A candidate fusion pair carries charges

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

Z_1=1

Z_2=2

and reduced mass energy

\mu_{\rm red}c^2 = 1120\,\mathrm{MeV}

. Tabulated

S

-factor measurements for this pair are

S(50\,\mathrm{keV})=42\,\mathrm{keV\,b}

S(150\,\mathrm{keV})=68\,\mathrm{keV\,b}

S(250\,\mathrm{keV})=130\,\mathrm{keV\,b}

and

S(400\,\mathrm{keV})=240\,\mathrm{keV\,b}

. A separate raw cross-section measurement at

50\,\mathrm{keV}

gives

\sigma(50\,\mathrm{keV}) = 5.1\times10^{-5}\,\mathrm{b}

. ((a)) Using

E_G = 2\mu_{\rm red}c^2(\pi\alpha_{\rm fs}Z_1Z_2)^2

with

\alpha_{\rm fs} = 1/137.036

, calculate

E_G

for this pair. ((b)) Convert the raw cross section at

50\,\mathrm{keV}

into an

S

-factor value and check it against the tabulated entry. ((c)) At

k_BT = 50\,\mathrm{keV}

, calculate the Gamow peak energy

E_0

and the

1/e

full width

\Delta_{\rm G}

. ((d)) Using the tabulated

S(E)

values that fall inside the window

E_0\pm\Delta_{\rm G}/2

, decide whether the saddle-point (constant-

S

) rate-coefficient formula can be trusted here, and state what should be done instead if it cannot.