Light materials with small atomic mass (light or heavy water, graphite, and so on) are usually used as a neutron reflector and moderator. The present paper proposes using a new, heavy element as neutron moderator and reflector, namely, “radiogenic lead” with dominant content of isotope ^{208}Pb. Radiogenic lead is a stable natural lead. This isotope is characterized by extremely low micro crosssection of radiative neutron capture (~0.23 mb) for thermal neutrons, which is smaller than graphite and deuterium crosssections. The reflectorconverter for a fast reactor core is the structure capable of transforming some part of prompt neutrons leaked from the core into the reflected neutrons with properties similar to those of delayed neutrons, that is, sufficiently large contribution to reactivity at the level of effective fraction of delayed neutrons and relatively long lifetime, comparable with lifetimes of radionuclidesemitters of delayed neutrons. It is evaluated that the use of radiogenic lead makes it possible to slow down the chain fission reaction on prompt neutrons in the fast reactor. This can improve the fast reactor safety and reduce some requirements to the technologies used to fabricate fuel for the fast reactor.
Importance of such physical characteristic as prompt neutron lifetime for nuclear reactor safety is well known for a long time and was reflected in numerous publications, for example, in one of such fundamental works as [
The positive role to be played by the radiogenic lead, that is, lead with a dominant content of isotope ^{208}Pb, as a coolant for fast reactor safety was first noted in works [
Extension of prompt neutron lifetime in fast reactors with the radiogenic lead as a neutron reflector was first proposed in works [
The longer prompt neutron lifetimes can substantially improve kinetic response of the fast reactor to a jumplike insertion of relatively large (~1 $ or even more) positive reactivity. For simplicity, let effective fraction
The curves presented in Figure
Dependencies of the reactivity jump required to provide the power excursion with asymptotic time period
These curves demonstrate that the shorter lifetime of prompt neutrons results in the faster power excursion at the same reactivity jump. It seems helpful to reformulate the statement as follows. If prompt neutron lifetime became longer, then the power excursion with a certain asymptotic time period
For example, based on onepoint model of neutron kinetics, it may be concluded that application of thick ^{208}Pb reflector in the fast reactor BREST allowed us to reach prompt neutron lifetime of ~1 ms. If 1$ reactivity jump (!) occurs in the fast reactor, then its power increases with the excursion period nearly 1 s instead of 14 ms in the fast reactor reflected by natural lead. If prompt neutron lifetime in the fast reactor BREST reflected by thick ^{208}Pb layer is prolonged up to 10 ms, then the power excursion period will be longer than 1 s even at the reactivity jumps up to 2$. So long the power excursion periods give sufficient time for coolant to remove the heat from fuel rods. This means that feedbacks on coolant temperature and coolant density could be apparently actuated.
The dependencies shown in Figure
Within the frames of onepoint kinetic model (
As has been noted in [
According to the set of equations (
The balance relationship defines prompt neutron lifetime in a system as a sum of prompt neutron lifetimes in all system components (in the reactor core plus in the reflector, for instance) with the weighing coefficients that characterize contributions of these components to total criticality.
It may be concluded from (
One else important circumstance consists in the following fact. The larger fraction of delayed neutrons depends mainly on neutron leakage from the reactor core and, thus, may be chosen as a developer wills, while fraction of nucleiemitters of delayed neutrons may be chosen only within very stringent constraints. Evidently, generation rate of these “delayed” neutrons substantially depends on leakage rate of fast and resonance neutrons from the reactor core. That is why application of thick neutron reflector is a reasonable option not only for fast reactors but also for the reactors with resonance and even thermal spectra, with small sizes of the reactor core, that is, for the reactors with significant leakage of fast and resonance neutrons from the reactor core.
The inversehour equation (
This model of neutron kinetics can be used by considering a neutron reflector as a whole (as one zone in twopoint model) or by considering a neutron reflector as a set of annular (nonintersecting) layers (multipoint model depending on the number of these layers):
The further results were obtained for sixlayer reflector (thickness of the first layer is 50 cm, thicknesses of the next layers are equal to 1 m each). Kinetics parameters such as
Dependence of reactivity gain and prompt neutron lifetime on thickness of ^{208}Pb reflector in the fast BRESTtype reactor for various models of neutron kinetics.
Twopoint model  


Thickness of ^{208}Pb reflector, m  1  2  3  4  5  6 
Reactivity gain caused by the reflector, $  3.01  5.30  6.51  7.21  7.58  7.78 
Prompt neutron lifetime in the reflector, s  2.90 · 10^{−5}  1.04 · 10^{−3}  5.71 · 10^{−3}  1.21 · 10^{−2}  1.82 · 10^{−2}  2.33 · 10^{−2} 


Multipoint models  


The number of points  2  3  4  5  6  7 
Annular layer in ^{208}Pb reflector, m  0.5 
1 
2 
3 
4 
5 
Increase of the reactivity gain caused by thicker reflector, $  3.01  2.29  1.21  0.69  0.37  0.20 
Prompt neutron lifetime in the reflector layer, s  2.90 · 10^{−5}  2.36 · 10^{−3}  2.62 · 10^{−2}  7.25 · 10^{−2}  1.36 · 10^{−1}  2.16 · 10^{−1} 
It can be seen that, in the case of 6 m reflector (sevenpoint model), prompt neutron lifetime in the last 1 m thick reflector layer is considerably longer (about one order of magnitude) than that for twopoint model (one point for the reflector as a whole). This means that more correct mathematical models must be used to provide proper accounting for neutron transport effects in the fast reactors surrounded by physically thick and weakly absorbing neutron reflectors.
In order to simplify explanation of continuous neutron kinetics model, it seems reasonable to consider a spherically symmetrical reactor with a physically thick neutron reflector. The reactor core is described by one point in the continuous model. When the number of the reflector layers becomes infinite, multipoint model naturally converts into the continuous model where summing operation should be replaced by integration:
The inversehour equation and relationship for determination of prompt neutron lifetime for continuous model of the reflector in spherically symmetrical geometry can be rewritten in the following new forms:
Some relevant physical parameters of neutron reflector are considered here within the frames of spherically symmetrical continuous model of neutron kinetics and presented in Figure
Physical parameters of the fast BRESTtype reactor depending on thickness of neutron reflector made of ^{208}Pb.
It follows from the curves shown in Figure
The following two terms of (
As the reflector becomes thicker, difference between prompt neutron lifetime in the reflector
When physical parameters of physically thick neutron reflector were analyzed by using continuous neutron kinetics model, the following dependencies were calculated:
Radial distribution (within the reflector thickness) of the contribution given by annular layer of unitary thickness into the reactivity gain
Radial distribution (within the reflector thickness) of prompt neutron lifetime in annular layer of unitary thickness
This means that each annular layer of unitary thickness with inner radius
Radial distribution of the contribution given by neutrons of the reflector with lifetimes as long as
Component of the reactivity gain—the second summand of the inversehour equation (
Differential contribution into the reactivity jump required to provide the power excursion with asymptotic time period
To analyze the contributions of neutrons with various lifetimes in the reflector into the reactivity gain, it may be helpful to transform integration over radial coordinate into integration on neutron lifetime in the reflector
Component of the reactivity gain required to provide the power excursion with asymptotic time period
Component of the reactivity gain required to provide the power excursion with asymptotic time period
As it follows from Figures
If the asymptotic period of the power excursion is comparable or longer than time constant of fuel rods, then thermal energy can receive a long enough time period for its partial removal from fuel rods by coolant. Therefore, the process of fuel heating up slows down, and an opportunity arises for feedbacks on coolant density and temperature to actuate. Otherwise, if asymptotic period of the power excursion is substantially shorter than time constant of fuel rods, then fuel can be overheated and melted down. As the severest consequence, reactor can lose its ability of working. As for typical time constants of fuel rods, in experimental fast breeder reactor EBRII (USA) fuel rods had time constants about 0.11 s [
It is demonstrated that the use of radiogenic lead as a neutron reflectorconverter makes it possible to slow down the chain fission reaction on prompt neutrons in the fast reactor and it can improve the nuclear safety of fast reactor.
Multipoint model of neutron kinetics in nuclear reactors was applied to analyze timedependent evolution of neutron population in fast reactor with physically thick neutron reflector made of weak neutron absorber (radiogenic lead with dominant content of ^{208}Pb).
Multipoint model of neutron kinetics was applied to investigate a possibility for substantial elongation of prompt neutron lifetime with correct accounting for time of neutron staying in physically thick, weak neutronabsorbing reflector.
The paper demonstrates how multipoint discrete model of neutron kinetics can be transformed into continuous model of neutron kinetics in the reactor core surrounded by the neutron reflector.
Numerical analyses of neutron kinetics were carried out with application of multipoint and continuous models and demonstrated that physically thick and weakly neutronabsorbing reflector is able to prolong prompt neutron lifetime in the fast BRESTtype reactor by several orders of magnitude (roughly, from 0.5
Advanced design of fuel rods with quick heat transport from fuel to coolant can remarkably enhance the reactor resistance against the power excursions with short asymptotic periods caused by large reactivity jumps.
If the asymptotic period of the power excursion caused by positive reactivity jump is comparable with thermal constant of fuel rods, then heat would have a time to flow down from fuel to coolant, its density would decrease and it would be able to impact remarkably on the power excursion process (see Appendix
The term radiogenic lead is used here for designation of lead that is produced in radioactive decay chains of thorium and uranium isotopes. After a series of alpha and beta decays, ^{232}Th transforms into stable lead isotope ^{208}Pb, ^{238}U into stable lead isotope ^{206}Pb, and ^{235}U into stable lead isotope ^{207}Pb. Therefore, uranium ores contain radiogenic lead consisting mainly of ^{206}Pb, while thorium and mixed thoriumuranium ores contain radiogenic lead consisting mainly of ^{208}Pb. Sometimes, the presence of natural lead in uranium and thorium ores can change isotope composition of radiogenic lead. Anyway, isotope composition of radiogenic lead depends on the elemental composition of the ores from which this lead is extracted.
Radiogenic lead consisting mainly of stable lead isotope ^{208}Pb can offer unique advantages, which follow from unique nuclear physics properties of ^{208}Pb. This lead isotope is a doublemagic nuclide with completely closed neutron and proton shells. The excitation levels of ^{208}Pb nuclei (Figure
The excitation levels of lead nuclei.
Inelastic scattering crosssection of lead isotopes as a function of neutron energy.
The unique nuclear properties of ^{208}Pb can be used to improve parameters of chain fission reaction in a nuclear reactor. First, since energy threshold of inelastic neutron scattering by ^{208}Pb (~2.61 MeV) is substantially higher than that by natural lead (~0.8 MeV), ^{208}Pb can soften neutron spectrum in the highenergy range to a remarkably lower degree. Second, neutron radiative capture cross section of ^{208}Pb in thermal point (~0.23 mb) is smaller by two orders of magnitude than that of natural lead (~174 mb) and even smaller than that of reactorgrade graphite (~3.9 mb). These differences remain large within sufficiently wide energy range (from thermal energy to some tens of kiloelectronvolts). Energy dependence of neutron absorption cross sections is presented in Figure
Capture cross sections of various nuclides as a function of neutron energy (JENDL4.0).
If radiogenic lead consists mainly of ^{208}Pb (~90% ^{208}Pb plus 9% ^{206}Pb and ~1% ^{204}Pb + ^{206}Pb), then such a lead absorbs neutrons is as weak as graphite within the energy range from 0.01 eV to 1 keV. Such isotope compositions of radiogenic lead can be found in thorium and in mixed thoriumuranium ores.
Thus, on one hand, ^{208}Pb, being heavy nuclide, is a relatively weak neutron moderator both in elastic scattering reactions within full neutron energy range of nuclear reactors because of heavy atomic mass and in inelastic scattering reactions with fast neutrons because of high energy threshold of these reactions. On the other hand, ^{208}Pb is an extremely weak neutron absorber within wide enough energy range.
Some nuclear characteristics of light neutron moderators (hydrogen, deuterium, beryllium, graphite, and oxygen) and heavy materials (natural lead and lead isotope ^{208}Pb) are presented in Table
Neutronphysical characteristics of some materials.
Nuclide 

Number of collisions (0.1 MeV 



^{ 1}H  30.1  12  332  149 
^{ 2}D  4.2  17  0.55  0.25 
^{ 9}Be  6.5  59  8.5  3.8 
^{ 12}C  4.9  77  3.9  1.8 
^{ 16}O  4.0  102  0.19  0.16 
Pb_{nat}  11.3  1269  174  95 
^{ 208}Pb  11.5  1274  0.23  0.78 
One can see that elastic cross sections of natural lead and ^{208}Pb do not differ significantly from the others nuclides, being between the corresponding values for hydrogen and other light nuclides. Neutron slowingdown from 0.1 MeV to 0.5 eV requires from 12 to 102 elastic collisions with light nuclides, while the same neutron slowingdown requires ~1270 elastic collisions with natural lead or ^{208}Pb. The reason is the high atomic mass of lead in comparison with the other light nuclides. From this point of view neither natural lead nor ^{208}Pb are effective neutron moderators.
Since ^{208}Pb is a double magic nuclide with closed proton and neutron shells, radiative capture crosssection at thermal energy and resonance integral of ^{208}Pb is much smaller than the corresponding values of lighter nuclides. Therefore, it can be expected that even with multiple scattering of neutrons on ^{208}Pb during the process of their slowingdown, they will be slowed down with a high probability and will create high flux of slowed down neutrons.
So, thanks to very small neutron capture cross section, the moderating ratio (see Table
Properties of neutron moderators at 20°C.
Moderator  Average logarithmic energy loss 
Moderating ratio 
Neutron age 
Diffusion length 
Mean lifetime 

H_{2}O  0.95  70  6  3  0.2 
D_{2}O  0.57  4590  58  147  130 
BeO  0.17  247  66  37  8 
^{ 12}C  0.16  242  160  56  13 
Pb_{nat}  0.00962  0.6  3033  13  0.8 
^{ 208}Pb  0.00958  477  2979  341  598 
It is noteworthy that mean lifetime of thermal neutrons in ^{208}Pb is very large (~0.6 s). This effect could be used to essentially improve the safety of the fast reactor by slowing down progression of chain fission reaction on prompt neutrons [
It should be noted that ^{208}Pb is not the only of the kind nuclide in Mendeleev’s periodic system whose neutronphysical properties are very specifics. For example, nuclide ^{88}Sr has a very small neutron capture as well, inelastic cross section even smaller than that for ^{208}Pb nuclide, and values of nuclear moderating capabilities
Table
Neutronphysical characteristics of different materials.
Material 
Slowingdown probability 


Neutron lifetime (ms)  

Slowing down  Thermal  
Pb_{nat}  0.304  134  30  0.56  0.8 
^{ 208}Pb  0.993  134  835  0.56  598 
D_{2}O  0.999  19  360  0.01  130 
Graphite  0.998  31  138  0.03  13 


Radiogenic Pb from different deposits  


Brazil  0.930  134  186  0.56  29 
Australia  0.914  134  142  0.56  17 
The USA  0.885  134  129  0.56  14 
Ukraine  0.915  134  145  0.56  18 
It is worthy to note that, in both cases, that is, during the slowingdown process of fast neutrons in natural lead and in ^{208}Pb, mean distance of neutron transport and mean slowingdown time are approximately the same (~134 cm and 0.56 ms). As for thermal neutrons, mean distances of neutron diffusion until absorption are quite different for the two lead types (30 cm in natural lead and 835 cm in ^{208}Pb).
This means that, first, very small fraction of neutrons that were slowed down in natural lead reflector can come back into the reactor core. On the contrary, ^{208}Pb reflector gives them such a possibility. Second, mean lifetime of thermal neutrons in the infinite ^{208}Pb environment (0.6 s) is longer by three orders of magnitude than that in natural lead (0.8 ms). So, main process in ^{208}Pb reflector is a neutron slowing down, not neutron absorption, and these slow neutrons, after a diffusion (and time delay), have a probability to come back into the reactor core and sustain the chain fission reaction. Mean lifetimes of slow neutrons and, especially, thermal neutrons are substantially longer than those for fast and slowingdown neutrons. This constitutes a potential for significant extension of mean prompt neutron lifetime in the chain fission reaction.
It is noteworthy that neutronphysical parameters of radiogenic lead extracted from thorium and thoriumuranium ore deposits in Brazil, Australia, the United States, and the Ukraine are inferior to those of ^{208}Pb, but they are substantially better than those of natural lead (radiogenic lead compositions from different deposits are presented in the appendix). For comparison, Table
In nature there are two types of elemental lead with substantially different contents of four stable lead isotopes (^{204}Pb, ^{206}Pb, ^{207}Pb, and ^{208}Pb). The first type is a natural, or common, lead with a constant isotopic composition (1.4% ^{204}Pb, 24.1% ^{206}Pb, 22.1% ^{207}Pb, and 52.4% ^{208}Pb). The second type is a socalled radiogenic lead with very variable isotopic composition. Radiogenic lead is a final product of radioactive decay chains in uranium and thorium ores. That is why isotopic compositions of radiogenic lead are defined by the ore age and by elemental compositions of mixed thoriumuranium ores sometimes with admixture of natural (common) lead as an impurity. The isotopes ^{208}Pb, ^{206}Pb, and ^{207}Pb are the final products of the radioactive decay chains starting from ^{232}Th, ^{238}U, and ^{235}U, respectively:
It should be noted that neutron capture crosssections of ^{206}Pb, although larger than those of ^{208}Pb, are significantly smaller than those of ^{207}Pb and ^{204}Pb. Thus, at first glance it appears that the ores containing ~93% ^{208}Pb and 6% ^{206}Pb (Table
Main deposits of uranium, thorium, and mixed uraniumthorium ores. Elemental compositions of minerals and isotope compositions of radiogenic lead.
Deposit  U/Th/Pb (wt.%)  ^{ 204}Pb/^{206}Pb/^{207}Pb/^{208}Pb (at.%)  Age (×10^{9} yr) 

Monazite, Guarapari, Brazil  1.3/59.3/1.5  0.005/6.03/0.46/93.5  0.52 to 0.55 
Monazite, Manitoba, Canada  0.3/15.6/1.5  0.01/10.2/1.86/87.9  1.83 to 3.18 
Monazite, Mt. Isa Mine, Australia  0.0/5.73/0.3  0.038/5.44/0.97/93.6  1.00 to 1.19 
Monazite, Las Vegas, Nevada  0.1/9.39/0.4  0.025/9.07/1.13/89.8  0.77 to 1.73 
Uraninite, Singar Mine, India  64.3/8.1/8.9  —/89.4/6.44/4.18  0.885 
Monazite, South Bug, Ukraine  0.2/8.72/0.9  0.01/6.04/0.94/93.0  1.8 to 2.0 
So, radiogenic lead can be taken as a byproduct from the process of uranium and thorium ores mining. Until now, extraction of uranium or thorium from minerals had been followed by throwing radiogenic lead into tail repositories. If further studies will reveal the perspective for application of radiogenic lead in nuclear power industry, then a necessity arises to arrange byextraction of radiogenic lead from thorium and uranium deposits or tails. Evidently, the scope of the ores mining and processing is defined by the demands for uranium and thorium.
However, the demands of nuclear power industry for thorium are quite small now and will remain so in the near future. Nevertheless, there is one important factor that can produce a substantial effect on the scope of thorium and mixed thoriumuranium ore mining. In the majority of cases, uranium and thorium ores belong to the complexore category, that is, they contain minor amounts of many valuable metals (rareearth elements, gold and so on).
The paper by Sinev [
The balance relationship can be used to evaluate mean prompt neutron lifetime in a multizone fast reactor with the reactor core surrounded by neutron reflectorconverter to slow down the chain fission reaction under accidental power excursions. As is expected, the balance relationship allows us to determine spatial (zonewise) contributions into mean prompt neutron lifetime.
To make the derivation as clear as possible, the simplest spherical twozone model (Figure
Twozone model of nuclear reactor.
As is wellknown [
Spaceenergy distributions of neutron flux and its adjoint function are defined by the following operator equations:
As applied to the twozone model under consideration here (the reactor core surrounded by the neutron reflector), the following designations will be used below: the bracket
Two problems must be solved to evaluate prompt neutron lifetimes in the reactor zones. The first problem consists in solving (
Model spatial distributions of neutron flux in the reflected core
Some fraction of those neutrons which, being generated in the reactor core, escaped the core, went into the neutron reflector and then came back into the reactor core, could contribute to timedependent evolution of the chain fission reaction. Just these neutrons can be characterized by a relatively longer lifetime due to the lengthy processes of their diffusion, scattering and slowing down in the weakly absorbing neutron reflector, due to the processes of their coming back to the reactor core and contributing into the chain fission reaction.
Equation (
Now we can formulate the balance relationship for twozone reactor model. The relationship can link mean prompt neutron lifetime
If the neutron reflector is divided into several annular layers, then the approach presented above can be applied to determine mean lifetime of those neutrons, which visited each layer of the reflector. Thus, dependency of mean neutron lifetime on depth of neutron penetration into the reflector can be calculated. Then, the approach can be generalized for continuous model of the neutron reflector divided into infinite number of infinitesimally thin annular layers. In this case, the balance relationship for
The inversehour equation for the fast reactor core surrounded by the neutron reflectorconverter can be written in the following form with accounting for six groups of delayed neutrons:
For this case the roots of the inversehour equation (
Roots of the inversehour equation for two types of the reflected fast reactor core (
Fast reactor with fertile fuel breeding zone
Fast reactor with neutron reflectorconverter
As the neutron reflectorconverter adjoins closely to the reactor core, neutron lifetime in the nearest reflector layer differs slightly from
The neutron kinetics equations for the fast reactor core surrounded by the neutron reflectorconverter (without accounting for any feedback effects) can be written in the following form:
Unfortunately, one else problem takes place in numerical analysis of neutron kinetics in the fast reactor core surrounded by the neutron reflectorconverter. The matter is the neutrons coming back from the reflector into the reactor core and contributing into the power excursion process are characterized by lifetimes that continuously cover the range from
The following specific feature must be also taken into consideration. The feature is related with time dependency of the integrand in the second summand of the inversehour equation (
Roots of the inversehour equation for FR with the neutron reflectorconverter (
FR with the neutron reflectorconverter
FR with the neutron reflectorconverter (the neutrons coming back from the reflector with lifetimes
Item “b” (plus the “zero prompt neutron lifetime” in the reactor core)
By using the standard mathematical operations (described in [
system (
time scale (lifetimes of neutrons and emitters of delayed neutrons) now covers a relatively longer range (from 10^{−4} s to 10 s).
Solution of system (
As to the function
Depending on the power excursion rate, some feedbacks can obtain a long enough time interval to change evolution of the process. Dopplereffect, for example, can change resonance neutron absorption at fuel warming up instantly. On the contrary, some time interval must elapse before the coolant density effect caused by its warming up actuates (heat must have a time to flow down from fuel to coolant). Thermal constant of fuel rods
Thermal constant
So, it can be
Neutron kinetics equations in the case of fast power excursion with accounting for Dopplereffect caused by fuel warming up can be written in the following form for the FR core surrounded by the neutron reflectorconverter:
According to the wellknown NordheimFuchsHansen (NFH) model [
Effective fraction of delayed neutrons
The right part of the first equation contains new summands which define the neutrons coming back from the reflector and delayed neutrons from decays of their emitters. These summands must be added because the time range for the procrastinated return of neutrons from the reflector stretches from neutron lifetime in the reactor core up to lifetime of the most shortlived group of delayed neutrons.
System of neutron kinetics equations must contain equations for the neutrons coming back from the reflector and for delayed neutrons.
The approach described in [
The authors declare that there is no conflict of interests regarding the publication of this paper.