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By Charles Rhodes, P.Eng., Ph.D.

The molecular binding energy Ebl of liquid lead is about:
Ebl = 2.23 ev / atom.
When the atomic kinetic energy exceeds the molecular binding energy the lead starts to adopt gas like characteristics.
Ebl = (Ml / 2) Vl^2 or
Vl = [2 Ebl / Ml]^0.5
= [(2 X 2.23 eV X 1.602 X 10^-19 J / eV) / (208 X 1.67 X 10^-27 kg)]^0.5
= [.1437 X 10^4 m / s]
= 1437 m / s

At this negative radial velocity the lead molecular kinetic energy exceeds the lead molecular binding energy. Thus near the center of the compression sphere the lead atoms act like a gas instead of like a liquid. Rapid heating of the lead atoms near the lead wall further enhances this gas like behavior.

In a collision between a tritium ion and a stationary lead atom the amount of momentum transferred to the lead atom is given by:
2 Mt Vtw = Ml Vl

The kinetic energy transferred to a lead atom at the liquid lead face is:
(Ml / 2) Vl^2 = (Ml / 2) (2 Mt Vtw / Ml)^2
= (4 Mt / Ml)(Mt / 2) (Vtw^2)

Recall that:
Vtw = (2^0.5) Vi
= (2^0.5) (2 Ekd / Mt)^0.5

Thus the kinetic energy transferred to a lead atom at the liquid lead face is:
(Ml / 2) Vl^2 = (4 Mt / Ml)(Mt / 2) (Vtw^2)
= (4 Mt / Ml)(Mt / 2) [(2^0.5) (2 Ekd / Mt)^0.5]^2
= (4 Mt / Ml)(Mt / 2) [(2) (2 Ekd / Mt)]
= 8 (Mt / Ml) Ekd

To ensure no sputtering this energy has to be less than the binding energy per atom of liquid lead. However, that simply is not the case. Hence the system must function well in the presence of heavy sputtering.

Consider a tritium ion that knocks off a lead atom, travels across the plasma the distance 2 Ri at velocity Vi and then knocks off another lead atom. In order for the lead vapor to not accumulate the first lead atom must be re-absorbed in time interval:
(2 Ri / Vi).
In that time interval the first lead atom travels at most:
[2 (Eia + (4 / 208) Eta) / Ml]^0.5 (2 Ri / Vi)

In the same time interval the liquid lead wall has velocity profile:
Vi = Via + (dVia / dT) (T - Ta)
and a distance profile:
Via (4 Ri / Vi) + (dvia / dT)(1 / 2) (4 Ri / Vi)^2

Equate these two expressions and solve for dvia / dT

However, such reabsorption is only possible if the average lead surface temperature is sufficiently low. Clearly we must be concerned about the rate of heat conduction away from the surface of the liquid lead wall as compared to the rate of heat absorption by the liquid lead wall due to hydrogen isotope ion impacts.

The MTF and the PIF concepts both rely on liquid lead continuing to provide adiabatic compression far into their evaporation /sputtering region. However, any accumulation of lead vapor in the hydrogen isotopes would undermine the process both by increasing the number of contained particles and by slowing the rate of hydrogen isotope fusion.

Thus liquid lead evaporation / sputtering is potentially a major constraint on the MTF and PIF processes. During the adiabatic compression kinetic energy flows from the liquid lead wall to the plasma. However, the liquid lead near the liquid lead wall also gets very hot. Towards the end of the compression the lead wall may in fact be a compressed lead vapor wall.

This web page last updated January 26, 2015.

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