As the fuel temperature increases the resonance self-shielding in the fuel isotopes decreases. The resulting change in reactivity is called the Doppler effect because it is due to the dependence of the effective energy of a neutron on the relative motion of the nucleus with which it interacts. The decrease in self-shielding results in an increase in all the cross-sections of the fuel isotopes at energies below about 20 keV.

Although all the cross-sections increase, including scattering, the most significant changes are in £c and £ f. In all breeder reactors the increase in capture in 238U dominates over the increase in fission in 239Pu and the Doppler effect is negative, but the same is not always true for consumer reactors.

The Doppler effect is not independent of temperature. The dependence of J (equation 1.39) on T is complex. If fi/f0, where f0 is the peak value of f, is large (corresponding to a resonance with a low peak at higher energy) then J a T-1/2 and dJ/dT a T-3/2, but if fi/f0 is small J a T and dJ/dT is constant. Thus the Doppler effect varies in a different way with temperature for different resonances, but the total coefficient, involving a weighted sum of dJ/dT for all the resonances, varies roughly as 1 / T. As a result it is convenient to define a “Doppler constant”, — T dp/dT.

Figure 1.27 shows the contribution to the Doppler reactivity effect from various isotopes in different neutron energy groups for a breeder reactor with low enrichment. The most important effects are around

1 keV at which energy the resonances are not resolved. The effect of the iron resonance at 1 keV can be seen. This is the only resonance in a non-fissile or fertile isotope that makes any significant contribution to the Doppler effect.

In an operating reactor the fuel is not all at the same temperature, and as the Doppler coefficient is a function of temperature there is a question of the correct average temperature to use. In a cylindrical fuel element with constant thermal conductivity the volume-weighted mean temperature T is halfway between the central and surface tem­peratures. Because the Doppler coefficient is higher at low temper­atures the effective temperature Teff for the Doppler effect is slightly lower than T, but the difference is small. If the temperatures in the fuel element are in equilibrium, it can be shown that

(1.47)

where T0 is the fuel surface temperature. Temperature variation across the core can be taken into account by weighting changes with fgf* as indicated in equation 1.25.