Introduction VIII. Neutron Moderation and the Six Factors 130 We continue our quest to calculate the multiplication factor (keff) and the neutron distribution (in position and energy) in nuclear reactors. In the previous chapter we found that we could solve a diffusion k-eigenvalue equation for the thermal flux in bare homogeneous reactors. We obtained expressions for the multiplication factor and the thermal-flux spatial distribution (fundamental mode) for several simple geometries. We found that:, (1) but to get to this point we employed special definitions of some of the terms: D means D th, the thermal diffusion coefficient. Σ a means Σ a,th Σ s,th fast, the thermal absorption + upscattering cross section. Σ f means Σ f,th, the thermal fission cross section. ν means Thus, Eq. (1) really says the following: (2) This is consistent with the six-factor formula as it was derived in Chapter III, because: and. (3) If we recall the definition of the fast-fission factor (ε), we can rewrite our expression for the multiplication factor as follows: k eff = P FNL p 1 1+ L 2 2 th B g ν th Σ f,th Σ a,th + P FNL ( 1 p)u F η F. (4) In this chapter we develop useful approximate expressions for the factors that are related to the neutron slowing-down process: P FNL (fast non-leakage probability), p (resonance-escape probability), and u F η F (fast utilization times fast reproduction factor). In the next chapter we shall address the factors related to thermal neutrons (those that have already slowed down).
131 Some Physics In thermal reactors, neutron energies range from almost zero to more than 10 7 ev (10 MeV). We can divide this energy range into three segments, each characterized by different kinds of neutron-nucleus interactions. (The boundaries between segments are fuzzy.) Energy Range 0 to 1 ev 1 ev to 10 5 ev 10 5 ev to 10 7 ev Interaction Physics: Interaction Results: Upscattering Molecular Effects Quantum-Mech. Effects (diffraction) Neutron No Upscattering Elastic Scattering from free nuclei; isotropic in CM frame (s-wave) Resonance Absorption (resolved resonances) Neutron (slowing down) Fission Source Elastic Scattering from free nuclei; anisotropic in CM frame (p-wave) Inelastic Scattering Resonance Absorption (unresolved resonances) Fast Fission; Moderation In this chapter we will focus mostly on the middle (slowing-down) regime and somewhat un the upper (highest-energy) regime. Some Math Recall the continuous-energy diffusion equation and consider the k-eigenvalue problem: (5) Given a homogeneous reactor (one in which the material properties are the same at every spatial location), the D and the cross sections do not depend on position, and we have: (6) If we consider a bare reactor (one that is surrounded by vacuum), then the following extrapolated boundary condition is perhaps reasonable: φ(r s + de n, E) = 0, r s on boundary. (7)
132 In reality, it would be best to let the extrapolation distance, d, be different for different neutron energies. But to keep the math tractable we shall declare it independent of energy for purposes of our explorations in this chapter. Let us try separation of variables for the energy-dependent scalar flux: If we insert this guess into the energy-dependent diffusion equation and divide through by the product (Dfψ), we obtain (8) (9) Observe that and the right-hand side the left-hand side Thus, each side must equal a constant. In keeping with previous chapters, we shall call this constant B 2. We therefore obtain two separate problems: Spatial problem: Energy problem: (10) (11) We recognize that the problem given by Eqs. (10) is the Helmholtz eigenvalue problem that we have solved previously. We know that B 2 eigenvalue is the one associated with the the and with the
133 This smallest B 2 eigenvalue is the of the reactor. Recall that geometric buckling = and depends very little on its material properties. (The dependence on material properties is only through the extrapolation distance, d, in the boundary condition. We have noted that a good value to use for this distance is 2D, where D is the diffusion coefficient of the reactor material. But for a large reactor, the value used for the extrapolation distance has very little effect on the multiplication factor or on the fundamental-mode distribution.) If we put these pieces together, we see that the spatial part of fundamental-mode solution satisfies:. (12) The energy-dependent equation, Eq. (11), is exactly like an k-eigenvalue equation except for the term. This term plays the same role in the equation as It accounts for outleakage from the reactor, and can be thought of as a (Recall that we are considering bare reactors, which means that net outleakage through the reactor surface is the same as outleakage, because inleakage is zero.) Note that if f is a solution of Eq. (11), then so is any constant times f. That is, this equation says nothing about f s amplitude. It gives information only about f s shape (which is the energy distribution of the neutrons in the reactor). Thus, without losing any information, we can normalize f(e) such that the following is true: This allows us to rewrite our equation:. (13) (14)
Solving for the Energy Distribution 134 Hydrogen as the main neutron slower Consider a water-moderated reactor. The vast majority of neutrons energy loss in such a reactor comes from scattering off of There are three main reasons for this: 1) The scattering cross section for hydrogen is much larger than that for oxygen or for fissionable nuclides. 2) There is more hydrogen than anything else. 3) The average energy loss per scattering collision is much greater for scattering from hydrogen than from heavier nuclei. (On average, a fast neutron loses half of its energy in a collision with hydrogen, but only around 10% in a collision with oxygen and less than 1% in an elastic collision with uranium.) We are fortunate that we can solve Eq. (14) analytically if we ignore the energy loss from collisions with all nuclei except hydrogen. In this case, the differential scattering cross section becomes that for hydrogen, which has an exceptionally simple form: [for hydrogen, H-1] (15) The analytic solution of Eq. (14), given the differential scattering cross section (15), is, [H-1] (16) where we have defined = probability that a neutron born with energy E (17) p nl (E,E') = probability that a neutron that scatters at energy E (18)
135 p re (E,E') = probability that a neutron that scatters at energy E (19) Here E 0 is an energy above which there are essentially no neutrons in the reactor (something in the range of 10-12 MeV would be a good number for this). We can use this solution to generate approximate expressions for the factors that we seek. Recall that these factors are: P FNL (fast non-leakage probability), p (resonance-escape probability), and u F η F (fast utilization times fast reproduction factor). Before we do this, though, we generalize the solution to allow for slowing down caused by nuclides other than hydrogen. Including scattering from nuclides with A>1 This case is too difficult to solve analytically, but it is not terribly difficult to create an accurate approximate solution. This is:, (20) where we have defined = probability that a neutron born with energy E scatters before it is absorbed or leaks. (21) p nl (E,E') = probability that a neutron that scatters at energy E does not leak before it reaches energy E. (22)
136 p re (E,E') = probability that a neutron that scatters at energy E does not get absorbed before reaching energy E. (23) Note that the only difference in the solutions is the inclusion of the factor of several terms. This factor is in the denominator average logarithmic energy decrement per scattering event = (24) = average lethargy gain per scattering event The symbol here is the Greek letter xi. We call it worm-bar. (Some unsophisticated nuclear engineers have been known to call it squiggle. We shall not stoop to such a level.) If you compare to the previous solution, which said that hydrogen was the only nuclide that could cause neutrons to lose energy, you would conclude that. for pure H-1 must be (25) This is correct. Only hydrogen has such a large value of worm-bar. Worm-bar for some other moderators is given in Table 1 below, along with some other interesting measures of moderator performance. Table 1. Slowing-down parameters for common moderators. Avg. number Moderator A of collisions from 2 MeV Σ s to 1 ev H 2 0 1 and 16 0.920 1.35 D 2 0 2 and 16 0.509 0.176 He 4 0.425 C (graphite) 12 0.158 0.06 U 238 238 0.008 0.003 Σ s / Σ a Obtaining the Factors Now we shall use what we have learned to create useful approximate expressions for the factors that we need. Fast Non-Leakage Probability We are interested in the probability that a neutrons slows to thermal without leaking or getting absorbed. We shall define an energy,
137 to be the thermal cutoff energy. If a neutron slows to E th, we say that it has become thermal. From Eq. (22) we have: p nl (E th,e') = probability that a neutron that scatters at energy E does not leak before it reaches energy E th. (26) We shall not prove the following, but it turns out that the integral in the exponent is related to a slowing-down distance : (27) where τ(e E) (1/6) of the average squared distance from the point where a neutron to the point where We are often very interested in the distance from the neutron s birthpoint (from a fission event) to the point where it becomes thermal. We define: τ th (1/6) of the average squared distance from the point where to the point where (28) With this definition and the previous equations, we obtain our useful approximate formula for the fast non-leakage probability: P FNL [approximation for fast non-leakage prob.] (29) This is the approximation you should use for the fast non-leakage probability in a large, bare, homogeneous reactor! If you use this expression (correctly) and it tells you that P FNL is far from unity (say, less than 0.8), then your reactor is probably not large enough for this to be a very accurate approximation. For historical reasons (going back to Fermi and a very clever approximation that he made to do the first nuclear reactor analysis),
138 and Remember that Age is proportional to a mean-squared slowing-down distance, and thus has units of Age to thermal characterizes distance from fast-neutron birth point to thermalization point, in the same way that L th 2 characterizes distance from thermal-neutron birth point to absorption point. One more point about P FNL : If P FNL is close to 1, then its exponent is small, and P FNL. [if B g 2 τ th << 1] (30) Resonance-Escape Probability We found earlier that p re (E th,e 0 ) (31) This is the probability of a neutron s escaping absorption all the way from birth at energy E 0 to slowing down to the thermal energy E th. Because of the presence of E in the denominator of the integrand and because of the relatively low value of the absorption cross section for highenergy neutrons, this probability is relatively insensitive to the value chosen for E 0. We can therefore pick a value for E0 and then use the resulting expression as the resonance-escape probability for all neutrons born from fission. We now take a closer look at the denominator of the integrand that appears in the exponent. If we multiplied Σ t by the scalar flux, we would obtain the collision-rate density. If we multiplied DB g 2 by the scalar flux, we would obtain the net outleakage rate density. It follows that = (32) for a neutron of energy E. In a large reactor, this ratio will be Thus, we shall ignore the leakage term in the denominator and use the approximation:
139 p. (33) We recall now that in the slowing-down energy range, the dominant interaction between neutrons and light nuclei (which includes nuclei in all moderators) is This is elastic scattering off of the potential of the nucleus, which means the neutron does not actually penetrate the nucleus. The potential scattering cross section is essentially independent of the energy of the neutron, and we give it the symbol Σ p. We multiply and divide by this cross section, recognizing that it is independent of energy, to obtain: p. (34) We define the resonance integral for absorption for nuclide i in some given mixture of nuclides as follows:. (35) (There is a corresponding resonance integral for fission.) In the general case of an arbitrary mixture of fuel and moderator nuclides, this integral would need to be evaluated on a case-bycase basis, because it would depend upon the details of the mixture. However, there is a limiting case that simplifies things considerably: in which there are lots, lots more moderator nuclei than fuel (or other absorbing) nuclei. In this limit, note that Σ t (E ) = (36) In this case, the resonance integral becomes independent of the details of the mixture, and we have: = (37) Note the infinity superscript that denotes the infinitely dilute limit. Thus, in the dilute limit (lots more moderating atoms than absorbing atoms), the resonance integral can be replaced by the infinitely-dilute resonance integral, and our expression for the resonance-escape probability becomes:
140 p [dilute limit](38) If there are lots more moderator atoms than absorber atoms, this is the approximation to use for resonance-escape probability. If you use this expression (correctly) and it tells you that p is very small (say, less than 0.5), then your mixture is not dilute enough for it to be very accurate. Fast Utilization times Fast Reproduction Factor The product of these two factors is the number of fission neutrons emitted That is, u F η F =. (39) Most of the absorptions and fissions will take place at energies at which most of the neutrons present are neutrons that have scattered, not neutrons that are freshly born from fission. (Most fission neutrons are born with energies above 100 kev; most absorption takes place below this energy.) For these energies, the direct contribution of c(e) to f(e) is small, and our approximate solution for f(e) can be further approximated as:, (40) where A is a constant. (Here we have used the fact that D(E)B g 2 << Σ t (E), as discussed above.) We insert this into Eq. (39) to obtain:
141 u F η F. (41) We now assume that: 1) p re p nl can be replaced in each integral with an average value, and 2) this average value is the same for both integrals. We also multiply numerator and denominator by the potential scattering cross section of the moderator. We obtain: u F η F (42) Note that the resonance integrals have appeared again! Recall that the only resonance integrals we have readily available to us are the resonance integrals, in which the ratio of Σ p for the moderator to Σ t for the mixture is taken to be unity. Recall that the resonance integrals approach this limit when the ratio of fuel atoms to moderator atoms is very small.
142 Summary In this chapter we have developed several useful formulas that will help us estimate the multiplication factor for a bare homogeneous reactor. They are summarized here. P FNL. (43) p. (44) u F η F. (45)