Thermonuclear reaction rates and Gamow windows

Assume nonrelativistic, nondegenerate nuclei in local thermal equilibrium at one temperature. Separating center-of-mass and relative motion gives a Maxwell distribution with reduced mass $\mu=m_i m_j/(m_i+m_j)$. The thermal reactivity has dimensions of volume per time. In cgs nuclear-network work it is often expressed in cubic centimetres per second; tabulations may instead multiply it by Avogadro's constant. A tabulated molar reactivity cannot be inserted into a number-density formula without undoing that change of convention.

Distinct reacting species supply one pair for every choice of one nucleus of each type. Identical species require division by two because exchanging the labels does not create a new pair. This changes the event rate, while each identical-particle event still destroys two reactants. Thus a factor removed from pair counting reappears in the reactant depletion equation. Finite systems use the exact unordered count $N(N-1)/2$; the density-squared approximation assumes a macroscopic population and neglects correlations beyond the adopted plasma model.

For a smooth, nonresonant charged-particle reaction, write the cross section as an astrophysical S factor divided by energy times a Coulomb penetration exponential. The S factor removes the dominant barrier dependence and has units of energy times area. It is not automatically constant: thresholds, subthreshold structure, and resonances can produce important variation. If it changes slowly across the contributing energies, the exponential competition between the Maxwell tail and barrier suppression locates the dominant collision-energy region.

Differentiate the sum of the two suppression exponents with respect to energy and set it to zero. This gives the conventional Gamow-peak energy; a quadratic expansion around that stationary point gives a Gaussian width. The resulting window is an approximation for this particular integrand, not a universal detector energy band. Its accuracy improves when the peak is well separated from threshold and sufficiently sharp. Neutron reactions have no entrance-channel Coulomb barrier, so applying the same charged-particle window formula to them is inappropriate.

An isolated narrow resonance contributes through its resonance energy and strength rather than through a constant-S approximation. The narrow formula requires the width to be small relative to the thermal variation scale and other factors, with no troublesome overlap or threshold distortion. Broad or interfering resonances require integrating an appropriate cross-section model. The location of the smooth Gamow peak alone cannot decide whether a sharp resonance matters, because a localized enhancement can dominate the weighted integral.

Plasma screening changes the relation between an isolated-nucleus cross section and the stellar rate. In a weakly coupled plasma, a small approximately constant barrier lowering can motivate a multiplicative enhancement exponential. That expression is not a general formula for strongly coupled or degenerate matter, nor is laboratory electron screening automatically the same effect. A useful prediction therefore declares whether its input is a bare-nucleus or stellar reactivity and whether internal nuclear excited states are thermally populated and included.