# Conversion Factor – Breeding Ratio

## Nuclear Fuel Breeding

Fuel breeding or fuel conversion plays a significant role in the fuel cycle of all commercial power reactors. During fuel burnup, the fertile materials (conversion of 238U to fissile 239Pu known as fuel breeding) partially replace fissile 235U, thus permitting the power reactor to operate longer before the amount of fissile material decreases to the point where reactor criticality is no longer manageable.

It must be added natural uranium primarily consists of isotope 238U (99.28%). All commercial light water reactors contain both fissile and fertile materials. For example, most PWRs use low enriched uranium fuel with enrichment of 235U up to 5%. Therefore more than 95% of the content of fresh fuel is fertile isotope 238U.

There is potential to increase the recoverable energy content from the world’s uranium and thorium resources by almost two orders of magnitude by converting the fertile isotopes uranium-238 and thorium-232 into fissile isotopes.

Plutonium 239 breeding

Neutron capture may also be used to create fissile 239Pu from 238U, the dominant constituent of naturally occurring uranium (99.28%). Absorption of a neutron in the 238U nucleus yields 239U. The half-life of 239U is approximately 23.5 minutes. 239decays (negative beta decay) to 239Np (neptunium), whose half-life is 2.36 days. 239Np decays (negative beta decay)  to 239Pu.

Uranium 233 breeding

232Th is the predominant isotope of natural thorium. If this fertile material is loaded in the nuclear reactor, the nuclei of 232Th absorb a neutron and become nuclei of 233Th. The half-life of 233Th is approximately 21.8 minutes. 233Th decays (negative beta decay) to 233Pa (protactinium), whose half-life is 26.97 days. 233Pa decays (negative beta decay)  to 233U, which is a very good fissile material. On the other hand, proposed reactor designs must attempt to physically isolate the protactinium from further neutron capture before beta decay can occur.

## Conversion Factor – Breeding Ratio

A quantity that characterizes this conversion of fertile into fissile material is known as the conversion factor. The conversion factor is defined as the ratio of fissile material created to fissile material consumed either by fission or absorption. If the ratio is greater than one, it is often referred to as the breeding ratio, for then the reactor is creating more fissile material than it is consuming.

## Comparison of cross-sections

Source: JANIS (Java-based nuclear information software)  http://www.oecd-nea.org/janis/

When C is unity, one new atom is produced per one atom consumed. It seems fertile material can be converted in the reactor indefinitely without adding new fuel. Still, the content of fertile uranium 238 decreases in real reactors, and fission products with significant absorption cross-section accumulate in the fuel as fuel burnup increases.

If we use a simplified model, which includes only uranium and plutonium-239, the conversion factor is:

This equation indicates that increased fuel enrichment results in a decreased value of C(0), the initial conversion factor. As the content of fissile material decreases with fuel burnup, the conversion factor increases. As this happens an increasing fraction of the fission comes from plutonium.

## Conversion Factor and Excess of Neutrons

The role of the reproduction factor, η, is evident. The rate of transmutation of fertile-to-fissile isotopes depends on the number of neutrons more than those needed to maintain the chain fission reaction that is available. For conversion to occur, it is necessary that η must be greater than unity. Almost all reactors operate, at least to some extent, as converters. For fuel breeding to occur, it is necessary that η must be greater than 2. One neutron to sustain a chain reaction and one or more neutrons on average must be absorbed in fuel and must produce another fissile atom for natural uranium in the thermal reactor, η = 1.34. As a result of the ratios of the microscopic cross-sections, η increases strongly in the region of low enrichment fuels. This dependency is shown in the picture. It can be seen there is a limit value about η = 2.08.

Table of reproduction factors

The fertile-to-fissile conversion characteristics depend on the fuel cycle and the neutron energy spectrum. For a thermal neutron spectrum (E < 1 eV) and the uranium fuel cycle, fuel breeding (C>1) is not feasible, although η for both isotopes is greater than 2. This is due to the fact η is not large enough to compensate for the neutron leakage and its parasitic capture.

The situation is considerably better for a thermal neutron spectrum (E < 1 eV) and the thorium fuel cycle. Due to the very low capture-to-fission ratio, the reproduction factor for uranium 233 is about η = 2.25. From this point of view, is thorium fuel cycle is promising, and a thermal reactor of this type could successfully be made to breed.

For a fast neutron spectrum, there are differences in both the number of neutrons produced per one fission and, of course, in the capture-to-fission ratio, which is lower for fast reactors. The number of neutrons produced per one fission is also higher in fast reactors than in thermal reactors. These two features are important in the neutron economy and contribute to the fact that fast reactors have a large excess of neutrons in the core. The superior neutron economy of a fast neutron reactor makes it possible to build a reactor that, after its initial fuel charge of plutonium, requires only natural (or even depleted) uranium feedstock as input to its fuel cycle. Russian BN-350 liquid-metal-cooled reactor was operated with a breeding ratio of over 1.2.

## Conversion Factor for LWRs

All commercial light water reactors breed fuel, but they have low breeding ratios. In recent years, the commercial power industry has emphasized high-burnup fuels (up to 60 – 70 GWd/tU), typically enriched to higher percentages of U-235 (up to 5%). As burnup increases, a higher percentage of the total power produced in a reactor is due to the fuel bred inside the reactor.

At a burnup of 30 GWd/tU (gigawatt-days per metric ton of uranium), about 30% of the total energy released comes from bred plutonium. At 40 GWd/tU, that percentage increases to about forty percent. This corresponds to a breeding ratio for these reactors of about 0.4 to 0.5. Light water reactors with higher fuel burnup (up to 60 GWd/tU) have a conversion ratio of approximately 0.6. That means about half of the fissile fuel in these reactors is bred there. This effect extends the cycle length for such fuels to nearly twice what it would otherwise be. MOX fuel has a smaller breeding effect than 235U fuel and is thus more challenging and slightly less economical to use due to a quick drop-off in reactivity through cycle life.

Effect of Fuel Temperature on Nuclear Breeding
In LWRs, the fuel temperature influences the rate of nuclear breeding (the breeding ratio). In principle, the increase in the fuel temperature affects primarily the resonance escape probability, which is connected with the phenomenon usually known as the Doppler broadening (primarily 238U). The impact of this resonance capture reaction on the neutron balance is evident, the neutron is lost, and this effect decreases the effective multiplication factor. On the other hand, this capture leads to the formation of unstable nuclei with higher neutron numbers. Such unstable nuclei undergo a nuclear decay, leading to the formation of other fissile nuclei. This process is also called nuclear transmutation and is responsible for new fuel breeding in nuclear reactors.

From this point of view, the neutron is utilized much more effectively when captured by 238U than when captured by the absorber because the effective multiplication factor must in every state equal to 1 (Note that in PWRs, the boric acid is used to compensate an excess of reactivity of reactor core along the fuel cycle). In other words, it is better to capture the neutron (lower excess of reactivity) by 238U rather than by 10B nuclei.

At HFP (hot full power) state, the fuel temperature is directly given by:

• Local linear heat rate (W/cm) is given by neutron flux distribution. See also: Power Distribution
• Fuel-cladding gap. As the fuel burnup increases, the fuel-cladding gap reduces. This reduction is caused by the swelling of the fuel pellets and cladding creep. Fuel pellets swelling occurs because fission gases cause the pellet to swell, resulting in a larger volume of the pellet. At the same time, the cladding is distorted by outside pressure (known as the cladding creep). These two effects result in direct fuel-cladding contact (e.g., at burnup of 25 GWd/tU). The direct fuel-cladding contact causes a significant reduction in fuel temperature profile because the overall thermal conductivity increases due to conductive heat transfer.
• Core inlet temperature. System parameters in steam generators directly give the core inlet temperature. When steam generators are operated at approximately 6.0MPa, the saturation temperature is equal to 275.6 °C. Since there must always be ΔT (~15°C) between the primary circuit and the secondary circuit, the reactor coolant (in the cold leg)have about 290.6°C (at HFP) at the inlet of the core. As the system pressure increases, the core inlet temperature must also increase, and this increase causes a slight increase in fuel temperature.

It can be summarized that fuel breeding is lower when the reactor is operated at lower power levels. Note that additional absorbers must be inserted inside the core to lower the reactor power. The fuel breeding is higher  (e.g., 1 EFPD surplus) when the core inlet temperature of the reactor coolant is higher (e.g., 1°C for 300 EFPDs). It must be added the inlet temperature is limited, and it cannot be changed arbitrarily.

Reactors with Spectral Shift Control
Instead of increasing fuel temperature a reactor can be designed with so-called “spectral shift control”.  The main idea of the spectral shift is based on the neutron spectrum shifting from the resonance energy region (with lowest p – resonance escape probability) at the beginning of the cycle to the thermal region (with the highest p – resonance escape probability) at the end of the cycle. In pressurized water reactors, chemical shim (boric acid) and burnable absorbers are used to compensate for an excess of reactivity of reactor core along the fuel burnup (long-term reactivity control). From the neutronic utilization aspect, compensation by absorbing neutrons in poison is not ideal because these neutrons are lost. For better utilization of the neutrons, these neutrons can be absorbed by fertile isotopes to produce fissile nuclei (in radiative capture). These fissile nuclei would contribute to obtaining more energy from the fuel.