(a) Use Laplace's method to obtain an approximation to , and state its fractional…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Plasma Physics · Fusion plasmas

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

Problem

(a) Use Laplace's method to obtain an approximation to

5! = \int_0^\infty x^5e^{-x}\,\mathrm{d}x

, and state its fractional error compared with the exact value

120

. Compare that error with the leading correction

1/(12n)

n=5

. (b) Starting from the plasma power balance

\frac{\mathrm{d}W}{\mathrm{d}t} = P_\alpha + P_{\rm ext} - P_{\rm brems} - P_{\rm imp} - \frac{W}{\tau_E},

with thermal energy density

W = 3nk_BT

and alpha heating

P_\alpha = \tfrac14 n^2\langle\sigma v\rangle E_\alpha

for an equimolar deuterium-tritium plasma, derive the ignition condition

n\tau_E \ge 12k_BT/(\langle\sigma v\rangle E_\alpha)

by setting

P_{\rm ext} = P_{\rm brems} = P_{\rm imp} = 0

and

\mathrm{d}W/\mathrm{d}t = 0

. (c) Evaluate the ignition value of

n\tau_E

k_BT = 20\,\mathrm{keV}

, given

\langle\sigma v\rangle = 4.2\times10^{-22}\,\mathrm{m^3\,s^{-1}}

and

E_\alpha = 3.5\,\mathrm{MeV}