For deuterium-deuterium fusion with constant and

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Nuclear Physics · Fission, fusion and nuclear energy

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

Problem

For deuterium-deuterium fusion with constant

S = 125\,\mathrm{keV\,b}

and

E_G = 986\,\mathrm{keV}

, the rate coefficient at

k_BT = 10\,\mathrm{keV}

has already been evaluated:

E_0 = 29.1\,\mathrm{keV}

\tau \equiv 3E_0/k_BT = 8.73

\Delta_{\rm G} = 39\,\mathrm{keV}

, and the saddle-point (Laplace) estimate is

\langle\sigma v\rangle_{\rm saddle} = 1.10\times10^{-24}\,\mathrm{m^3\,s^{-1}}

. At fixed

S

, the closed form scales as

\langle\sigma v\rangle \propto (k_BT)^{-3/2}\,\Delta_{\rm G}\,e^{-\tau}

. ((a)) Compute

E_0

\tau

and

\Delta_{\rm G}

k_BT = 15\,\mathrm{keV}

. ((b)) Using the scaling relation above and the

10\,\mathrm{keV}

values as reference, find

\langle\sigma v\rangle_{\rm saddle}

15\,\mathrm{keV}

. ((c)) The leading correction to the saddle point is a factor

1 + 5/(12\tau)

. Use it to estimate the numerically integrated value at

15\,\mathrm{keV}

. ((d)) Find the effective temperature exponent

n = \mathrm{d}\ln\langle\sigma v\rangle/\mathrm{d}\ln T = (\tau-2)/3

15\,\mathrm{keV}