See also: Nuclear Resonance
The Doppler broadening of resonances is very important phenomenon, which improves reactor stability, because it accounts for the dominant part of the fuel temperature coefficient (the change in reactivity per degree change in fuel temperature) in thermal reactors and makes a substantial contribution in fast reactors as well. This coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature. The prompt temperature coefficient of most thermal reactors is negative.
Theory of Doppler Effect
A negative fuel temperature coefficient is generally considered to be even more important than a negative moderator temperature coefficient. Especially in case of reactivity initiated accidents (RIA), the Doppler coefficient of reactivity would be the first and most important effect in the compensation of the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the Doppler coefficient is effective almost instantaneously. Therefore the moderator temperature cannot turn the power rise for several seconds, whereas the fuel temperature coefficient starts adding negative reactivity immediately. The fuel temperature coefficient αf may be defined as:
The Doppler effect arises from the dependence of neutron cross sections on the relative velocity between neutron and nucleus. The probability of the radiative capture depends on the center-of-mass energy, therefore depends on the kinetic energy of the incident neutron and the velocity of target nucleus. Target nuclei are themselves in continual motion owing to their thermal energy. As a result of these thermal motions neutrons impinging on a target appears to the nuclei in the target to have a continuous spread in energy. This, in turn, has an effect on the observed shape of resonance. Raising the temperature causes the nuclei to vibrate more rapidly within their lattice structures, effectively broadening the energy range of neutrons that may be resonantly absorbed in the fuel. The resonance becomes shorter and wider than when the nuclei are at rest.
Although the shape of a resonance changes with temperature, the total area under the resonance remains essentially constant. But this does not imply constant neutron absorbtion. Despite the constant area under resonance, the resonance integral, which determines the absorbtion, increases with increasing target temperature. The broadened resonances result in a larger percentage of neutrons having energies that are susceptible to capture in the fuel pellets. On the other hand with colder fuel, only neutrons very close to the resonance energy are absorbed.
Moreover, there is a further phenomenon closely connected with Doppler broadening. The vicinity of the resonance causes an increase in the neutron absorption probability, when a neutron have an energy near a resonance. This results in a reduction of the effective absorption per nucleus due to the depression of the energy dependent flux Φ(E) near the resonance in comparison to a flat flux. At energies just below the resonance, where Σa(E) becomes small again, the neutron flux reaches almost to the same value just above the resonance. This reduction in the energy dependent neutron flux near the resonance energy is known as energy self-shielding. These two phenomena provides negative reactivity feedback against fuel temperature increase.
Doppler Coefficient in Power Reactors
In power reactors, the Doppler coefficient is always negative. It is ensured by the fuel composition. In PWRs, the Doppler coefficient can range, for example, from -5 pcm/K to -2 pcm/K. The value of the Doppler coefficient αf depends on the temperature of the fuel and also depends on the fuel burnup.
- Fuel Temperature. The dependency on the fuel temperature is determined by the fact the rate of broadening diminishes at higher temperatures. Therefore the Doppler coefficient becomes less negative (but always remains negative) as the reactor core heats up.
- Fuel-cladding gap. There is also one very important phenomenon, which influences the fuel temperature. As the fuel burnup increases the fuel-cladding gap reduces. This reduction is caused by the swelling of the fuel pellets and cladding creep. Fuel pellets swelling occurs because fission gases cause the pellet to swell resulting in a larger volume of the pellet. At the same time, the cladding is distorted by outside pressure (known as the cladding creep). These two effects result in direct fuel-cladding contact (e.g. at burnup of 25 GWd/tU). The direct fuel-cladding contact causes a significant reduction in fuel temperature profile, because the overall thermal conductivity increases due to conductive heat transfer.
- Fuel Burnup. During fuel burnup the fertile materials (conversion of 238U to fissile 239Pu known as fuel breeding) partially replace fissile235U. The plutonium content raises with the fuel burnup. For example, at a burnup of 30 GWd/tU (gigawatt-days per metric ton of uranium), about 30% of the total energy released comes from bred plutonium. But with the 239Pu raises also the content of 240Pu, which has significantly higher resonance cross-sections than 238U. Therefore the Doppler coefficient becomes more negative as the 240Pu content raises.
In summary, it is difficult to define the depedency, because these effects have opposite directions.
In fast neutron reactors, the Doppler effect becomes less dominant (due to the minimisation of the neutron moderation), but strongly depends on the neutron spectrum (e.g. gas-cooled reactors have harder spectra than other fast reactors) and the type of the fuel matrix (metal fuel, ceramic fuel, etc.) In fast reactors, other effects, such the axial thermal expansion of fuel pellets or the radial thermal expansion of reactor core, may also contribute to the fuel temperature coefficient. In thermal reactors, these effects tend to be small relative to that of Doppler broadening.