The Physics of Energy, page 73
Figure 19.1 provides a visual summary of these first two steps in the thermal-neutron cycle; in the figure .
Figure 19.1 A sketch showing the physics behind the neutron reproduction factor η and the fast fission factor ϵ in a thermal reactor. measures the probability that a thermal neutron (blue) produces a fast neutron (red) after encountering a uranium nucleus. measures the enhancement in the number of fast neutrons resulting from fast neutrons that cause a nucleus to fission. Note that the likelihood of fission by a thermal neutron is negligible.
p – the resonance escape probability To induce fission of another nucleus, a fast neutron must be slowed to thermal velocity by multiple collisions with the moderator. As it slows, the neutron must avoid capture by a nucleus, which has resonances in its radiative capture cross section – especially in the neutron energy range between ~10 eV and ~10 keV (see Figure 18.4). The resonance escape probability p measures the probability that the neutron reaches thermal equilibrium without being captured. This factor is not easy to calculate (see Figure 19.2 for a pictorial representation of the process). The analysis simplifies somewhat for a homogeneous reactor, but it is still complicated. An approximate, empirical result for a homogeneous system with is [91]
(19.9)
where is the number density of in the reactor, is the macroscopic scattering cross section for the moderator plus fuel in barns per unit volume, and is the logarithmic energy decrement for the mixture of moderator and fuel. When , the moderator dominates so that .
Figure 19.2 A sketch showing the physics behind the resonance escape probability p. Fast neutrons propagating through a homogeneous mixture of moderator and fuel slow down to thermal velocities by repeatedly scattering, primarily from moderator nuclei. The neutron flux is reduced by radiative capture at energies that coincide with resonances in the radiative capture cross section.
As one would expect for an escape probability, p given by eq. (19.9) increases with ξ (better moderation) and with (more encounters with the moderator), and decreases with (higher density of absorbers).
The resonance escape probability for an infinite, homogeneous reactor increases monotonically as a function of the moderator-to-fuel ratio y but depends only weakly on the enrichment x for the small values of x encountered in thermal reactors.
f – the thermal utilization factor The thermal utilization factor f measures the probability that a neutron, once thermalized, is absorbed by a uranium nucleus rather than by the moderator, as illustrated in Figure 19.3,
(19.10)
For fixed moderator-to-fuel ratio y, f increases with enrichment because is a more efficient absorber than ; for fixed enrichment x, on the other hand, f decreases as the moderator-to-fuel ratio y increases, since the higher the fraction of moderator, the greater the chance it will absorb the neutron. With this step, the thermal neutron has entered the fuel and the cycle is complete. Note that (19.10) determines f for an idealized reactor containing only fuel and moderator; if other materials (such as fission poisons, §19.1.7) that absorb neutrons are present, their cross section must be added to the denominator.
Figure 19.3 Thermal neutrons wander through the moderator–fuel mixture until they encounter a uranium nucleus and begin the neutron reproduction cycle again (Figure 19.1) or they are absorbed by the moderator.
Neutron Multiplication Factor
The neutron multiplication factor k measures the number of neutrons of the generation produced by each neutron of the generation. In an infinite reactor,
where η measures the average number of fast neutrons generated by each thermal neutron that is absorbed by a uranium nucleus; ϵ accounts for extra neutrons produced by fast fission; p measures the probability that a fast neutron escapes capture and thermalizes; and f measures the probability that a thermal neutron is absorbed by the fuel (leading to capture or fission) as opposed to the moderator or other material. Generally, while .
k must be kept very close to one in order to have a controlled, self-perpetuating chain reaction in a nuclear fission reactor.
Example 19.1 illustrates how the four factors can be used to estimate the k factor and reactivity of a reactor for a realistic choice of moderator-to-fuel ratio and a range of enrichments. Figure 19.4 shows several examples of the dependence of and its components on enrichment and choice of moderator.
Example 19.1 The Reactivity of a Water-moderated, Infinite, Homogeneous Reactor
What is the reactivity of a water-moderated, uranium-fueled reactor with moderator-to-fuel ratio as a function of enrichment from to ?
Recall that the reactivity (19.2) is measure of the neutron multiplication factor and must be zero in order to sustain a chain reaction in steady state. To compute , we first compute the reproduction factor η, which depends only on the enrichment x. Substituting the necessary cross sections from Table 19.1 into eq. (19.8),
As discussed in the text, for an infinite homogeneous reactor, we approximate the fast fission factor by . Next, the resonance escape probability p must be estimated using eq. (19.9). Since and , we have . Thus,
Using and and substituting cross sections from Tables 19.1 and 19.2, we find
For , p is only weakly dependent on x and can be approximated by its value at , which is . Finally, is obtained from eq. (19.10),
The figure at right shows the resulting and reactivity obtained from the four-factor formula. The reactivity becomes positive around , which means a fission reaction can, in principle, be sustained in a homogeneous, water-moderated, infinite reactor at enrichment levels above 1.3%.
Figure 19.4 factors and their components, η, p, and f, for three infinite, homogeneous, uranium reactors . (a) A graphite-moderated reactor fueled with natural uranium; (b) same as (a), enriched to 1.6% ; (c) a water-moderated reactor fueled with uranium enriched to 3.7%, appropriate for the Oklo natural reactor (see Box 19.1). must be greater than one for a chain reaction to be possible.
From these analyses of the four-factor formula, we can see some of the tradeoffs involved in the choice of moderator and enrichment level for a reactor. While D2O is an excellent moderator and works with natural uranium, it is quite expensive. As a result, graphite and ordinary water are the moderators used in most commercial reactors. Figures 19.4(a) and (b) show that it is not possible to have a graphite-moderated, infinite, homogeneous reactor fueled with natural uranium, but that enhancement to 1.6% will work. Note that the ratio of moderator to fuel is very large for a graphite-moderated reactor. This is because moderating a neutron in graphite requires many more collisions than for D2O or H2O (Table 18.6), so for the resonance escape probability p to have a high value, the moderator to fuel ratio y must be large. As a result, graphite-moderated reactors are generally larger than water-moderated reactors, an important design consideration. While water requires fewer collisions to moderate a neutron than graphite, it is also a strong neutron absorber, so f drops rapidly with y. A functioning water-moderated reactor therefore requires a relatively high enrichment level to compensate for the small value of f by increasing η. Figure 19.4(c) shows the components of for a water-moderated reactor at 3.7% enrichment, a value typical of modern power reactors. 3.7% was also approximately the fractional abundance of when the Oklo natural reactor went critical some two billion years ago (see Box 19.1).
Box 19.1 The Oklo Natural Reactor
Uranium occurs naturally in the environment, and ordinary water is a good moderator. The two could combineto produce a natural reactor except for the fact that the percentage of in natural uranium is too low (see the figure in Example 19.1) to support a self-sustaining reaction. was more abundant in the past, however; the fraction of in natural uranium has gradually decreased over time because its half-life is a factor of six shorter than that of . In 1956 Japanese nuclear chemist Paul Kuroda [92] suggested that natural fission reactors may have existed in rich uranium deposits in Earth's past. Very early in Earth's history, oxygen was rare. The chemically reduced uranium compounds that existed at that time are not readily soluble in water, limiting the extent to which they might be concentrated by geological processes. As time went on, plants added oxygen to Earth's atmosphere, and by about two billion years ago, soluble uranium salts had become more common, settled out of solution, and accumulated into relatively rich deposits. The stage was set for the ignition of a natural fission reactor.
In 1972 French scientists measuring the uranium enrichment of ore samples from the Oklo uranium deposits in Gabon discovered that the proportion of was 0.7171% rather than the normal concentration of 0.7253%. Even this slight difference is significant because the only known way to reduce the relative abundance of is by induced fission. Further investigation indicated that the abundance of various isotopes of neodymium and ruthenium in the ore was biased toward isotopes that are produced as fission fragments. Subsequent studies have found regions at Oklo where the / ratio is a low as 0.292% [93]. There is now little doubt that natural fission reactors evolved in Oklo ores over extended periods beginning about 1.7 billion years ago [94], and there by depleted the content of the ores.
The Oklo uranium deposits show evidence of fission activity in at least 16 different sites. All the necessary ingredients for a fission reactor were present: the ores were rich in uranium, in places containing over 90% UO2 [93]. The percentage of was 3.67% at the time the reactors began. The ores were embedded in porous sandstones that were periodically submerged or saturated with water in a wet climate. The ores were also relatively low in nuclei with large neutron-absorption cross sections (other than ). Furthermore, the ore bodies were large enough that neutron leakage did not prevent the chain reaction. The occasional spontaneous fission of a or nucleus provided the seed neutrons to initiate a fission chain reaction when the water saturation level was high enough to provide the necessary moderation.
The reaction apparently generated temperatures as high as 1000℃ at the core and even 250–450℃ at theboundaries of the reactor zone. These temperatures were high enough to drive away the water moderator, thereby ending the chain reaction. Geologists have evidence that suggest cycles of hydration, reaction, desiccation, and rehydration on time scales of several hours lasting for hundreds of thousands of years. Over the time scaleof these cycles, fission poisons presumably were present at relatively constant background levels, limiting the power of the reactor.
The Oklo story represents a remarkable bit of Earth history.
19.1.5 Power and Fuel Consumption
In this section we give a simplified analysis of the rate of power production and fuel depletion in an infinite homogeneous reactor, to give a sense of the quantities and time scales involved. When a fission reactor is started up – usually by partially removing control rods – the reactivity ρ is made positive and kept positive long enough for a fission chain reaction to become established. The flux of neutrons in the reactor increases while , along with the fission rate. As the reactor heats up, an increasing amount of thermal power can be extracted from the reactor core. When the reactor reaches the desired power, ρ is decreased back to zero and kept there so that the reactor runs at a steady rate.
At first a uranium-fueled reactor burns only . As time goes on, a significant amount of is converted to fissile , which also serves as fuel. A detailed calculation of power output and fuel consumption focuses on the neutron flux, which determines the rates of depletion of , of conversion of into , and of fission of . Here for simplicity we consider the situation where only and are present, as is the case just after a uranium-fueled reactor is first powered up. This allows us to circumvent the detailed analysis of the neutron flux, and relate the power generated by the reactor directly to the rate of fission ,
(19.11)
Here MeV includes all the energy released in the fission of except for the energy that escapes on neutrinos. Note, in particular, that it includes the MeV given off (on average) when prompt neutrons are radiatively captured by other nuclei in the reactor. The rate at which is being depleted in the reactor is proportional to the fission rate
(19.12)
where the factor takes account of the fact that is depleted by radiative capture of thermal neutrons as well as by fission.2 The cross sections and change rapidly with neutron energy (see, e.g., Figure 18.5(a)) and should be averaged over the neutron energy distribution in the reactor. The ratio, , however, varies only slowly over the range of neutron energies found in a thermal-neutron reactor, so the cross sections can be taken to be those given in Table 19.1 for purposes of computing this ratio, independent of the precise temperature or neutron flux in the reactor. Combining eqs. (19.11) and (19.12), the reactor power can be related to the rate of depletion of fuel,
(19.13)
Since a reactor is typically run with constant power output, the rate of depletion of is constant. Thus, at start-up, when only is being burned, eq. (19.13) allows us to relate the reactor power to the mass of uranium fuel (enriched to a fraction x of ) that is used in a time
(19.14)
where m is the molar mass of uranium.
Fission Reactor Power
As a rule of thumb, the thermal power per ton of fuel, generated by a fission reactor burning uranium enriched to a fraction x of , and designed to have a fuel consumption time constant of years, is roughly
When losses during enrichment and to incomplete fuel consumption are included, this is consistent with the estimate of 200 t/GWe of natural uranium quoted in §16.
During the course of its operation, produced in the reactor increases the supply of fissile material and therefore decreases the rate of consumption. In addition, fast-neutron fission of contributes to the power output without depleting the supply of . Together, these effects enhance the power relative to the fuel consumption by a factor of ~1.5. Thus the power-to-fuel-consumption ratio over the entire period that the fuel is in the reactor can be approximated by multiplying eq. (19.14) by this factor,
(19.15)
where we have substituted values for the cross sections, , and m. For example, running a 3 GWth reactor for one year consumes t of (). This can be shown to be consistent with the statement in §16 that roughly 200 t of natural uranium are required to run a 3 GWth reactor for a year (Problem 19.10).
The fission rate is also directly related to the average flux of thermal neutrons by eq. (19.7). At startup, when only is present
(19.16)
where denotes an average over the volume of the reactor and the neutron energy distribution. Since is constant when a reactor is run at constant power, eq. (19.16) requires that the thermal-neutron flux must be increased with time as the amount of fissile material in the reactor is depleted.
To estimate the neutron flux in a thermal reactor, consider a 3 GWth reactor fueled with one tonne of at startup, and suppose that the reactor has reached its operating temperature of 320℃. The fission cross section for drops by roughly 1/3 as the temperature increases from 20℃ to 320℃ (see Figure 18.5), so we take b, and find cm s. Thermal-neutron fluxes of this order can easily be maintained in modern reactors, and there is considerable latitude in adjusting to keep constant as the amount of fissile material in the reactor decreases.
19.1.6 Reactor Time Scales and Delayed Neutrons
The operator of a nuclear reactor, whether human or machine, must be able to react on the time scale over which conditions in a reactor can change. In particular, since the number of neutrons grows exponentially when the reactivity is greater than one (c.f. eq. (19.1))
(19.17)
the time that it takes each generation of neutrons to produce the next is the critical time scale for reactor control.3 We can identify as the time required for the longest step in the chain of events that takes one generation of neutrons to the next. Many of the processes that contribute to are so rapid that they are negligible by any human standards. For example, the neutrons that accompany fission are emitted within seconds of the fission event. The other candidates for are (i) the time that it takes a fast neutron to slow to a thermal speed, or (ii) the time it takes a (prompt) thermal neutron to find another fissionable nucleus as the neutron diffuses through the reactor. Typically . For example, in a water-moderated thermal reactor, s while s [79]. We can ignore the shorter time scale and conclude that s for such a reactor. This sounds like bad news for reactor control: if were in fact of order s and if ρ were even as small as 0.001, the number of neutrons in the reactor would grow by roughly a factor of in just one second. So an operator would have to be able to adjust control rods on a time scale of milliseconds in order to prevent a runaway chain reaction! Even the most modern feedback and control systems are not up to this challenge.
So far we have examined only the prompt neutrons that are emitted in the fission event itself. As described in §18.3.5, however, a small fraction d of the fission neutrons ( for ) are emitted by fission fragments long after the fission takes place. In §18.3.5 we mentioned one mode that results in a neutron delayed by s in 0.14% of fissions. There are in fact six different fission fragments that emit delayed neutrons on time scales from 0.23 to 55 seconds. A proper analysis treats each of these delayed neutron groups separately. For a semi-quantitative estimate of the effect of delayed neutrons, it is sufficient to lump them all together with the weighted average lifetime of s. So there are two distinct time scales that are relevant, one for prompt neutrons () governing a fraction of the neutrons, and another () governing the remaining, delayed neutrons. If , then the delayed neutrons are not “in play” on the short time scale and the chain reaction does not grow exponentially on the time scale . Only after the time has passed are all the neutrons available (Problem 19.8).
