The 4-Factor Formula for k 

1. The 4-Factor Formula for k B. Rouben McMaster University EP 4D03/6D03 Nuclear Reactor Analysis 2008 Sept-Dec

2. Contents • We derive the 4-factor formula for the reactor multiplication constant.

3. Two Energy Groups • Up to now we have considered in detail the neutron-diffusion equation in one energy group. • In the 1-group model, all neutrons are treated as if they had the same energy. • However, we know that in fact the neutron energy in thermal reactors spans many orders of magnitude: from the MeV range for neutrons born in fission to small fractions of an eV for thermal neutrons. • Although the one-group formalism helps to understand the underlying principles, an improved methodology must be used for the most accurate reactor calculations. cont’d

4. Two Energy Groups (cont’d) • Because a very large fraction of the fissions is in the low-energy range in thermal reactors, a 2-energy-group diffusion equation is often appropriate for the calculation of the flux shape in the reactor. • In the 2-group formalism, the 2 groups are: • Group #1, the “fast” group (actually, the “slowing-down” group) • Group #2, the “thermal” group • The energy boundary which separates the two groups depends somewhat on the computer code used in the analysis; however, the value 0.625 eV is often used. cont’d

5. The Four-Factor Formula • An early formalism for analyzing the multiplication factor in two groups is the 4-factor formula for k, the infinite-lattice multiplication constant. • We derive it here, by following what happens to neutrons as they follow the cycle from one neutron generation to another. • Because the four-factor formula is for the infinite lattice, there is no need to consider leakage.

6. The Four-Factor Formula (cont’d) • Imagine N neutrons born in thermal fission. • There is a small number of fast-neutron fissions which occur (neutron-induced fission in U-238 has a neutron-energy threshold of ~0.7 MeV). The number of fast fissions is a small fraction of all fissions (a few percent). • A factor  is defined which is the ratio of total number of fission neutrons to the number born in thermal fissions. •  is called the fast-fission ratio (1) • Therefore total number of fission neutrons = N (2) cont’d

7. The Four-Factor Formula (cont’d) • In thermal reactors, these fission neutrons are thermalized by the moderator. • However, during slowing down, a number of neutrons will be lost in resonance captures. • A factor p is defined which gives the probability of neutrons not being captured in the resonance-energy range. • p is called the resonance-escape probability (3) • Therefore the total number of neutrons surviving to the thermal-energy range = Np (4) cont’d

8. The Four-Factor Formula (cont’d) • The thermal neutrons can be absorbed in fuel, or in other components of the lattice. • The neutrons absorbed in materials other than fuel are simply lost, as far as the chain reaction is concerned. • A factor f is defined which gives the fraction of thermal neutrons absorbed in fuel. • f is called the thermal or fuel utilization (5) • Therefore the total number of neutrons which survive to the thermal-energy range and are absorbed in the fuel = Npf (6) cont’d

9. The Four-Factor Formula (cont’d) • Some (not all) of the thermal neutrons which are absorbed in the fuel will induce fission in the fissile nuclides (U-235, Pu-239, Pu-241). • The factor  is defined as the number of fission neutrons produced per thermal-neutron absorption in fuel. • is called the reproduction factor (7) • Therefore the total number of fission neutrons which are born in the next generation = Npf (8)

10. The Four-Factor Formula (cont’d) • The infinite-lattice multiplication factor is the ratio of fission neutrons in one generation to the number in the previous generation. • Therefore the 4-factor formula for k results:

11. Interactive Discussion/Exercise • The derivation in the past few slides started from the identification of the fast-fission factor, ,in the neutron cycle . Show that the four-factor formula can also be derived by starting at any of the different points in the cycle, i.e., starting from p, f, or .

12. Typical Values of Four Factors in CANDU • Because the basic lattice has complex geometry, and because it has materials with high absorption cross section (the fuel) as well as materials with high scattering cross section (the moderator), diffusion theory cannot be used for calculation within the cell. • We must use transport-theory codes, e.g. WIMS-IST or DRAGON, or empirical codes (i.e., codes built upon the results of careful measurements), e.g., POWDERPUFS-V, to model the “microscopic” flux distribution within the basic lattice, its reactivity, and the quantitative trend (depletion) with burnup. • [The nuclear properties thus obtained can then be used in diffusion codes to calculate the “macroscopic” flux distribution in the core.] • Typical values of the 4 factors, obtained from POWDERPUFS-V lattice calculations, are shown in the next Table.

13. Typical Values of Four Factors in CANDU

14. Graphing the Four Factors in CANDU

15. The Four Factors (cont’d) • From the Table and graphs, we can see that   1.026, and p  0.906, and that they change very little (< 1 part per thousand, i.e., < 1 mk) with fuel burnup (or irradiation). This is understandable since  and p originate with U-238, and the U-238 fraction in the fuel changes very little. • f0.940. It increases by a few parts per thousand (a few mk) with burnup. This is due mostly to the additional absorption in the accumulating fission products in the fuel. • The biggest change with burnup is in  (range 1.11-1.24): • At first (up to the plutonium peak, ~1,200 MW.h/Mg(U)),  increases a bit. This is due to the initial high rate of Pu-239 production. • It then decreases more substantially with burnup, beyond the plutonium peak. This is due to the overall decrease in fissile inventory (U-235 depletes, while Pu-239 keeps building, but at a slower net rate). • The total change in  is a decrease of about 30 parts per thousand from fresh fuel to exit-burnup (~8,000 MW.d/Mg(U)) fuel. • In summary, it is mostly the change in , and less so the change in f, which drives the change in k-infinity.

16. Need the Diffusion Equation in 2 Groups • The 4-factor formula can give us information about the reactor multiplication constant (or system reactivity), but it does not provide any information about the flux shape in the reactor. • To compute the flux distribution in the core, we need the diffusion equation, in two energy groups (or more).

17. END