X. Assembling the Pieces 179 Introduction Our goal all along has been to gain an understanding of nuclear reactors. As we ve noted many times, this requires knowledge of how neutrons are produced and lost. We ve also noted that production and loss rates depend on a multitude of factors, most of which we have discussed in some detail. Perhaps the most basic thing we d like to know about any given reactor is its criticality: is it subcritical, supercritical, or critical? The multiplication factor, k eff, tells us the answer. In Chapter III we developed a 6-factor formula for k eff, and in later chapters we learned that k eff is an eigenvalue of the [Loss] 1 [Production] operator. If we accept the diffusion approximation, this means k eff is an eigenvalue of a differential (or integro-differential) equation. We have now developed solutions of our differential and integro-differential equations, using the diffusion approximation and sometimes using further approximations (all of which are accurate for large homogeneous reactors with lots more moderator than fuel). These solutions now enable us to calculate each of the six factors in the six-factor formula for k eff. We also found, in Chapter VII of these notes, solutions for how neutrons distribute themselves spatially in a bare homogeneous reactor. We now know this for basic shapes such as spheres, cylinders, and rectangular parallelepipeds. In Chapter VIII we learned how neutrons distribute themselves in energy as they slow to thermal energies in infinite reactors, discovering that the dominant behavior is a 1/E distribution in the slowing-down range. We found that absorption perturbs this distribution, causing downward spikes at resonances and a lowering of the flux just below each resonance (compared to the value it would have had without the resonance). We developed approximate expressions for P FNL and p that are accurate for dilute concentrations of fuel in moderators and for large reactors. In Chapter IX we found that neutrons in the thermal range tend toward a Maxwellian distribution in energy. We saw that absorption can perturb this distribution fairly significantly, but that the perturbed distribution looks a lot like a Maxwellian at a higher temperature, which we call the neutron temperature. We also discovered that for 1/v absorbers, which include all light nuclides, it is very easy to compute thermal absorption rates, and that for non-1/v absorbers it is easy to get approximate absorption (and fission) rates by using a non-1/v factor. We also found relationships between 2200-m/s cross sections and thermal cross sections. In this chapter we put this all together and show step by step how to use it to estimate the multiplication factor of a bare homogeneous reactor. We provide tables of needed data (such as thermal diffusion length and age to thermal for various moderators). We also show that we can calculate any reaction rate in any part of the reactor, because we know the spatial and energy distribution of the neutrons.
Multiplication Factor of Bare Homogeneous Reactor 180 For the remainder of this chapter let us agree that k is the multiplication factor of the reactor. That is, we drop the eff subscript. We are not concerned in this chapter with any of the other (smaller) k-eigenvalues, so there should be no confusion. The six-factor formula says k = P FNL εpp TNL η T f (1) In Chapters III and VIII we noted that we could write an expression without the fast-fission factor, ε: k = P FNL p P TNL f η T + P FNL (1 p) u F η F. (2) In Chapters VII and VIII we obtained expressions for P TNL, (f η T,) and (u F η F ): (3) Note that resonance integrals in the second term play exactly the same role as the thermal cross sections in the first term. Our goal in this section is to review how to estimate each term in this equation using what we have learned throughout the course. Let s do it. Fast Non-Leakage Probability In Chapter VIII we agreed to use P FNL (4) But where do we find the age to thermal, τ th? This quantity is tabulated for different moderators. Here we reproduce data from a table given in the Lamarsh text. Table: Approximate Ages (in cm 2 ) of Fission Neutrons to Thermal Energies. Also approximate average logarithmic energy decrements and potential scattering cross sections. Moderator Approximate τ th [cm 2 ] σ p [barns] H 2 O 27 0.920 44 D 2 O 131 0.509 10.5 Be 102 0.209 6.1 Graphite 368 0.158 5
181 Based in part on data given in L. S. Kothari and V. P. Duggal, Scattering of Thermal Neutrons from Solids and Their Thermalization Near Equilibrium, Advances in Nuclear Science and Technology, Vol. 2. New York: Academic Press, 1964. What about a mixture of fuel with one of these moderators? We noted in Ch. VIII that an approximation for age to thermal is. (5) If the mixture is dilute (lots more moderator atoms than fuel atoms), then only at resonance energies will the fuel s presence significantly change the cross sections. These are very narrow ranges compared to the full range of integration. In these ranges the cross sections will be higher than the moderator-only value, and thus in these tiny ranges the contribution to τ th will decrease. We conclude that: In a dilute mixture of the age to thermal is only slightly Thus, in our calculations of large reactors with dilute homogeneous mixtures, Resonance-Escape Probability In Chapter VIII we agreed to use we will use the age to thermal for the moderator. p (6) Here Σ p is the potential scattering cross section of the moderator. If you explore how we got Eq. (6) you will see that Σ p is the scattering cross section evaluated at energies where most of the resonance absorption is taking place, which means from around an ev to a few kev. Resonance Integrals Where do we find the resonance integrals for the absorbing nuclides? If we are in the dilute limit (lots more moderator atoms than fuel atoms), we can get them from the chart of the nuclides. The table below provides values for a few interesting nuclides, and it includes the resonance integral for fission as well, which is needed in the expression for k. Table: Infinitely-Dilute Resonance Integrals. (From http://atom.kaeri.re.kr/) Nuclide Resonance Integrals (barns) Capture + Fission Fission U-235 411 278 U-238 280 2
182 Neutrons Released Per Fission Nuclide Resonance Integrals (barns) Capture + Fission Fission Pu-239 484 303 Pu-240 8112 9 Fe (natural) 1.4 0 Zr (natural) 1.2 0 O-16 0.0006 0 H-1 0.15 0 H-2 (D) 0.0003 0 H 2 O 0.3006 (.15+.15+.0006) 0 D 2 O 0.0012 (0.0003+0.0003+0.0006) 0 Experiments and theory agree that a particular fission event may release 0 or 1 or 2 or 3 or 4 or... neutrons. In reactor analysis we are rarely interested in how many fissions release 0, how many release 1, etc.; all we usually care about is how many are released on average from fissions of a given nuclide, caused by the absorption of neutrons of a given energy. Experiments find that this value fits a very simple formula over wide ranges of incident neutron energy: where and ν 0 and a are constants for a given nuclide and a given range of E, E = energy of the neutron that caused the fission. (8) A table of these constants is given below for various interesting nuclei, sometimes in two energy ranges to accurately fit the measured data. The Constants, v 0 and a, for Eq. (7) Nuclide v 0 = ν th a, [MeV -1 ] Energy range Th 232 1.87 0.164 all E 2.48 0.075 0 E 1 MeV U 233 2.41 0.136 E > 1 MeV 2.43 0.065 0 E 1 MeV U 235 2.35 0.150 E > 1 MeV U 238 2.30 0.160 all E 2.87 0.148 0 E 1 MeV Pu 239 2.91 0.133 E > 1 MeV From G. R. Keepin, Physics of Nuclear Kinetics, Reading, Mass.: Addison-Wesley, 1965. (7)
183 Note that this function varies so slowly in energy that the correct value to use for thermal neutrons is Note in particular that this is 2.43 for U-235. You should remember this number! What about ν f, the average number of neutrons released per fission caused by a non-thermal (fast) neutron? In a thermal reactor, recall that most non-thermal fissions occur in resonances the slowing-down energy range, where the flux is roughly proportional to 1/E. Because the resonance range for the important fissile and fissionable nuclides is roughly from 1 ev to a few kev, and because of the 1/E flux, a reasonable average to use for ν f would be the ν value evaluated at 100 ev or so. This would give ν f = ν 0 + (0.0001 MeV)a, (10) which gives. (11) Results are similar for other nuclides: In a thermal reactor, we can use ν f = ν th. (9) Thermal Non-Leakage Probability We now have everything except the thermal-neutron term in our equation for k. This term is:. (12) Here we see the thermal non-leakage probability that we have identified before:. (13) We can evaluate this term if we know L th, the thermal diffusion length. We recall its definition and rearrange that definition: (14) So we begin with the (square of the) diffusion length for the moderator and now examine the two multiplicative correction factors. An example will help clarify the validity of the approximations we are about to propose. Let us keep in mind the example of a mixture of U-235 and H 2 O, with 1 U-235 atom per 100 H 2 O molecules. Let us first consider the absorption ratio:
184 = (15) Obviously we cannot ignore a factor this far from 1: We must correct L th for absorption by fuel atoms! (Note that even if the U/water ratio had been 1/1000, this correction would still be around ½.) Now consider the transport-cross-section ratio. To get an estimate of the magnitude of this ratio, we shall look at the ratio of total cross sections, which is almost the same as the ratio of the transport cross sections. We found above that the temperature and sqrt(π)/2 ratios divide out of our cross-section ratios, so here we will jump straight to 2200-m/s values. We note that the 2200-m/s scattering cross section in U-235 is around 15 b, whereas it is around 100 b for H 2 O. Thus: = (16) This is close enough to unity that we can ignore it, especially since it is going to be multiplied by the small number B g 2 and added to unity (see expression for P TNL ). We conclude that for reasonably dilute mixtures of fuel and moderator, we can use: (17) Note that the mixture s diffusion length is always smaller than the moderator s. We need a table of diffusion lengths for common moderators. We take it from Lamarsh: Thermal Neutron Diffusion Parameters of Common Moderators at 20 C Moderator Density (g/cm 3 ) D th (cm) Σ a,th (cm -1 ) L th 2 (cm 2 ) L th (cm) H 2 O 1.00 0.16 0.0197 8.1 2.85 D 2 O 1.10 0.87 2.9 x 10-5 3.0 x 10 4 170 Be 1.85 0.50 1.04 x 10-3 480 21 BeO 2.96 0.47 6.0 x 10-4 790 28 Graphite 1.60 0.84 2.4 x 10-4 3500 59
185 Based on Reactor Physics Constants, U.S. Atomic Energy Commission Report ANL 5800, 2nd ed., 1963, Section 3.3. The diffusion properties of D 2 O are sensitive to the amount of H 2 O present. Graphite often contains impurities that affect its diffusion properties; the values given here are for highly-purified graphite. As the note following the table indicates, for materials with teeny tiny absorption cross sections (like graphite or heavy water), the addition of an absorber has a dramatic effect on the diffusion properties. You can see this from our Eq. (17) above we developed this to correct for the presence of fuel, but it is equally valid to correct for the presence of an impurity. In our problems, this won t make any difference unless the impurity s macroscopic absorption cross section is comparable to the fuel s. This is exceedingly unlikely! So in our problems we shall assume that absorption in fuel dominates absorption in impurities, and we will stick with Eq. (17). The Remaining Term The only term we have not addressed is a pretty simple one: = (18) Here we have noted that the temperature and sqrt(π) factors divide out. Given a mixture of an arbitrary number of moderating and fissioning nuclides, our expression becomes. (19) Example: k of Uranium + D 2 O Consider a mixture with Q uranium atoms per D 2 O molecule. Assume a neutron temperature of 100 C. Suppose the U is entirely U-238 and U-235, with W U-235 atoms per total uranium atom. Find k of this mixture as a function of Q and W. Using a spreadsheet, plot this for a few interesting values of W. We begin with our k equation, Eq. (3), with the non-leakage probabilities set to 1. This is:. (20) We begin with expression for p, Eq. (6), and start filling in data: p
186 From our table of ages etc., we find: and. From our table of resonance integrals we find: and. So we have: p (21) We plot this below for natural uranium (W = 0.007) and a typical value of enrichment (W = 0.035) as a function of Q. As you can see, the enrichment doesn t make much difference in p, because for low enrichments p is dominated by absorption in U-238. 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 p (W=.007) p (W=0.035) 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Figure: Resonance-escape probability as a function of Q = U/D 2 O ratio for two different values of W = U-235/(total U) ratio. Next we recognize that ν f ν th = 2.43. Now the resonance-integral ratio:
187 = (22) Note that we have ignored the resonance integral in D 2 O. From our table of resonance integrals we see that the resonance integral in D and O are 0.0003 b and 0.0006 b, respectively. These values are too tiny to make a significant contribution, so we don t worry about them. Now the thermal ratio:. Let s get numbers for these ratios under the assumption of natural uranium (W=0.007):. With these, and with W=0.007 plugged into our expression for p, we have: (23) A plot:
188 1.2 1 0.8 k 0.6 0.4 0.2 0 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 Q Figure: Infinite-medium multiplication factor as a function of U/D 2 O ratio for natural uranium. Example 2: k of (Uranium + D 2 O) cylinder Consider a finite cylindrical reactor composed of natural uranium mixed with D 2 O. Again assume a neutron temperature of 100 C, and assume a density of 1.1 g/cm 3. Consider a bare cylinder of this material of radius 2 meters and height 3 meters, and assume the near-optimal ratio of 1 uranium atom for every 500 D2O molecules (Q = 0.002). What is the multiplication factor? From the previous section we have k = 1.0785. Now we need the non-leakage probabilities. For each of them we will need the geometric buckling, so let s get it: = For the fast non-leakage probability we will need the age to thermal. The age to thermal in D 2 O is approximately, according to our table above. We will also need the square of the thermal diffusion length. Our formula for this is Eq. (17) above:. (24)
189 According to our table, The other term is: = 3.0 10 4 cm 2. (That s pretty big!) = (25) Putting this together, we have: P FNL = = P TNL = = (26) Note that the huge diffusion length in D 2 O (which is the result of having almost no absorption in D2O) causes lots of thermal neutrons to leak out! This makes the reactor quite subcritical, as we see when we insert our numbers: (27) When we crunch the numbers we obtain k (28) We conclude that while it is possible to make a critical reactor out of a homogeneous mixture of natural uranium and heavy water, the reactor must be really large! This is why CANDU reactors (Canadian reactors fueled by natural uranium and moderated by heavy water) are much larger than light-water reactors. Another Trick You now should see how to estimate the multiplication factor for a bare homogeneous reactor if you are given its size, shape, and material composition, at least if there are lots more moderator atoms than fuel atoms.
190 But what if you are asked to design a critical reactor out of a given material? That is, suppose you know that you want k=1 and you know the material properties, and your unknown is size and/or shape? Recall that the only number that matters regarding size and shape is So you should first treat B g 2 as your unknown and try to solve for it. But when you do so, you encounter a difficulty a Recall the expression we are using for k, and see what we need to solve for B g 2 if we want a critical reactor:. The presence of B g 2 in an exponent and in a denominator makes this problematic! So here is a trick that gets around the problem: exp{ B g 2 τ th } 1 B g 2 τ th. (29) This approximation is accurate when B g 2 τ th is small, which is the same limit in which our approximation for P FNL is accurate. If you plug this into your criticality expression, you will be able to solve algebraically for B g 2. Then you simply have to find a size and shape that produces this value of B g 2. Spectrum in Reactors Continuing the theme of putting pieces together, let s assemble our estimate of the neutron flux spectrum (i.e., the neutron distribution in energy) that we d expect to see in a large thermal reactor. We know that in the high-energy range (a few hundred kev to 10 MeV), the dominant phenomenon is that neutrons are born from fission and then scatter down to lower energies. The result is that in this energy range, the scalar-flux spectrum looks a lot like a fission spectrum. (In Chapter VIII, Eq. (20), the χ(e) term on the right-hand side dominates in this energy range.) In the slowing-down range (around an ev to a few hundred kev), the χ(e) term of Chapter VIII Eq. (20) is negligible, and the other term is very nearly proportional to 1/E, with sharp drops at resonances (where the Σ t term in the denominator has sharp increases). These drops are in such narrow energy ranges that they are basically invisible on a plot that covers a wide energy range.
191 In the thermal range (below approximately 1 ev), we learned in Chapter IX that the scalar flux is essentially Maxwellian, although at an effective neutron temperature that will usually be slightly higher than the temperature of the reactor material. We put this together in the figure below: f(e) 1E+14 1E+13 1E+12 1E+11 1E+10 1E+9 1E+8 1E+7 1E+6 1E+5 1E+4 1E+3 1E+2 1E+1 1E+0 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+2 1E+3 1E+4 1E+5 1E+6 E (ev) Figure. Rough sketch of typical neutron flux spectrum in a large thermal reactor operating near room temperature.