Some useful parameter estimates about inertial confinement fusion can be obtained by an analysis of selected particle kinetics and energy transfer processes. Consider, therefore, a spherically symmetric pressure wave converging towards the pellet centre. Suppose that ignition and bum conditions are attained when the radius of the compressed pellet is Rb and hold over a bum time tb, during which the pressure generated by the shock waves and the heating due to fusion reactions causes the pellet to expand to an extent where the density, and hence the fusion reaction rate, have decreased to insignificant levels.

At any time during this nuclear bum period, we take the total number of ions in the burning part of the pellet to be Nb. This total ion population decreases with time because of fusion reactions which occur at the rate

Here Rfu is the fusion reaction rate density with each fusion event destroying two ions; the integration is over the pellet bum volume Vb. Substituting for the fusion reaction rate density involving deuterium and tritium gives

= ~2j Nd(t)N,(t)<OV>dt d3r

Vb

The variables can now be separated and integrated to give

Nu л 1 4

= <U’5)

L nU>) 2 о

Here Ni>0 is the ion density at the beginning of the bum, t = 0, and Nj f is the fuel ion density at the end of the bum, t = xb. Integration and rearrangement of the terms yields for the bum time

where < OV >dt denotes the fusion reactivity parameter averaged over the bum period according to

____________ у ч

<(Tv>dt=—<CTv>dt(t)dt. (11.7)

TbJ0

It will be useful to introduce the symbol fb for the fraction of fuel burned during Tb,

Ni,0 — Nij

Thus

and we also use

Pb ~ Ni, omi (119)

where рь is the pellet density during the bum, m,- is the average ion mass, and the mass contributions of the electrons have been neglected, attributable to щ. being three orders of magnitude smaller than m,. Substitution then yields the explicit expression for Ть, Eq.(l 1.6), as

where Vis is the speed at which the core mass moves outward. The corresponding kinetic energy is of the order of the ion thermal energy, Eq. (2.19c), so that we may use

1 2 . Jtn vdis ■

and therefore we derive here

This specifies the conditions on the pellet density and pellet radius at the beginning of the fusion bum that are required for a specified bum fraction with the reaction occurring at some average temperature. A useful numerical value for

this parameter is obtained by taking < av >dt = <ov>dt (T = 20 keV) and an estimate for vdls also at this temperature, yielding for a 50% bum fraction

PbRb>3 g-cm"1 • (11.16)

For Rb ~ 1 mm this demands a density pb = 30 g-cm3 and is indeed very high compared to d-t liquid density of pe ~ 0.2 g-cm’3. Thus, the compression of the
initial fuel pellet by a factor of about 103 to 104-relative to liquid density-appears to be necessary for a satisfactory bum.