Doppler Broadening of Resonances

Doppler effect

Doppler effect improves reactor stability. Broadened resonance (heating of a fuel) results in a higher probability of absorbtion, thus causes negative reactivity insertion (reduction of reactor power).

In general, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of kinetic energies of molecules or atoms. In reactor physics a particular case of this phenomenon is the thermal Doppler broadening of the resonance capture cross sections of the fertile material (e.g. 238U or 240Pu) caused by thermal motion of target nuclei in the nuclear fuel.

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.

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:

doppler coefficient

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.

Self-shielding - neutron cross-section

An increase in temperature from T1 to T2 causes the broadening of spectral lines of resonances. Although the area under the resonance remains the same, the broadening of spectral lines causes an increase in neutron flux in the fuel φf(E), which in turn increases the absorption as the temperature increases.

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 αdepends 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.

Doppler Broadening
Source: J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
Doppler Broadening
Comparison of resonance capture cross-sections of 238U and 240Pu.
In LWRs, the fuel temperature influences the rate of nuclear breeding (the breeding ratio). In principle, the increase is the fuel temperature affects primarily the resonance escape probability, which is connected with the phenomenon usually known as the Doppler broadening (primarily 238U).  The impact of this resonance capture reaction  on the neutron balance is evident, the neutron is lost and this effect decreases the effective multiplication factor. On the other hand, this capture leads to formation of unstable nuclei with higher neutron number. Such unstable nuclei undergo a nuclear decay, which may lead to formation of another fissile nuclei. This process is also referred to as the nuclear transmutation and is responsible for new fuel breeding in nuclear reactors.

From this point of view, the neutron is utilized much more effectively when captured by 238U than when captured by absorbator, because the effective multiplication factor must in every state equal to 1 (Note that in PWRs the boric acid is used to compensate an excess of reactivity of reactor core along thefuel cycle). In other words it is better to capture the neutron (lower an excess of reactivity) by 238U, rather than by 10B nuclei.

At HFP (hot full power) state, the fuel temperature is directly given by:

  • Local linear heat rate (W/cm), which is given by neutron flux distribution. See also: Power Distribution
  • Fuel-cladding gap. 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.
  • Core inlet temperature. Core inlet temperature is directly given by system parameters in steam generators. When steam generators are operated at approximately 6.0MPa, it means the saturation temperature is equal to 275.6 °C. Since there must be always ΔT (~15°C) between the primary circuit and the secondary circuit, the reactor coolant (in the cold leg)have about 290.6°C (at HFP) at the inlet of the core. As the system pressure increases, the core inlet temperature must also increase. This increase causes slight increase in fuel temperature.

It can be summarized, the fuel breeding is lower, when the reactor is operated at lower power levels. Note that, in order to lower the reactor power, additional absorbators must be inserted inside the core.  The fuel breeding is higher  (e.g. 1 EFPD surplus), when the core inlet temperature of the reactor coolant is higher (e.g. 1°C for 300 EFPDs). It must be added, the inlet temperature is limited and it cannot be changed arbitrarily.

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.

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