The reaction rate per unit volume for two distinct nuclear species labelled 1 and 2

PHYSICS · Subjective · Fermi University Physics · Nuclear Physics · Nuclear astrophysics and applications · Nuclear astrophysics and applications

Stars from the Inside, Ch Thermonuclear energy generation: reaction rates, burning stages and the elements they make

Problem

The reaction rate per unit volume for two distinct nuclear species labelled 1 and 2, present with number densities

n_1

and

n_2

in a non-degenerate gas at temperature

T

, is

r_{12} = n_1n_2\langle\sigma v\rangle

. In the centre-of-mass frame the relative speed

v

of a reacting pair, of reduced mass

\mu_{red}

, follows the Maxwell-Boltzmann distribution

f(v)\,\mathrm{d}v = 4\pi v^2\left(\frac{\mu_{red}}{2\pi k_BT}\right)^{3/2} e^{-\mu_{red}v^2/(2k_BT)}\,\mathrm{d}v,

normalised so that

\int_0^\infty f(v)\,\mathrm{d}v = 1

. The cross section is written through the astrophysical S factor as

\sigma(E) = \frac{S(E)}{E}\,e^{-\sqrt{E_G/E}},

where

E = \tfrac12\mu_{red}v^2

is the relative kinetic energy and

E_G

is the Gamow energy of the pair. (a) Starting from

\langle\sigma v\rangle = \int_0^\infty \sigma(v)\,v\,f(v)\,\mathrm{d}v

, change the integration variable from

v

E

and show that

\langle\sigma v\rangle = \sqrt{\frac{8}{\pi\mu_{red}}}\,(k_BT)^{-3/2}\int_0^\infty \sigma(E)\,E\,e^{-E/k_BT}\,\mathrm{d}E.

(b) Insert the S-factor form of

\sigma(E)

into this result to obtain the reaction-rate integral

\langle\sigma v\rangle = \sqrt{\frac{8}{\pi\mu_{red}}}\,(k_BT)^{-3/2}\int_0^\infty S(E)\,e^{-E/k_BT-\sqrt{E_G/E}}\,\mathrm{d}E.