Fission Reactors Physics 1

Chapter eutronics

A nuclear engineer has to be able to monitor the fission chain reaction of a reactor by following the production of neutrons from nuclear reactions and their decreasing through capture and/or leakage. In addition, he has to know how to extract the energy produced inside the core; here, knowledge of thermodynamics, hydrodynamics, and heat transfer are crucial, as well as activation by radiation, in order to guarantee safety to the personnel and the public. Thus, a nuclear reactor ends up being a multi-physics system, quite tough to manage. We shall start this course, aimed at the comprehension of these subjects, by studying the phenomena involving neutrons inside the reactor: the so-called neutronics.

As a brief introduction, notice that the typical neutron density in a fission reactor reaches the value of 10¹³ neutrons per cm³, while the nuclei density is of the order of 10²² nuclei per cm³. Such numbers do not allow for a "point-to-point" description of the state inside the reactor core but are quite well served by a statistical approach. Thus, the most suitable way to describe neutron behavior is to use probabilities of reaction. In particular, we can distinguish between two types of reaction:

• Spontaneous disintegration of nuclei, in which we classify α, β, and γ decay, and that depends just on the nature of the disintegrating nuclei;
• Induced reaction, produced from the collision of particles, such as nucleus-nucleus or particle-nucleus, which depends not only on the nature of the particles but also on their dynamical properties.

Radioactive decay

Aiming to give a description useful to the study of the effects of radiation in a reactor, we will now discuss radioactive decay and activation of materials. The different types of decays are all described by the empirical formula in which the rate of decay is assumed linearly proportional to the number of decaying atoms, through a constant value, λ, called the decay constant:

dN/dt = -λ N(t)

The solution of this equation is quite trivial, given by a simple decreasing exponential:

N(t) = N(0) e^-λt

From this expression, we can define the rate of decay, or activity, simply by:

A(t) = λ N(0) e^-λt

This quantity is measured in decays per second, or becquerels (Bq). Some other quantities can be defined noting that the decay process is statistical; hence, we introduce the mean lifetime of a nucleus as:

t = 1/λ

and the radioactive half-life, as the time needed to half the decaying nuclei:

T_1/2 = ln(2)/λ

However, in many applications of our concern, there may be a source of radioactive nuclide. If the rate of such production is indicated by R(t), we can write the decay law as:

dN/dt = -λ N(t) + R(t)

This problem is somehow more complicated, especially if R is a function of time. If we can consider it constant, the trend of N in time is the following: an initial start with a linear evolution, with slope equal to R, followed by an asymptotic increase to a value N* = R/λ. In formula, for N(0) = 0:

N(t) = R/λ [1 - e^-λt]

We could say that this behavior corresponds to some kind of storage mechanism, such as the one seen in a typical RC circuit: a source provides energy (R(t)), a capacitor (dN/dt) stores it while a resistor (-λN) dissipates it. This relation can, of course, be used to study a decay chain, that is the decay of a nuclide into another and the subsequent decay of the latter:

λ_X → λ_Y → λ_Z ...

This chain can be described by a set of differential equations of the type above, considering the source contributions of the decaying nuclide:

dN_X/dt = -λ_X N_X + R_X, dN_Y/dt = -λ_Y N_Y + λ_X N_X + R_Y, ...

Example of application to estimate the moderator activation by the core radiation: Let us consider the following model for the circulation of moderator through the reactor core: the fluid is subjected to a constant rate of activation R during the time in which it passes through the core; otherwise, it simply decays with decay constant λ. We want to evaluate the activity induced after one loop, after m loops, and the maximum activity reached. The model we use is quite simple and goes as follows:

• From the entrance in the reactor, at time t₁, to its exit, the moderator decay is given by dN/dt = -λ N(t) + R.
• Solved by N(t) = N(0) e^-λt + R/λ [1 - e^-λt].
• After the transition, the moderator decays, with the same evolution as the former point, but with a different initial number of radioactive nuclei.

This repeats itself for each loop. For the first loop, with N(0) = 0, we obtain that the number of active nuclei at the exit from the core (time t₁) is:

N(t₁) = R/λ [1 - e^-λt₁]

while at the end of the loop (time t₂) we are left with:

N(t₂) = R/λ [1 - e^-λt₁] e^{-λ(t₂-t₁)}

At the m-th loop, it can be shown that the number of active nuclei at the exit from the core is given by:

N(t_m) = R/λ [1 - e^-λt₁] Σ_k=0^m-1 e^{-λk(t₁+t₂)}

Evaluating the series, we obtain:

N(t_m) = R/λ [1 - e^-λt₁] (1 - e^{-mλ(t₁+t₂)})/(1 - e^{-λ(t₁+t₂)})

The activity is simply given, in both cases, by a multiplication by λ. Its maximum value is reached for an infinite number of loops and is equal to:

A_max(t) = R [1 - e^-λt₁]/(1 - e^{-λ(t₁+t₂)})

Nuclear reactions

If we are ultimately interested in the study of the reaction of neutrons with other particles, particularly with nuclei, we have to study these reactions first. Primarily, we are interested in knowing the energy absorbed or released by a nuclear reaction. Such interaction can be written in a very generic way as:

a + b → c + d

in which a and b are the reagents and c and d the products. Using the relativistic relation between energy and mass, we can write that the energy of the reaction, also called Q-value, is equal to:

Q = [(M_a + M_b) - (M_c + M_d)]c²

If Q > 0, then the reaction is said to be exothermic and it will release energy. If Q < 0, then we say the reaction is endothermic, with absorption of energy, supplied by the reagent (as kinetic energy). As said, reactions between neutrons and nuclei are of most interest for our study; they include:

• Nuclear fission: n + X₁^A → X₂^A1 + X₃^A2 + neutrons + 200 MeV
• Radiative capture: n + X₀^A → (X₀^A1)* → X₀^A+1 + γ
• Scattering: n + X₀^A → n + X₀^A (elastic); n + X₀^A → n + X₀^A* + γ (inelastic)

In which a neutron simply scatters off the nucleus, retaining all of its energy (elastic), or transferring it to the nucleus (inelastic). The importance of fission is obvious, while radiative capture and scattering play a major role in the neutron economy and hence in the chain reaction.

Microscopic cross sections

The probability that a nuclear reaction will occur is characterized by a parameter called nuclear cross section. This quantity can be operationally defined by considering a monoenergetic collimated beam of particles (say neutrons), of intensity I, impinging on a target with nuclear density N (nuclei per unit area). We assume this target to be a monoatomic layer, so that no nuclei in it will be shielded by the others. In this case, we would expect the reaction rate to be proportional to either the intensity of the beam and the density of the target; calling this constant σ, we thus write:

R = σ I N_A

From this equation, we can see that the units of σ shall be those of an area. If the incident neutrons and target nuclei could be visualized as classical particles (or hard spheres), σ would naturally correspond to the cross-sectional area presented by each target nucleus to the beam. Hence σ is known as the microscopic cross-section, characterizing the probability of interaction. Considering, in this approach, the cross-sectional area of an atom of typical radius (10^-12 cm), it will be equal to 10^-24 cm²; such order of magnitude is usually called barn: 1b = 10^-24 cm². However, this quantity can be much larger or much smaller than the cross-sectional area of an atom due to resonance effects, which, in turn, are a consequence of the quantum mechanical nature of the neutron and the nucleus. A slightly more formal definition thus can be:

σ = R/I N_A

Thus far, we have been discussing the concept of a nuclear cross-section in a rather abstract sense without actually specifying the type of reaction we have in mind. Such cross-sections can be used to characterize any type of nuclear reaction. We can define a microscopic cross-section for each type of neutron-nuclear reaction and each type of nuclide, of which the total cross-section is the sum. Since it is merely a matter of categorization, we shall only report a scheme, illustrating some of the reactions we are interested in. Moreover, even if we didn't mention it, it is certainly conceivable that such cross-section will vary, depending on the incident neutron energy and direction.

Macroscopic cross section

In the previous discussion, we considered a beam of neutrons incident on a very thin target. However, in reality, a certain amount of thickness is present, such that the nuclei deeper within the target would tend to be shielded from the incident beam by the superficial ones. Considering a differential thickness between x and x+dx, we can obtain an expression for the beam attenuation inside the target. Hence, since dx is a thin layer, we know the relation between the interaction rate and the thickness:

dR = σ_t I dN_A = σ_t I N_t dx

Such rate will be equal to the decrease in beam intensity between x and x+dx:

-dI(x) = σ_t I N_t dx

and dividing by dx we find a differential equation for the intensity I(x):

dI/dx = -σ_t I N_t

This simple equation, with I₀ the intensity at x = 0, is solved by an exponential attenuation of the beam:

I(x) = I₀ e^-N_tσ_tx

The product of the atomic number density N_t and the microscopic cross-section is often denoted as the macroscopic cross-section:

Σ_t = N_t σ_t

Looking at its units of measure, cm^-1, it is natural to interpret such quantity as the probability per unit path length traveled that the neutron will undergo a reaction. With this interpretation, the mean free path (mfp) a neutron travels before interacting is simply given by:

x̄ = 1/Σ_t

In a similar fashion, calling v the neutron speed, the frequency with which reactions occur is given by the product Σ_t v. This quantity is usually referred to as the collision frequency for the neutron in the sample. Its reciprocal, 1/(Σ_t v), is therefore interpretable as the mean time between neutron reactions.

It should be noted that, even if one can formally define the macroscopic cross-section for specific reactions, our discussion of neutron penetration into a thick target applies only to the total macroscopic cross-section. We can calculate these specific reaction probabilities only after a more complete consideration of neutron transport in materials.

The concept of a macroscopic cross-section can also be generalized to homogeneous mixtures of different nuclides. For example, if we have a homogeneous mixture of three different species of nuclide, X, Y, and Z, with respective atomic number densities N_x, N_y, and N_z, then the total macroscopic cross-section characterizing the mixture is given by:

Σ_t = N_X σ_tX + N_Y σ_tY + N_Z σ_tZ

where σ_tX is the microscopic total cross-section for nuclide X, and so on.

The macroscopic cross-section can depend on additional variables, other than the energy and direction of the incident neutrons. For example, suppose that the target material does not have a uniform composition. Then the number density N will depend on the position r in the sample, and hence the macroscopic cross-sections themselves will be space-dependent. In a similar manner, the number density might depend on time - suppose, for example, that the nuclide of interest was unstable. Therefore, in the most general case, we would write:

Σ(r, E, t) = N(r, t) σ(E)

Characteristics of neutron-nuclear cross section

Before considering in better detail the various types of neutron-nuclear reactions, it is useful to have a brief discussion of some of the relevant physics underlying the behavior of these cross-sections. There are two aspects involved in the analysis of neutron cross-sections: (a) the kinematics of two-particle collisions and (b) the dynamics of nuclear reactions.

The kinematics of two-body collision should be familiar and will be briefly recalled in Appendix A. The dynamics of nuclear reactions is concerned with the fundamental physical mechanisms involved in such collision events. The two mechanisms of most interest in nuclear reactor applications are those of potential scattering, in which the neutron merely bounces off the force field of the nucleus without actually penetrating the nuclear surface, and compound nucleus formation, in which the incident neutron is absorbed by the nucleus to form a new nucleus of mass number A+1 which then decays by emitting a γ ray, a neutron, or perhaps fissioning. The first type of collision event is very similar to that which would occur between two hard spheres, and the cross-section for such a reaction is essentially just the geometrical cross-section of the nucleus. Potential scattering cross-sections are characterized by a rather flat energy dependence from about 1eV up to the MeV range.

Compound nucleus formation occurs in many neutron-nuclear reactions of interest to the reactor engineer, including fission, radiative capture, and certain types of scattering. That such a mechanism must be involved in these reactions can be inferred from the relatively long times. It would take a slow neutron some 10^-17 s to cross the nucleus, neutron-nuclear reactions such as fission occur on a time scale of some 10^-14 s — or 1000 transit times. The long lifetime of the compound nucleus implies that the disintegration process is essentially independent of the original mode of formation. This process actually corresponds to a so-called resonance reaction, in which the incident neutron energy matches one of the energy levels in the compound nucleus.

Radiative capture reactions are quite significant for reactor analysis since they remove neutrons from the chain reaction; in fact, they see the formation of a compound nucleus, which subsequently decays by emitting a cascade of high-energy gammas. For this reason, the functional dependence of the capture cross-section on the neutron kinetic energy E exhibits a resonance behavior at those energies at which the center of mass energy E_c plus the neutron binding energy E_b match an energy level of the compound nucleus.

For resonances (or energy levels) which are spaced widely apart, it is possible to describe the energy dependence of the absorption cross-section by a very simple expression, known as the Breit-Wigner single-level resonance formula:

σ(E) = σ₀ E^1/2Γ_γ/( (E - E₀)² + Γ_c²/4 )

where E₀ is the energy at which resonance occurs, Γ_c is the so-called total line width of the resonance that characterizes the width of the energy level, while Γ_γ is the radiative line width essentially characterizing the probability that the compound nucleus will decay via gamma emission; σ₀ is the total cross-section at the resonance energy.

It should be noted that since resonance absorption is primarily of importance in heavy nuclei, one can usually approximate E_c ~ E. For low energies E << E₀, the cross-section behaves as essentially E^-1/2 or 1/v. For large energies E >> E₀, the cross-section drops off quite rapidly as E^-5/2. It is also important to note that such absorption cross-sections are largest at low energies. The energy levels in heavy nuclei become relatively more closely spaced at higher energies. Indeed for energies above roughly 1 keV in heavy nuclei such as U-238, the absorption resonances become so closely spaced that they cannot be resolved by experimental measurements. The treatment of neutron absorption in such unresolved resonances is a very difficult but important task in nuclear reactor analysis.

Nuclear fission is another of those reactions that proceeds by compound nucleus formation. We might again expect the cross-section for nuclear fission to exhibit a resonance structure very similar to that characterizing radiative capture. This is certainly the case for fission cross-sections characterizing nuclei such as U-233, U-235, and Pu-239. However, the fission cross-sections for other heavy nuclei such as Th-232 and U-238 exhibit a somewhat different structure in that they are essentially zero until the incident neutron energy exceeds a threshold of roughly a MeV.

Inelastic scattering in an inelastic scattering reaction, the incident neutron is first absorbed by the nucleus to form a compound nucleus. This nucleus then subsequently decays by reemitting a neutron. However, the final nucleus is left in an excited state. Such reactions usually occur only for relatively high neutron energies.