At low temperature heat is conducted mainly by the diffusion of phon­ons through the crystals. As the temperature increases the density of phonons increases and, as the behaviour of the crystal lattice is slightly
nonlinear, the probability of interaction between phonons increases, their mean-free-path decreases and the conductivity decreases. At higher temperature still electrons become more mobile and make a growing contribution to the conductivity.

For mixed oxide the conductivity is affected by the chemical com­position, which can be described as (PuaU(1-a))O2+x, where a rep­resents the plutonium fraction and x, which can be either positive or negative, the departure from the stoichiometric oxygen content. It is usually in the range -0.03 < x <0 .03.

The effect of stoichiometry on conductivity is as follows. If there is no plutonium (a = 0), below about 1400 °C Kf decreases steadily as x increases, but above 1600 °C Kf is a maximum for x = 0 and is reduced by departure from the stoichiometric composition in either direction. This seems to be because at high temperature an excess or a deficiency of oxygen, not exceeding a few per cent, is accommodated by vacancies or interstitials in the lattice. These defects reduce the phonon mean — free-path and hence the conductivity. At low temperatures, however, excess uranium (x < 0) is precipitated as the metal and the resulting free electrons increase the conductivity.

The situation is quite different if plutonium is present. Uranium and plutonium have nearly the same ionic radius and stoichiomet­ric UO2 and PuO2 form solid solutions in whatever ratio they are mixed (for all values of a). But if the mixed oxide is not stoichiometric (x < 0) the excess metal is accommodated by means of lattice defects for x > -0.02 and by precipitation of Pu2O3 for x < -0.02. Thus for mixed oxide with a Pu fraction typical of a large fast reactor (a ~ 0.2) the conductivity decreases as the composition departs from stoi­chiometric in either direction (x < 0 and x > 0) at all temperatures. The effect is quite large: for x = +/-0.02, Kf is reduced to about 75% of its value for x = 0. For x = 0 the conductivity decreases slowly as a increases, so that in the range 400-1200 °C, for a = 0.2 Kf is about 13% smaller than for a = 0.

This picture is confused in practice by structural effects that can be much more important than the effects of composition. Principal among these is the effect of porosity, which reduces the effective conductivity. As explained later (section 2.3.2) the fuel may be manufactured with 10% or more porosity (i. e. 10% of the overall volume of the fuel may be occupied by voids). Porosity of 10% can reduce the thermal effective conductivity by as much as 25%.

Estimation of the effect of porosity in an operating fuel element is made very difficult because, as explained in section 2.4.1, the pores move under the influence of the temperature distribution, so that the conductivity changes with time and in different ways in different parts of the fuel. Further confusion is provided by the cracks in the fuel that are formed under the influence of differential thermal expansion and that have a large but unpredictable effect on conduction. Finally fission products, which themselves move through the fuel (section 2.4.6), and changes in crystal structure (section 2.4.1) affect conductivity in a manner that is not accurately known.

The maximum linear heat rating q is set by the requirement that the fuel should not melt. The melting points of UO2 and PuO2 are about 2850 °C and 2430 °C respectively, and the solidus temperature for a = 0.2 is about 2730 °C. Figure 2.1 shows the variation of Kf (T) and f Kf (T)dT with T for unirradiated stoichiometric (U0.8Pu0.2)O2 from 500 °C to the melting point. The integral from a fuel surface temperature Tfs of 1000 °C up to the melting point is about 4.9 kWm-1 for stoichiometric oxide.

Equation 2.3 shows that f Kf (T)dT — 4.9 kWm-1 implies q ~ 62 kWm-1 if the fuel has no central hole. But if there is a hole with radius 20% of the fuel radius (a = .04) equation 2.5 shows that q can be increased to about 71 kWm-1. In practice, because of the uncertainty of the effects of cracking, porosity, stoichiometry, changes in composition with burnup and the conductance between fuel and cladding (section 2.2.3), q is limited to about 50 kWm-1.