$$\langle \sigma v \rangle_{3\alpha} \;\propto\; \frac{\Gamma_\alpha}{\Gamma}\,\Gamma_{\mathrm{rad}}\; \exp\!\left(-\frac{E_R}{k_{B}T}\right)$$

Equation 1 · Evolutionary Nuclear Physics

The Resonance That Selected Carbon, and the Universes That Couldn't

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The underlying reaction-rate dependence that ties the Hoyle-state’s radiative width directly to the resonance’s position above threshold is the same relation used throughout the modern literature on this reaction, of the schematic resonant form $\langle \sigma v \rangle_{3\alpha} \;\propto\; \frac{\Gamma_\alpha}{\Gamma}\,\Gamma_{\mathrm{rad}}\; \exp\!\left(-\frac{E_R}{k_{B}T}\right)$. where $\Gamma_\alpha$ and $\Gamma_{\mathrm{rad}}$ are the alpha-decay and radiative partial widths, $\Gamma$ their sum, and $E_R$ the resonance energy above the reacting threshold [ 5 ] . The exponential term is why a shift in the resonance’s position has outsized leverage on the reaction rate, and why the disputed radiative-width measurements of the previous section and the sensitivity calculations here are, in the end, two views of the…

Meanings in this article

• $\alpha$: the alpha-decay and radiative partial widths, $\Gamma$ their sum.
• $\Gamma_\alpha$: the alpha-decay and radiative partial widths, $\Gamma$ their sum.
• $\Gamma_{\mathrm{rad}}$: the alpha-decay and radiative partial widths, $\Gamma$ their sum.
• $E_R$: the resonance energy above the reacting threshold [ 5 ].

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