# What is Fuel Temperature Coefficient – Doppler Coefficient – DTC – Definition

The fuel or doppler temperature coefficient – FTC or DTC is defined as the change in reactivity per degree change in the fuel temperature. Reactor Physics

## Fuel Temperature Coefficient – Doppler Coefficient

The fuel temperature coefficient – FTC or DTC is defined as the change in reactivity per degree change in the fuel temperature.

αf = dTf

It is expressed in units of pcm/°C or pcm/°F. The magnitude and sign (+ or -) of the fuel temperature coefficient is primarily a function of the fuel composition, especially the fuel enrichment. In power reactors, in which low enriched fuel (e.g.  PWRs and BWRs require 3% – 5% of 235U) is used, the Doppler coefficient is always negative. The value of the Doppler coefficient αdepends on the temperature of the fuel and also depends on the fuel burnup. In PWRs, the Doppler coefficient can range, for example, from -5 pcm/°C to -2 pcm/°C.

This coefficient is of the highest importance in the reactor stability. The fuel temperature coefficient is generally considered to be even more important than the moderator temperature coefficient (MTC). Especially in case of all reactivity initiated accidents (RIA), the fuel temperature coefficient will be the first and the most important feedback, that will compensate the inserted positive reactivity. The time for heat to be transferred to the moderator is usually measured in seconds, while the fuel temperature coefficient is effective almost instantaneously.

Therefore this coefficient is also called the prompt temperature coefficient because it causes an immediate response on changes in fuel temperature.

Since the phenomenon, which is dominantly responsible for the fuel temperature coefficient, is the Doppler broadening of resonances, it is also known as the Doppler coefficient. The Doppler broadening of resonances is very important phenomenon, which improves reactor stability.

The total amount of reactivity, which is inserted to a reactor core by a specific change in the fuel temperature, is usually known as the doppler defect and is defined as:

dρ = α . dT

Example: doppler defect
Assume, that the Doppler coefficient in a reactor is -3 pcm/°C.

Calculate the reactivity defect that results from the fuel temperature increase of 100°C.

Solution:
dρ = α . dT = -3 * 100 = -300 pcm

The reactivity addition due to the fuel temperature increase is negative about -0.5 \$ (for reactor core with βeff = 0.006).

Capture cross-section of 238U and 240Pu

## Theory of Doppler Coefficient

↑Tf ⇒ ↓keff = η.ε.  ↓p  .f.Pf  .Pt

Change in the fuel temperature affects primarily the resonance escape probability  via the thermal Doppler broadening of the resonance capture cross sections of the fertile material (e.g. 238U or240Pu) caused by thermal motion of target nuclei in the nuclear fuel.

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 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 Coefficient in Fast Reactors

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.

Example: Reactor stability
The negative MTC and the negative fuel temperature coefficient (DTC) ensure stable power operation of the reactor core. If the temperature of the moderator is increased, negative reactivity is added to the core. This negative reactivity causes reactor power to decrease, which acts against any further increase in temperature or power. As the thermal power decreases, the power coefficient acts against this decrease and the reactor returns to the critical condition. The reactor power stabilize itself. This effect is shown on the picture below. Let assume all the changes are initiated by the changes in the core inlet temperature.

## Fuel Temperature Coefficient and Fuel Design

In general, the fuel temperature coefficient is primarily a function of the fuel enrichment. There are two phenomena associated with Doppler effect. It results in increased neutron capture by fuel nuclei (both fissile and fissionable) in the resonance region, on the other hand it results in increased neutron production from fissile nuclei. Therefore DTC may be also positive. A net positive Doppler coefficient requires higher enrichments of fissile nuclei. Some calculations show, that enrichments higher than 30% are associated with slightly positive DTC, but this enrichment results in a harder neutron spectrum. Low enriched uranium oxide fuels  usually provide a large negative DTC, because of the elastic scattering reaction of oxygen nuclei (soften spectrum). Another way to produce more negative DTC is to soften the neutron spectrum by addition of moderator nuclei directly into the fuel matrix. For example, the TRIGA reactor uses uranium zirconium hydride (UZrH) fuel, which has a large negative fuel temperature coefficient. The rise in temperature of the hydride increases the probability that a thermal neutron in the fuel element will gain energy from an excited state of an oscillating hydrogen atom in the lattice. As the neutrons gain energy from the ZrH, the thermal neutron spectrum in the fuel element shifts to a higher average energy (i.e., the spectrum is hardened). This spectrum hardening is used differently to produce the negative temperature coefficient.

In the United States, the Nuclear Regulatory Commission will not license a reactor unless αPrompt (FTC) is negative. All licensed reactors are thereby assured of being inherently stable. This requirement is followed by many other countries.

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.