The temperature coefficient aTp at fuel temperature TF is considered. The Doppler effect is shown through the resonance escape probability p and the fuel expansion effect is shown through f and p.

(1) Fuel temperature coefficient of resonance escape probability The resonance escape probability can be given by [14, 15]

(1.58)

where Nf and VF are the atomic density and volume fraction of the fuel, and Zm^sm and VM are the moderating power and volume fraction of the moderator. I is the resonance integral for the normalized moderated neutron flux which behaves as 1/E beyond the resonance, given by the following.

I =f <J„F (E) ф(Е)clE

An experimental formula [16] for the temperature dependence I of is known as

I ( Tf)=I(300K) [ 1+іЗ’Ш-М)]

/ (300K)=11.6+22.8 77- MF

238U02 : /Ґ’=61Х 10-4+47X 10-4-^7“

MF

232Th02 : /?"=97 X10-4+120 X 10-4

where TF is the fuel temperature (unit: K), and SF and MF are, respectively, the surface area (unit: cm2) and mass (unit: g) of the fuel element. Thus, the temperature dependence of I indeed shows the Doppler effect. That is, since the vibration of the nuclei increases with the fuel temperature, the relative velocity of the interacting neutron to the nucleus changes and hence the neutron energy range for the resonance reaction is broadened. Then, the Doppler effect mitigates the depression of the neutron flux due to large resonance absorption cross sections of a large amount of nuclides in the reactor (the energy self-shielding effect) and increases the resonance absorption reaction.

The temperature coefficient aTFDoppler) of the resonance escape probability p due to the Doppler effect can be obtained by differentiating I with TF as the following.

(1.61)

Since p < 1, ln(l/p) > 0 and a^Doppler) becomes negative. It can be noted that the magnitude of the absolute value is about 10_4-10_5^k/k/K.

Next, the effect of the fuel expansion is considered assuming the ratio of the fuel volume fraction is constant. Suppose that the fuel will axially expand causing a decrease in the fuel atomic density. The temperature coefficient of the atomic density can be represented by the coefficient of linear expansion as Eq. (1.62).

(1.62)

The linear expansion coefficient of a material in the cube of length l is generally defined as

Equation (1.62) is derived from the relation between axial length and atomic density

The temperature coefficient of the resonance escape probability due to the fuel expansion can be given by taking note of the variation in fuel atomic density due to a temperature change and by using Eq. (1.62) as

Since ln(1/p) > 0, aTFExpansion) is positive. The magnitude of the absolute value is small, about 10_5dk/k/K for a solid fuel. This shows physically that the decrease in the density of resonance nuclides leads to the increase in the resonance escape probability.