The aneutronic reaction has reduced mass and charge product

PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · Nuclear reactions · Fission, fusion and nuclear energy

The Nucleus Resolved, Ch Fusion: the cross section computed and the power balance made honest

Problem

The aneutronic reaction

d + {}^3\mathrm{He} \to {}^4\mathrm{He} + p

has reduced mass

\mu_{\rm red}c^2 = 1124.5\,\mathrm{MeV}

and charge product

Z_1Z_2 = 2

. Two measured cross sections are

\sigma(100\,\mathrm{keV}) = 71\,\mathrm{mb}

\quad \sigma(200\,\mathrm{keV}) = 266\,\mathrm{mb}

. No resonance is reported below

250\,\mathrm{keV}

. ((a)) Compute the Gamow energy

E_G

for this pair. ((b)) Convert both measured cross sections into the astrophysical

S

factor and show they agree to within rounding. ((c)) Use the flat

S

to extrapolate the cross section down to

30\,\mathrm{keV}

. ((d)) For a plasma at

k_BT = 20\,\mathrm{keV}

, compute the Gamow peak energy

E_0

and the

1/e

full width

\Delta_{\rm G}

, and decide whether the saddle-point (closed-form) rate-coefficient formula may be used, given the range over which

S

has been shown to be flat.