.2) .. (5.0,0.75)
  .. controls (6.6,0.45) and (7.6,0.35) .. (8.6,0.3);
% nuclear well inside r_n
\draw[acc, very thick] (1.4,3.5) -- (1.4,-1.9) -- (0.55,-1.9) -- (0.55,0);
% energy level E
\draw[black, densely dashed] (0,0.9) -- (8.4,0.9);
\node[black!70, anchor=west] at (8.0,1.15) {E};
% turning point marker
\fill[black] (3.75,0.9) circle (1.6pt);
\node[black!70, anchor=south] at (4.0,0.95) {turning point};
% barrier-top label
\node[acc, anchor=west] at (1.55,3.5) {barrier top};
% contact radius
\draw[black, densely dotted] (1.4,0) -- (1.4,3.5);
\node[black, anchor=north] at (1.4,-0.05) {$r_n$};
\node[black, anchor=north] at (3.75,-0.05) {$r_c$};
% decaying wavefunction across the barrier (schematic)
\draw[black, thin]
  (3.75,2.7) sin (3.55,2.85) cos (3.35,2.7) sin (3.15,2.55) cos (2.95,2.7)
  -- (2.95,2.7);
\draw[black, thin] (1.4,2.55)
  .. controls (2.2,2.62) and (3.0,2.58) .. (3.75,2.7);
\node[black, anchor=south] at (2.6,2.85) {tunneling};
\end{tikzpicture}
$$

## Barrier penetration and the Gamow factor

The probability that a pair of relative energy $E$ tunnels through the Coulomb
barrier follows from the WKB approximation. The transmission coefficient is the
exponential of minus twice the action accumulated between the classical turning
point $r_c = Z_1 Z_2 e^2/(4\pi\epsilon_0 E)$ and the nuclear radius,

$$
P \approx \exp\!\left[-\frac{2}{\hbar}\int_{r_n}^{r_c}
\sqrt{2\mu\bigl(V(r) - E\bigr)}\ \d r\right],
$$

with $\mu = m_1 m_2/(m_1 + m_2)$ the reduced mass. For $E \ll E_{\rm C}$ the lower
limit can be set to zero and the integral evaluated in closed form, giving the
**Gamow factor**

$$
P \approx e^{-2\pi\eta}, \qquad
\eta = \frac{Z_1 Z_2 e^2}{4\pi\epsilon_0 \hbar v}
= \alpha\, Z_1 Z_2 \frac{c}{v},
$$

where $\eta$ is the Sommerfeld parameter, $v = \sqrt{2E/\mu}$ is the relative
velocity, and $\alpha = e^2/(4\pi\epsilon_0\hbar c) \approx 1/137$ is the
fine-structure constant. Writing the exponent in terms of energy,

$$
2\pi\eta = \sqrt{\frac{E_{\rm G}}{E}},
\qquad
E_{\rm G} = 2\mu c^2\,(\pi\alpha Z_1 Z_2)^2,
$$

defines the **Gamow energy** $E_{\rm G}$, a fixed property of the reacting pair. For
two protons $\mu c^2 = 469\ \text{MeV}$ and $E_{\rm G} = 0.49\ \text{MeV}$; the
tunneling probability $e^{-\sqrt{E_{\rm G}/E}}$ falls precipitously as $E$
decreases, since low-energy pairs face a wider barrier.

## The reaction rate and the astrophysical S-factor

The cross section for a nuclear reaction factorizes into three pieces: a geometric
term set by the de Broglie wavelength, $\pi\hbar^2/(2\mu E) \propto 1/E$; the tunneling
probability $e^{-\sqrt{E_{\rm G}/E}}$; and the intrinsically nuclear probability
that, once the nuclei are in contact, they actually react. Collecting the smooth
nuclear physics into a single function defines the **astrophysical S-factor**,

$$
\sigma(E) = \frac{S(E)}{E}\,\exp\!\left(-\sqrt{\frac{E_{\rm G}}{E}}\right).
$$

For a non-resonant reaction $S(E)$ varies only weakly with energy, because the two
rapidly varying factors — the $1/E$ geometric term and the exponential barrier
penetration — have been divided out. This is what makes $S(E)$ the quantity
laboratories measure and tabulate: the cross section itself is unmeasurably small at
stellar energies, but $S(E)$ can be measured at accessible energies of hundreds of
keV and extrapolated smoothly down to the few-keV stellar window.

$$
% caption: The cross section (left ordinate) plunges by many orders of magnitude
% toward low energy, while the S-factor (right ordinate) is nearly flat; laboratory
% data lie above the stellar Gamow window and are extrapolated down along S(E).
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% Gamow window band (background)
\fill[acc!10] (1.6,0) rectangle (2.6,4.2);
\draw[black, densely dotted] (1.6,0) -- (1.6,4.2);
\draw[black, densely dotted] (2.6,0) -- (2.6,4.2);
\draw[->, black] (0,0) -- (9.2,0) node[right, black!70] {energy E};
\draw[->, black] (0,0) -- (0,4.4) node[above, black!70] {log cross section};
% steeply falling cross section (toward low E, i.e. small x, it plunges)
\draw[acc, very thick] (0.6,0.3) .. controls (1.8,0.55) and (2.8,1.6) .. (4.0,2.7)
  .. controls (5.4,3.6) and (6.8,4.0) .. (8.6,4.15);
\node[acc, anchor=north west] at (5.5,3.25) {cross section};
% nearly flat S-factor (dashed) at a mid level
\draw[black, densely dashed, thick] (0.6,2.15) .. controls (3.0,2.05) and (6.0,2.1) .. (8.6,2.15);
\node[black!70, anchor=south west] at (0.7,2.25) {S-factor};
\node[black, anchor=south, align=center] at (2.1,4.2) {stellar\\window};
% lab data region
\draw[black, densely dotted] (6.2,0) -- (6.2,4.2);
\node[black, anchor=south] at (7.4,3.4) {lab data};
\end{tikzpicture}
$$

Averaging the cross section over the Maxwell–Boltzmann distribution of relative
velocities at temperature $T$ gives the thermally averaged rate coefficient. In
terms of energy,

$$
\langle\sigma v\rangle
= \left(\frac{8}{\pi\mu}\right)^{1/2}\frac{1}{(kT)^{3/2}}
\int_0^\infty S(E)\,
\exp\!\left(-\frac{E}{kT} - \sqrt{\frac{E_{\rm G}}{E}}\right)\d E .
$$

The number of reactions per unit volume per unit time between species 1 and 2 is
then

$$
r_{12} = \frac{n_1 n_2}{1 + \delta_{12}}\,\langle\sigma v\rangle,
$$

the factor $1/(1+\delta_{12})$ preventing an identical pair from being counted
twice. The energy generation rate per unit mass is
$\epsilon = r_{12}\,Q/\rho$, with $Q$ the energy released per reaction.

## The Gamow peak

The integrand carries two competing exponentials. The Maxwell–Boltzmann factor
$e^{-E/kT}$ falls with increasing energy: high-energy pairs are rare. The tunneling
factor $e^{-\sqrt{E_{\rm G}/E}}$ rises with increasing energy: high-energy pairs
penetrate the barrier more easily. Their product is sharply peaked at an
intermediate energy, the **Gamow peak**, that carries almost the entire integral.
Maximizing the exponent $f(E) = -E/kT - \sqrt{E_{\rm G}/E}$ gives

$$
f'(E) = -\frac{1}{kT} + \frac{1}{2}\frac{\sqrt{E_{\rm G}}}{E^{3/2}} = 0
\quad\Longrightarrow\quad
E_0 = \left(\frac{E_{\rm G}\,(kT)^2}{4}\right)^{1/3}.
$$

The peak energy scales as $E_0 \propto (Z_1 Z_2)^{2/3}\,T^{2/3}$: hotter gas and
lower charges shift the window upward, but only weakly. For proton–proton fusion at
the solar center, $E_0 \approx 5.9\ \text{keV}$ — well above the mean thermal energy
$kT_c \approx 1.3\ \text{keV}$, yet a hundred times below the barrier top. Fusion is
carried by the rare pairs in the high-energy tail that also happen to tunnel, not by
typical particles.

$$
% caption: The Maxwell tail (falling) and the tunneling probability (rising) overlap
% only in a narrow band; their product, the Gamow peak, sits at E_0, far above kT and
% far below the Coulomb barrier, and supplies nearly all fusion reactions.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (9.4,0) node[right, black!70] {energy E};
\draw[->, black] (0,0) -- (0,4.3) node[above, black!70] {relative probability};
% Maxwell-Boltzmann factor: falling exponential
\draw[black, thick] (0.3,4.0) .. controls (1.4,1.9) and (2.6,0.9) .. (4.2,0.45)
  .. controls (6.0,0.2) and (7.6,0.12) .. (8.8,0.1);
\node[black, anchor=west] at (5.2,0.6) {Maxwell factor};
% tunneling factor: rising from zero
\draw[black, densely dashed, thick] (0.5,0.05) .. controls (1.6,0.15) and (2.6,0.7) .. (4.0,1.6)
  .. controls (5.6,2.6) and (7.4,3.5) .. (8.8,4.0);
\node[black, anchor=south east] at (8.6,3.6) {tunneling factor};
% Gamow peak (product) as a shaded bump
\draw[acc, very thick] (1.3,0.05) .. controls (2.1,0.3) and (2.5,2.1) .. (3.0,2.15)
  .. controls (3.5,2.2) and (3.9,0.4) .. (4.9,0.05);
\fill[acc!12] (1.3,0.05) .. controls (2.1,0.3) and (2.5,2.1) .. (3.0,2.15)
  .. controls (3.5,2.2) and (3.9,0.4) .. (4.9,0.05) -- cycle;
\node[acc, anchor=south] at (3.0,2.2) {Gamow peak};
% E_0 marker
\draw[black, densely dotted] (3.0,0) -- (3.0,2.15);
\node[black, anchor=north] at (3.0,-0.05) {$E_0$};
\node[black, anchor=north] at (0.55,-0.05) {kT};
\end{tikzpicture}
$$

Expanding $f(E)$ to second order about $E_0$ replaces the integrand by a Gaussian.
The maximum value of the exponent is

$$
f(E_0) = -\frac{E_0}{kT} - \frac{2E_0}{kT} = -\frac{3E_0}{kT} \equiv -\tau,
\qquad
\tau = \frac{3E_0}{kT} = 3\left(\frac{E_{\rm G}}{4kT}\right)^{1/3},
$$

and the $1/e$ full width of the Gaussian is

$$
\Delta = \frac{4}{\sqrt{3}}\sqrt{E_0\,kT}.
$$

The peak is both narrow and far out on the thermal tail: for solar proton–proton
fusion $\tau \approx 13.7$ and $\Delta \approx 5.9\ \text{keV}$, comparable to $E_0$
itself. Carrying out the Gaussian integral gives the compact estimate

$$
\langle\sigma v\rangle \approx
\left(\frac{2}{\mu}\right)^{1/2}\frac{\Delta}{(kT)^{3/2}}\,S(E_0)\,e^{-\tau}
\;\propto\; T^{-2/3}\,\exp\!\left[-\left(\frac{E_{\rm G}}{4kT}\right)^{1/3}\!\cdot 3\right].
$$

> **Worked example.** For proton–proton fusion at $T_6 = 15$ (in units of
> $10^6\ \text{K}$), $E_{\rm G} = 0.49\ \text{MeV}$ and $kT = 1.29\ \text{keV}$ give
> $E_0 = \left[E_{\rm G}(kT)^2/4\right]^{1/3} = 5.9\ \text{keV}$. Then
> $\tau = 3E_0/kT = 13.7$ and $\Delta = (4/\sqrt3)\sqrt{E_0 kT} = 5.9\ \text{keV}$.
> The peak lies at $E_0/E_{\rm C} \approx 0.011$: fusion samples barely one percent
> of the way up the Coulomb barrier, and the reacting pairs sit at
> $E_0/kT \approx 4.6$ times the mean thermal energy.

## Temperature sensitivity

Because $\tau \propto T^{-1/3}$, the rate coefficient depends on temperature through
the combination $T^{-2/3}e^{-\tau}$, which is extraordinarily steep. Writing the
rate as a local power law $\langle\sigma v\rangle \propto T^{\nu}$ near a reference
temperature,

$$
\nu = \frac{\d\ln\langle\sigma v\rangle}{\d\ln T}
= -\frac{2}{3} + \frac{\tau}{3} = \frac{\tau - 2}{3}.
$$

The exponent $\nu$ is large precisely because $\tau$ is large, and $\tau$ grows with
the charge product $Z_1 Z_2$ through $E_{\rm G} \propto (Z_1 Z_2)^2$. Reactions
between more highly charged nuclei have taller barriers, sit at higher Gamow
energies relative to $kT$, and switch on far more abruptly with temperature. This
single relation orders the burning stages of stellar evolution.

| Reaction | $Z_1 Z_2$ | reference $T$ | $\tau$ | $\nu = (\tau-2)/3$ |
| --- | --- | --- | --- | --- |
| $p + p$ (pp chain) | $1$ | $1.5\times 10^7\ \text{K}$ | $13.7$ | $\approx 4$ |
| CNO cycle | $6$–$7$ | $1.5\times 10^7\ \text{K}$ | $\sim 62$ | $\approx 20$ |
| triple-$\alpha$ | $4$ (per step) | $10^8\ \text{K}$ | $\sim 124$ | $\approx 41$ |

$$
% caption: Energy generation rate versus temperature on log axes; the local slope is
% the exponent nu, which grows with the charge product: pp near 4, CNO near 20, and
% triple-alpha near 41, so each stage dominates over a narrow temperature range.
\begin{tikzpicture}[scale=1.0, font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->, black] (0,0) -- (8.6,0) node[right, black!70] {log temperature};
\draw[->, black] (0,0) -- (0,5.2) node[above, black!70] {log energy rate};
% pp: gentle slope
\draw[acc, very thick] (0.6,0.8) -- (7.6,2.4);
\node[acc, anchor=south] at (6.6,2.1) {pp, slope 4};
% CNO: steep
\draw[black, very thick] (2.4,0.5) -- (5.4,4.6);
\node[black, anchor=south east, rotate=54] at (4.1,2.7) {CNO, slope 20};
% triple-alpha: near vertical
\draw[black, densely dashed, very thick] (5.0,0.5) -- (6.2,4.8);
\node[black, anchor=west] at (6.05,4.4) {triple alpha};
\node[black, anchor=west] at (6.05,4.05) {slope 41};
\end{tikzpicture}
$$

## Resonances

The smooth S-factor picture holds only when no nuclear energy level of the compound
system sits near the Gamow window. When one does, the cross section is enhanced by
many orders of magnitude over a narrow band, and $S(E)$ develops a sharp
Breit–Wigner peak. For an isolated narrow resonance at energy $E_r$ with resonance
strength $\omega\gamma$ (a product of the statistical spin factor and the partial
widths), the thermal rate becomes

$$
\langle\sigma v\rangle
= \left(\frac{2\pi}{\mu kT}\right)^{3/2}\hbar^2\,(\omega\gamma)\,
e^{-E_r/kT},
$$

so the rate is set almost entirely by the Boltzmann population at the resonance
energy rather than by an integral over the Gamow peak. A single favorably placed
level can raise a reaction rate enough to control an entire burning stage. The
triple-$\alpha$ process depends on exactly such a coincidence, the Hoyle resonance in
carbon-12, treated in [helium burning and the triple-alpha
process](/astrophysics-cosmology/nuclear-astrophysics/helium-burning-and-the-triple-alpha-process).
Whether a reaction is resonant or not, the Gamow analysis fixes the temperature
window in which it can operate at all.

The Gamow peak turns the microscopic Coulomb barrier into the macroscopic ordering
of stellar burning. Hydrogen fuses at $10^7\ \text{K}$ because its barrier is
lowest; helium waits for $10^8\ \text{K}$; carbon and heavier fuels need $10^9\
\text{K}$ and above. The next lesson works out the two hydrogen-burning networks,
[the pp chains and the CNO
cycle](/astrophysics-cosmology/nuclear-astrophysics/hydrogen-burning-pp-chains-and-cno),
whose crossover in temperature is a direct consequence of the $\nu = (\tau-2)/3$
scaling derived here.

[^co-rates]: Carroll & Ostlie, §10.3 — Nuclear Reaction Rates: the Coulomb barrier, quantum tunneling and the Gamow factor, the astrophysical S-factor, the Gamow peak, and the temperature dependence of the energy generation rate.