Thermonuclear Reaction Rates and the Gamow Peak

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 $1 fm$ , 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 π ϵ _{0}} \frac{Z _{1} Z _{2} e ^{2}}{r},

which rises until the nuclei touch at the contact radius $r_{n} \approx 1.4 (A_{1}^{1/3} + A_{2}^{1/3}) fm$ , 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_{C} = \frac{1}{4 π ϵ _{0}} \frac{Z _{1} Z _{2} e ^{2}}{r _{n}} \approx \frac{1.44 MeV fm}{r _{n}} Z_{1} Z_{2} \approx 0.55 MeV

for $r_{n} \approx 2.6 fm$ . Compare this with the thermal energy at the center of the Sun, $T_{c} \approx 1.5 \times 1 0^{7} K$ :

k T_{c} = 1.38 \times 1 0^{- 23} J K^{- 1} \times 1.5 \times 1 0^{7} K = 2.1 \times 1 0^{- 16} J \approx 1.3 keV .

The ratio $E_{C} / k T_{c} \approx 430$ 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.¹

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.

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 $r_{c} = Z_{1} Z_{2} e^{2} / (4 π ϵ_{0} E)$ and the nuclear radius,

P \approx exp - \frac{2}{ℏ} r_{n} \int r_{c} 2 μ (V (r) - E) d r,

with $μ = m_{1} m_{2} / (m_{1} + m_{2})$ the reduced mass. For $E ≪ E_{C}$ the lower limit can be set to zero and the integral evaluated in closed form, giving the Gamow factor

P \approx e^{- 2 π η}, η = \frac{Z _{1} Z _{2} e ^{2}}{4 π ϵ _{0} ℏ v} = α Z_{1} Z_{2} \frac{c}{v},

where $η$ is the Sommerfeld parameter, $v = 2 E / μ$ is the relative velocity, and $α = e^{2} / (4 π ϵ_{0} ℏ c) \approx 1/137$ is the fine-structure constant. Writing the exponent in terms of energy,

2 π η = \frac{E _{G}}{E}, E_{G} = 2 μ c^{2} (π α Z_{1} Z_{2})^{2},

defines the Gamow energy $E_{G}$ , a fixed property of the reacting pair. For two protons $μ c^{2} = 469 MeV$ and $E_{G} = 0.49 MeV$ ; the tunneling probability $e^{- E_{G} / E}$ 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, $π ℏ^{2} / (2 μ E) \propto 1/ E$ ; the tunneling probability $e^{- E_{G} / E}$ ; 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,

σ (E) = \frac{S ( E )}{E} exp (- \frac{E _{G}}{E}) .

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.

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).

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,

⟨ σ v ⟩ = (\frac{8}{π μ})^{1/2} \frac{1}{( k T ) ^{3/2}} 0 \int \infty S (E) exp (- \frac{E}{k T} - \frac{E _{G}}{E}) 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 + δ _{12}} ⟨ σ v ⟩,

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

The Gamow peak

The integrand carries two competing exponentials. The Maxwell–Boltzmann factor $e^{- E / k T}$ falls with increasing energy: high-energy pairs are rare. The tunneling factor $e^{- E_{G} / E}$ 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 $f (E) = - E / k T - E_{G} / E$ gives

f^{'} (E) = - \frac{1}{k T} + \frac{1}{2} \frac{E _{G}}{E ^{3/2}} = 0 ⟹ E_{0} = (\frac{E _{G} ( k T ) ^{2}}{4})^{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, $E_{0} \approx 5.9 keV$ — well above the mean thermal energy $k T_{c} \approx 1.3 keV$ , 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.

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.

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}}{k T} - \frac{2 E _{0}}{k T} = - \frac{3 E _{0}}{k T} \equiv - τ, τ = \frac{3 E _{0}}{k T} = 3 (\frac{E _{G}}{4 k T})^{1/3},

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

Δ = \frac{4}{3} E_{0} k T .

The peak is both narrow and far out on the thermal tail: for solar proton–proton fusion $τ \approx 13.7$ and $Δ \approx 5.9 keV$ , comparable to $E_{0}$ itself. Carrying out the Gaussian integral gives the compact estimate

⟨ σ v ⟩ \approx (\frac{2}{μ})^{1/2} \frac{Δ}{( k T ) ^{3/2}} S (E_{0}) e^{- τ} \propto T^{- 2/3} exp [- (\frac{E _{G}}{4 k T})^{1/3} \cdot 3] .

Temperature sensitivity

Because $τ \propto T^{- 1/3}$ , the rate coefficient depends on temperature through the combination $T^{- 2/3} e^{- τ}$ , which is extraordinarily steep. Writing the rate as a local power law $⟨ σ v ⟩ \propto T^{ν}$ near a reference temperature,

ν = \frac{d ln ⟨ σ v ⟩}{d ln T} = - \frac{2}{3} + \frac{τ}{3} = \frac{τ - 2}{3} .

The exponent $ν$ is large precisely because $τ$ is large, and $τ$ grows with the charge product $Z_{1} Z_{2}$ through $E_{G} \propto (Z_{1} Z_{2})^{2}$ . Reactions between more highly charged nuclei have taller barriers, sit at higher Gamow energies relative to $k T$ , and switch on far more abruptly with temperature. This single relation orders the burning stages of stellar evolution.

Reaction	$Z_{1} Z_{2}$	reference $T$	$τ$	$ν = (τ - 2) /3$
$p + p$ (pp chain)	$1$	$1.5 \times 1 0^{7} K$	$13.7$	$\approx 4$
CNO cycle	$6$ – $7$	$1.5 \times 1 0^{7} K$	$\sim 62$	$\approx 20$
triple- $α$	$4$ (per step)	$1 0^{8} K$	$\sim 124$	$\approx 41$

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.

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 $ω γ$ (a product of the statistical spin factor and the partial widths), the thermal rate becomes

⟨ σ v ⟩ = (\frac{2 π}{μ k T})^{3/2} ℏ^{2} (ω γ) e^{- E_{r} / k T},

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- $α$ process depends on exactly such a coincidence, the Hoyle resonance in carbon-12, treated in 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 $1 0^{7} K$ because its barrier is lowest; helium waits for $1 0^{8} K$ ; carbon and heavier fuels need $1 0^{9} K$ and above. The next lesson works out the two hydrogen-burning networks, the pp chains and the CNO cycle, whose crossover in temperature is a direct consequence of the $ν = (τ - 2) /3$ scaling derived here.

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. ↩