## Resonance Escape Probability

The **fast fission factor** determined the number of fast neutrons produced by fissions at all energies, but in thermal reactors the bulk of fissions occurs **at thermal energies** and it is obvious these fission neutrons have to be **thermalized**. But during this thermalisation neutrons may collide not only with moderator nuclei, but also with fuel nuclei. Unfortunately, especially ^{238}U exhibits **resonance behaviour** between the fast region and the thermal region. While the neutrons are slowing down through **the resonance region** of ^{238}U, which extends from about **6 eV to 200 eV**, there is a chance that some neutrons will be captured. The probability that a neutron** will not be captured** in entire resonance region is called **the resonance escape probability**.

See also: Nuclear Resonance

**979**

**↓**

**p ~ 0.75**

**↓**

**734**

**The resonance escape probability**, symbolized by** p**, is the probability that a neutron will be slowed to thermal energy and will escape resonance capture. This probability is defined as the ratio of the number of neutrons that reach thermal energies to the number of fast neutrons that start to slow down.

**The neutron capture**is one of the possible absorption reactions that may occur. In fact, for

**non-fissionable nuclei**it is the only possible absorption reaction. Capture reactions result in the loss of a neutron coupled with the production of one or more gamma rays. This capture reaction is also referred to as a

**radiative capture**or

**(n, γ) reaction**, and its cross-section is denoted by

**σ**.

_{γ}**The radiative capture** is a reaction, in which the incident neutron is completely absorbed and compound nucleus is formed. The compound nucleus then** decays** to its ground state by gamma emission. This process can occur at all incident neutron energies, but the probability of the interaction strongly depends on the **incident neutron energy** and also on the **target energy** (temperature). In fact the energy in the center-of-mass system determines this probability.

See also: Radiative Capture

The compound nucleus is the intermediate state **formed in a compound nucleus reaction**. It is normally one of the excited states of the nucleus formed by the combination of the incident particle and target nucleus. If a target nucleus **X** is bombarded with particles **a**, it is sometimes observed that the ensuing nuclear reaction takes place with appreciable probability only if the energy of the particle **a** is in the neighborhood of certain definite energy values. These energy values are referred to as **resonance energies**. The compound nuclei of these certain energies are reffered to **as nuclear resonances**. Resonances are usually found only at relatively low energies of the projectile. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances between resonances and then no resonances can be observed. The narrowest resonances are usually the compound states of heavy nuclei (such as fissionable nuclei) and thermal neutrons (usually in (n,γ) capture reactions). The observation of resonances is by no means restricted to neutron nuclear reactions.

See also: Compound Nucleus Reactions

From the definition **p** is **always less than 1.0** when there is any amount of ^{238}U or ^{240}Pu present in the core, which means that resonance capture by these isotopes always removes some of the neutrons from the neutron flux.

It is obvious **the resonance escape probability** is strongly influenced by the **arrangement** and the **geometry** of the reactor core. In a **homogenous reactor core**, where fuel nuclei are surrounded by many of moderator nuclei, there is a significantly higher probability, that the resonance neutron will collide with the fuel nucleus. Therefore in homogenous reactor cores, the resonance escape probability is lower than in heterogeneous cores.

On the other hand, in a **heterogeneous reactor core**, all the fuel nuclei are in **fuel pellets** that are encapsulated within a fuel rod. This arrangement **increases the probability**, that the fast neutron will escape the fuel matrix. The neutron slows down in the moderator where there are no atoms of ^{238}U and ^{240}Pu present. Therefore in heterogeneous reactor cores, the resonance escape probability is significantly higher than in homogenous cores.

ξ_{CARBON} = 0.158N(**2MeV → 1eV**) = ln 2⋅10^{6}/ξ =14.5/0.158 = **92**

**The resonance escape probability** is for heterogeneous reactor cores about 0.75. But more than its own value, the ways, how can be this value changed **during reactor operation**, are crucial. The resonance capture, is the main phenomenon, which **contributes to the reactor stability** and makes the reactor core** inherently safe** and resistant to **prompt power changes** as in case of reactivity initiated accidents (RIA). This feature depends strongly on certain reactor design and also on certain fuel loading pattern. Therefore it must be verified during the reload safety assessment.

**The resonance escape probability for all resonances can be calculated according to following equation:**

The integral in this expression is called **the effective resonance integral** **Ι _{eff}.** In practical situations, this integral strongly depends on the geometry of the unit cell. The geometry of the core strongly influences the

**spatial and energy self-shielding**, that take place primarily in heterogeneous reactor cores. This phenomenon causes a significant increase in

**the resonance escape probability**(“p” from four factor formula) in comparison with homogeneous cores. Without the

**spatial self-shielding**provided by the separation of fuel and moderator, values of

**k**are possible with

_{eff}= 1**natural uranium fuel**only if

**heavy water**is used as the moderator. In the literature empirical relations have been developed for the effective resonance integral of the following form:

where **the resonance integrals** are in barns, rho is the material density in g/cm^{3}, and D is the fuel rod diameter in cm. This equation is valid for isolated rods with diameter higher than 0.2cm.

For tightly packed lattice in the fuel assembly self-shielding increases somewhat through what is called a **Dancoff correction**.

**elementary cell**of the fuel lattice (this cell contains fuel and moderator separated). In this elementary cell the neutron slowing down and thermalization problems can be treated with optical

**reflecting**or isotropic reflecting (white

**boundary condition**). However, even this is too complicated in the resonance energy range. In a closely packed lattice the in-current of resonance neutrons into the fuel is

**reduced**, as compared to the in-current into a single fuel rod in an infinite moderator, because of the

**shadowing effect**of adjacent rods.

As a first approximation in the resonance self-shielding calculations, a single fuel lump (usually a fuel rod) in an infinite moderator is considered, and the presence of other fuel rods is taken into account by applying a certain correction, generally called the **Dancoff correction**.

From the equation it is obvious, **the resonance escape probability** is dependent also on **the moderator-to-fuel ratio** – **N _{M} / N_{F}** (the term before I

_{eff}). With the change of the

**moderator-to-fuel ratio**changes also the neutron flux spectrum in the reactor core. Most of light water reactors are designed as so called

**undermoderated**and the

**neutron flux spectrum**is slightly

**harder**(the moderation is slightly insufficient) than in an optimum case. But this design provides important safety feature. This feature will be discussed in following section.

## Main operational changes, that affect this factor:

**Change in the moderator temperature.**It is known,**the resonance escape probability**is dependent also on the**moderator-to-fuel ratio**. As the**moderator temperature**increases the ratio of the moderating atoms (molecules of water) decreases as a result of the**thermal expansion of water**. Its density simply decreases. This, in turn, causes a**hardening of neutron spectrum**in the reactor core resulting in**higher resonance absorption**(lower p). Decreasing density of the moderator causes that neutrons stay at a higher energy for a longer period, which increases the probability of non-fission capture of these neutrons. This process is one of two processes, which determine the**moderator temperature coefficient (MTC)**. The second process is connected with the leakage probability of the neutrons. The moderator temperature coefficient must be for most PWRs**negative**, which improves the**reactor stability**, because a reactor core heating causes a**negative reactivity insertion**.

**Change in the fuel temperature**. The second operational change, which affects**the resonance escape probability**, is connected with the phenomenon usually known as**the Doppler broadening**. The effect of the Doppler broadening 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**. The Doppler broadening with the process of**self-shielding**causes, that the Doppler coefficient (or the**fuel temperature coefficient**) is for all power reactors always**negative**. Therefore an increase in the fuel temperature**promptly**causes an increase in the**resonance integral**(I_{eff}), which, in turn, causes a negative reactivity insertion.**It is of the highest importance in the reactor safety**.

See also: Doppler Broadening

See also: Self-shielding

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