The four- and six-factor formulas

Teodora's second assignment is a page from an old design manual with a column of six numbers on it, each between about 0.7 and 1.9, and their product at the bottom. Hana recognizes the six-factor formula from her undergraduate notes but has never used it on a real lattice. Teodora explains that nobody designs a modern core with it, yet every reactor physicist still reaches for it first, because it says in plain terms where the neutrons go and which part of the design is responsible for each loss.

Following one generation of neutrons

The six-factor formula follows a generation of neutrons through a thermal reactor stage by stage. Start with \(N\) thermal neutrons absorbed in fuel. They cause fission and release fast neutrons. A few of those fast neutrons cause fission in uranium-238 before slowing down, adding more. Some fast neutrons leak from the core. The survivors slow down, and on the way some are captured in uranium-238 resonances. The neutrons that reach thermal energy may leak while diffusing. The rest are absorbed, either in fuel, which starts the next generation, or in everything else. Each stage multiplies the population by a factor, and the product of all six is \(k_{\text{eff}}\).

The first four factors describe the material and together give \(k_\infty = \eta f p \varepsilon\), the four-factor formula. The last two are the fast and thermal non-leakage probabilities, which depend on the size and shape of the core and the migration properties of the lattice.

The reproduction factor, eta

The reproduction factor \(\eta\) is the average number of fission neutrons produced per thermal neutron absorbed in the fuel. It combines the neutrons per fission \(\nu\) with the chance that an absorption in fuel actually causes fission rather than capture:

Here the subscripts 25 and 28 are the traditional shorthand for uranium-235 and uranium-238. For pure uranium-235 at thermal energy, \(\eta \approx 2.43 \times 585/681 \approx 2.09\). Uranium-238 absorbs without fissioning at thermal energies, so diluting the fissile isotope lowers \(\eta\). For natural uranium the thermal value is about 1.33 to 1.35; at 3 to 5 percent enrichment it rises to about 1.85 to 1.93. These numbers set the basic economics of the two reactor families: a natural uranium lattice starts with only about 0.3 neutrons to spare per absorption in fuel, while an enriched lattice starts with nearly 0.9.

The thermal utilization factor, f

The thermal utilization factor \(f\) is the fraction of thermal neutrons absorbed in the fuel rather than in the moderator, coolant, cladding, structure or control material:

In a real lattice the thermal flux is depressed inside fuel pins, because fuel absorbs strongly, so the absorption rates use flux-weighted cross sections. The disadvantage factor, the ratio of average flux in the moderator to average flux in the fuel, enters \(f\) and is typically 1.05 to 1.3. Soluble boron in a pressurized water reactor and the light water in CANDU zone controllers both work mainly by lowering \(f\).

The resonance escape probability, p

While slowing down, neutrons pass through energies between about 6 eV and a few kilo-electronvolts where uranium-238 has very large capture resonances. The resonance escape probability \(p\) is the fraction of neutrons that reach thermal energy without being captured there. A widely used approximation relates it to the slowing-down power of the moderator and an effective resonance integral \(I\):

Lumping the fuel into rods or bundles, rather than mixing it uniformly with moderator, raises \(p\) greatly: neutrons at resonance energies are absorbed in the outer skin of the fuel, so the interior is shielded, an effect called self-shielding. That is why natural uranium can be critical in a heterogeneous heavy water or graphite lattice but never in a homogeneous mixture. The follow-on course, Reactor Physics II, treats resonance absorption in detail; here it is enough to know that more moderator per fuel atom raises \(p\) and lowers \(f\), and the best lattice balances the two.

The fast fission factor, epsilon

A few fast neutrons cause fission in uranium-238, whose fission threshold is about 1 MeV, before they slow down. The fast fission factor \(\varepsilon\) is the ratio of all fission neutrons to those from thermal fission. It is small in a CANDU lattice, about 1.02 to 1.03, because each bundle is surrounded by a large volume of heavy water and neutrons leave the fuel quickly. In a tightly packed light water lattice, where fuel occupies a larger share of the volume, it is larger, often quoted near 1.05 to 1.1 in textbook treatments, with higher values in some formulations that fold other fast effects into it.

The non-leakage probabilities

The fast non-leakage probability \(P_{\text{FNL}}\) is the fraction of fast neutrons that slow down inside the core, and the thermal non-leakage probability \(P_{\text{TNL}}\) is the fraction of thermal neutrons absorbed before leaking. Diffusion theory gives simple estimates: \(P_{\text{TNL}} \approx 1/(1 + L^2 B^2)\) and \(P_{\text{FNL}} \approx e^{-B^2 \tau}\), where \(L^2\) is the thermal diffusion area, \(\tau\) is the Fermi age (the slowing-down area) and \(B^2\) is the geometric buckling of the core. Lesson 5 derives these. Large cores have small \(B^2\) and therefore small leakage.

# Six-factor formula with illustrative textbook-style values for two lattices.
lattices = {
    # eta, f, p, epsilon, fast non-leakage, thermal non-leakage
    'CANDU-type (natural U, heavy water)': (1.32, 0.93, 0.89, 1.03, 0.985, 0.975),
    'PWR-type (low-enriched U, light water)': (1.65, 0.88, 0.75, 1.07, 0.97, 0.99),
}
for name, (eta, f, p, eps, PFNL, PTNL) in lattices.items():
    kinf = eta * f * p * eps
    keff = kinf * PFNL * PTNL
    print(name)
    print(f"  eta {eta:.2f}  f {f:.2f}  p {p:.2f}  epsilon {eps:.2f}")
    print(f"  k_infinity = {kinf:.4f}")
    print(f"  non-leakage = {PFNL * PTNL:.4f}")
    print(f"  k_effective = {keff:.4f}  (excess before control: {(keff - 1) / keff * 1000:.1f} mk)")

# Eta for natural and enriched uranium from thermal data (approximate)
nu, sf5, sa5, sa8 = 2.43, 585.0, 681.0, 2.68
for e in (0.0072, 0.03, 0.045):
    eta = nu * e * sf5 / (e * sa5 + (1 - e) * sa8)
    print(f"enrichment {e * 100:4.2f} %: eta = {eta:.3f}")

CANDU-type (natural U, heavy water)
  eta 1.32  f 0.93  p 0.89  epsilon 1.03
  k_infinity = 1.1253
  non-leakage = 0.9604
  k_effective = 1.0807  (excess before control: 74.7 mk)
PWR-type (low-enriched U, light water)
  eta 1.65  f 0.88  p 0.75  epsilon 1.07
  k_infinity = 1.1652
  non-leakage = 0.9603
  k_effective = 1.1190  (excess before control: 106.3 mk)
enrichment 0.72 %: eta = 1.353
enrichment 3.00 %: eta = 1.852
enrichment 4.50 %: eta = 1.927

Look at the last lines of the output. Raising enrichment from natural to 3 percent raises the thermal \(\eta\) from about 1.35 to 1.85, but going from 3 to 4.5 percent adds only another 0.08, because once uranium-235 dominates the absorption, \(\eta\) approaches its pure-isotope limit. The value 1.32 used in the heavy water example is slightly below the thermal estimate because spectrum effects are folded in.

Moderator-to-fuel ratio

The single most important lattice design choice is how much moderator to place around each unit of fuel. With too little moderator, neutrons slow down slowly through the resonances and \(p\) is low: the lattice is undermoderated. With too much, neutrons survive the resonances but too many are absorbed in the moderator itself, so \(f\) falls: the lattice is overmoderated. Plotting \(k_\infty\) against the moderator-to-fuel ratio gives a broad peak between these regions.

Light water reactors are designed slightly undermoderated on purpose. If the water heats up or boils and its density falls, the lattice moves further to the left of the peak and \(k\) drops, which is a built-in safety feature. CANDU lattices are designed with a large amount of heavy water around each fuel channel, but the coolant inside the channel is separate from the moderator. Losing coolant density in a CANDU channel raises \(k\) slightly, the positive coolant void effect, which the design handles with fast shutdown systems and careful safety analysis under the CNSC's requirements. Lesson 12 returns to this.

Using the factors to explain changes

The real value of the formula shows up when something changes. Teodora gives Hana four short cases from past work and asks her to name the factor that moves first in each one.

In the first case, the plant adds absorber to the core. Soluble boron in a pressurized water reactor, gadolinium in a fuel pin, or light water in a CANDU zone controller all add thermal absorption outside the fissile nuclei, so \(f\) falls. The other factors barely change, which is why reactivity worth of an absorber is roughly proportional to how much thermal absorption it adds in high-flux regions.

In the second case, the fuel heats up. The resonances of uranium-238 broaden with temperature, the Doppler effect, so more neutrons are captured while slowing down and \(p\) falls within milliseconds. This is the fastest and most reliable feedback in any thermal reactor, and lesson 12 is built around it.

In the third case, the coolant density drops. In an undermoderated light water lattice, less water means poorer moderation, so \(p\) falls more than \(f\) rises, and \(k\) drops. In a CANDU channel, where the moderator is separate, losing coolant changes the spectrum and fast fission inside the bundle; the net effect is a small positive change in \(k\) that safety analysis must account for.

In the fourth case, fuel burns. Uranium-235 is consumed, plutonium-239 and plutonium-241 are produced, and fission products that absorb neutrons build up. \(\eta\) changes because the fissile mixture changes, and \(f\) changes because fission products are absorbers that sit inside the fuel but do not fission. Reactor Physics II follows these changes over the life of the fuel.

Working through these cases, Hana notices that the factors almost never change alone. A change that lowers \(f\) usually shifts the spectrum slightly and nudges \(p\) and \(\eta\) too. The formula is still useful because one factor usually dominates, and identifying it tells the designer where to look in a detailed calculation.

Limits of the formula

The six-factor formula is a bookkeeping device built on a two-energy picture, fast and thermal. Modern core design uses many energy groups and detailed transport calculations in each fuel assembly, as Reactor Physics II explains. The individual factors also depend on how they are defined, so values from different textbooks are not always comparable; some authors fold fast fission and fast absorption differently between \(\varepsilon\) and \(p\). What survives is the insight. When a lattice calculation code reports a change in \(k\), a physicist can still ask which factor moved, and that question usually points to the cause.