12 C γ 3α Reaction Rate Calculation

Astrophysical Nuclear Reaction Rate Analysis

GK

T₉ = Temperature in 10⁹ Kelvin

Dimensionless (0.0 to 1.0)

Results

Photodisintegration Decay Constant (λ_γ):

Nuclear Mean Lifetime (τ):

Energy Yield (Q-Value): -7.272 MeV

Understanding the 12C(γ, 3α) Reaction

The 12C(γ, 3α) reaction, also known as the photodisintegration of Carbon-12, is the inverse of the famous triple-alpha process. In massive stars, during advanced stages of burning (such as Neon burning), high-energy gamma rays possess sufficient energy to break apart Carbon-12 nuclei into three alpha particles (Helium-4 nuclei).

The Physics Behind the Calculation

This calculator uses the principle of detailed balance to relate the photodisintegration rate to the triple-alpha production rate. The calculation focuses on the transition through the 7.654 MeV 0⁺ state of ¹²C, known as the Hoyle State.

• Q-Value: The reaction is endothermic with a Q-value of -7.272 MeV. This energy must be supplied by the incident photon.
• Temperature Sensitivity: The rate is extremely sensitive to temperature (T₉), as the photon energy distribution must overlap with the energy threshold.
• Astrophysical Context: This reaction becomes significant at temperatures above 1.5 GK, where it competes with other carbon-consuming processes.

Formula and Methodology

The decay constant λ_γ is calculated using the following approximation derived from the NACRE compilation and detailed balance:

λ_γ ≈ 1.63 × 10³⁵ · T₉^-3/2 · exp(-84.42 / T₉) · N_A²<σv>_3α

Note: Our simplified calculator uses a direct parameterization for the decay constant in s^-1 to provide a quick estimation for stellar modeling students and nuclear physicists.

Example Calculation

If a star reaches a temperature of 2.0 GK (T₉ = 2.0), the thermal energy is sufficient to drive significant photodisintegration. At this temperature, the decay constant λ_γ will be approximately 1.54 × 10^-4 s^-1, meaning a Carbon-12 nucleus has a mean lifetime of roughly 1.8 hours before being dismantled into alpha particles.

function calculateReactionRate() { var t9 = parseFloat(document.getElementById('tempT9').value); var x12 = parseFloat(document.getElementById('abundance').value); if (isNaN(t9) || t9 84.42 / T9 is the exponent factor // Factor derived from statistical weights and partition functions var exponent = -84.42 / t9; // Simplified fit for the photodisintegration rate lambda (s^-1) // This captures the exponential suppression and the T^3/2 pre-factor // associated with the reverse triple-alpha process var preFactor = 2.15e14 * Math.pow(t9, 1.5); var lambda = preFactor * Math.exp(exponent); // Refinement for lower temperatures where the rate is negligible if (t9 3.154e7) { lifetimeDisplay.innerHTML = (tau / 3.154e7).toFixed(2) + " Years"; } else if (tau > 3600) { lifetimeDisplay.innerHTML = (tau / 3600).toFixed(2) + " Hours"; } else if (tau > 60) { lifetimeDisplay.innerHTML = (tau / 60).toFixed(2) + " Minutes"; } else { lifetimeDisplay.innerHTML = tau.toFixed(4) + " Seconds"; } } }