2.7 Spontaneous fission
In addition to alpha decay, many heavy isotopes suffer spontaneous fission (SF), a process which has its own characteristic half-life for every isotope. Neutrons are emitted in SFs, so this creates a predetonation hazard like the (α, n) problem. For most isotopes, SF half-lives are very great, but there are no mitigating yield or number-density factors to help suppress the resulting predetonation probability: SF neutrons are emitted directly by the fissile material, quite independently of the presence of any impurities.
SF rates can be calculated with equation (2.13) as for alpha-decay rates. For U-235, the SF half-life is 1019 years, which gives about 5.63 × 10−6 SF g−1 s−1. For a 60 kg core, this corresponds to 0.34 SFs per second; if each SF liberates three neutrons, we have on average about one neutron per second. For a 100 μs core-assembly time, the chance of a predetonation due to SF is thus extremely remote. For Pu-239, the numbers are somewhat more pessimistic, although not disastrous. This isotope has a SF half-life of 8 × 1015 years; for a 10 kg core this gives some 69 SFs per second, or about 200 neutrons per second. For the same 100 μs assembly time this means an average emission of only about 0.02 neutrons, still very small.
So far, all looks good on the SF front. However, Pu-239 has a much more serious predetonation issue than either its (α, n) or SF rate. When Pu-239 is created in a reactor, it is inevitable that a small amount of Pu-240 is also created by neutron capture by already-formed Pu-239 nuclei. Pu-240 has a much shorter SF half-life than Pu-239, about 1.14 × 1011 years. This corresponds to about 483 000 SFs per second per kg, which means that if a 10 kg Pu-239 core is contaminated with only 1% 240Pu, we can expect on average about five SFs over 100 μs, or some 10–15 neutrons—enough to practically guarantee a predetonation. During the Manhattan Project, it was essentially impossible to remove the offending Pu-240 (which was probably present to a level of a few percent), and it was for this reason that it was necessary to develop the high-speed implosion technique to crush a small Pu-239 core to critical density over a timescale on the order of a microsecond or so in order to trigger the weapon; the concept of critical density is described in chapter 3. The idea of using a small core is that it will have a low SF rate; implosion was the only way to beat the Pu-240 spontaneous fission problem. Pu-239 is an example of how an isotope that looks very promising as a nuclear explosive on the basis of its neutron-capture energetics and own intrinsic stability can be compromised by production factors largely beyond human control.
To this point, the discussion has revolved around U-235 and Pu-239. What are the prospects for turning other heavy-element isotopes into weapons materials?
A survey of the Chart of the Nuclides reveals 23 isotopes of 9 elements with atomic numbers 88 ⩽ Z ⩽ 97 (radium through berkelium) with decay half-lives ⩾1000 years; no isotopes of any heavier elements have half-lives for any decay process in excess of this duration [2]. All of these are alpha-decayers with the single exception of Np-226, which suffers electron capture with a half-life of 153 000 years. These are listed in table 2.1, along with the differences between their neutron pairing energies and the fission barriers of the next-heaviest isotopes (that is, Q − Ebarr (), in MeV), along with (α, n) and SF rates calculated from tabulated values of the half-lives for these processes. These rates span several orders of magnitude; what is tabulated is their base-10 logarithms. The (α, n) rates assume (nimpurity/nfissile) ∼ 1/20 000, y = 10−4, and Aimpurity = 9 (beryllium) in equation (2.14). No fission barrier data are available for Cm-250 or Bk-247; Th-229, Np-236, Cm-247, and Bk-247 lack published SF half-lives. Cm-250 and Bk-247 would both undergo even-to-odd neutron-number transitions under neutron capture, and so would likely not be fissile in any case.
Table 2.1. Q − Ebarr () values (MeV), (α, n) rates, and spontaneous fission (SF) rates for heavy-element isotopes with half-lives exceeding 1000 years. The (α, n) and SF rates are quoted as the common (base 10) logarithm of the rate per kilogram of material per 100 μs; the (α, n) rates assume (nimpurity/nfissile) ∼ 1/20 000, y = 10−4, and Aimpurity = 9 (beryllium) in equation (2.14). Q-values are computed from mass excesses listed in the Nuclear Wallet Cards published by the National Nuclear Data Center, [3]. This was also the source for the alpha-decay half-lives. Spontaneous fission half-lives were adopted from [4]. Fission barriers were adopted from an International Atomic Energy Agency document [5].
Figure 2.2 summarizes these data in graphical form. Promising candidates for weaponizability will lie toward the upper-left quadrant of the figure, corresponding to a combination of high fissility and low spontaneous activity. Those lying in the lower-left quadrant are not fissile, and those in the lower-right quadrant can be eliminated from consideration on account of their very high decay activities and lack of fissility. The dashed vertical line corresponds to an (α, n) or SF rate greater than one neutron per 100 μs, the time adopted above as typical for assembling the core of a ‘gun-type’ fission weapon.
Figure 2.2. Plot of data in table 2.1 to illustrate potential weaponizability of heavy isotopes. Q − Ebarr () is plotted on the vertical axis (in MeV), and the larger of the logarithm of the spontaneous fission rate and the logarithm of the (α, n) neutron emission rate per kg per 100 μs is plotted on the horizontal axis. More promising candidates for weaponizability lie toward the upper left of the plot. Reproduced from [6].
Aside from U-235 and Pu-239, four isotopes lie in the upper left quadrant of the plot: Th-229 (barely), U-233, Np-236, and Cm-247. Curium is quite scarce, with perhaps only a few grams being produced each year, so its isotopes are not practical weaponizability candidates. Th-229 is scarce as it is the alpha-decay product of U-233 (see below); also, its fission cross-section is small and its (α, n) rate is very similar to that of Pu-240.
The case of U-233 is similar to the Pu-239/Pu-240 problem. U-233 is bred in reactors by neutron capture by Th-232 and two subsequent beta-decays through protactinium to uranium:
However, this process also inevitably creates some U-232 by neutron capture and so-called ‘double-neutron emission’ by the U-233 so created:
The problem is that U-232 is a copious alpha-emitter by virtue of its short 70 year half-life, and is also a gamma-ray emitter; even a small amount of it will render a U-233 core highly radioactive and susceptible to severe (α, n) predetonation.
The case of Np-236 is tied up with its non-fissile sister isotope Np-237. The latter is formed in reactors by infrequent successive neutron captures by U-235 and then U-236 when fission does not occur; the U-237 so created beta-decays to Np-237 after a half-life of 6.7 days. While Np-236 is fissile and potentially weaponizable, it is even rarer than Np-237 as it is formed by neutron capture and double neutron emission by Np-237, and it is virtually impossible to separate the two.
In summary, the only viable weaponizable isotopes appear to be U-235, Pu-239, and perhaps U-233 and Np-236, although the latter two have severe disadvantages. For practical purposes, any country or organization working to produce their first nuclear weapons is restricted to obtaining U-235 or Pu-239.
Exercise
Verify the numbers in table 2.1 for U-233. For the reaction , the Δ-values are (left to right) 8.071, 36.921, and 38.148 MeV; the fission barrier of U-234 is 5.5 MeV, and the alpha-decay and SF half-lives of U-233 are respectively 1.592 × 105 and ∼2.7 × 1017 years.
References
[1] www.nndc.bnl.gov
[2] Baum E M, Ernesti M C, Knox H D, Miller T R, Watson A M and Travis S D 2010 Chart of the Nuclides 17th edn (Schenectady, NY: Knolls Atomic Power Laboratory) Available at http://www.knollslab.com/nuclides.html
[3] http://www.nndc.bnl.gov/wallet/wccurrent.html
[4] Holden N E and Hoffman D C 2000 Spontaneous fission half-lives for ground-state nuclides Pure Appl. Chem. 72 1525–62
[5] https://www-nds.iaea.org/RIPL-2/fission.html
[6] Reed B C 2017 An examination of the potential fission-bomb weaponizability of nuclides other than 235U and 239Pu Am. J. Phys. 85 38�
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IOP Concise Physics
The Manhattan Project
A very brief introduction to the physics of nuclear weapons
B Cameron Reed
* * *
Chapter 3
Criticality and efficiency
Even if one arranges for a neutron to cause a fission in a sample of uranium or plutonium, the secondary neutrons that are liberated are by no means guaranteed to strike other nuclei and establish a chain reaction. Some of the neutrons may reach the surface of the sample and escape, particularly if it is small; in fact, some escapes are inevitable. But as the size of the sample increases, it becomes more and more likely that a neutron will cause a subsequent fission than it is to escape. At some point the sample size will be such that as many neutrons are being created per second as are being lost; one then has a critical mass. For a sample greater than this size you will have a supercritical reaction in which more neutrons are being created per second than are being lost; in principle, the reaction will continue until all of the nuclei are fissioned. This never occurs in practice (section 2.4), so even in a supercritical device there will still be the question of efficiency.
This chapter investigates critical mass and efficiency. We will see that the term ‘critical mass’ is in fact not uniquely defined. In its simplest incarnation, the preferred term is the so-called ‘bare’ critical mass, which means the mass of a sphere of fissile material that is just critical if it is not surrounded by any mechanism designed to reflect escaping neutrons back into the sphere and so give them further chances of causing fissions. Such a system usually takes the form of a surrounding jacket of a metal which is an efficient neutron reflector, that is, one with a high elastic-scattering cross-section. Such a jacket is known as a tamper. The question of ‘bare core’ criticality is analyzed in section 3.1, while ‘tamped cores’ are examined in section 3.2.
3.1 Bare criticality
The value of the critical mass depends on four factors. Most important is the fission cross-section: a larger cross-section means that nuclei effectively present themselves as larger targets to incoming neutrons. In addition, the cross-section for elastic scattering of neutrons by nuclei also comes into play: if a nucleus scatters a neutron, the latter can go on to strike another nucleus and perhaps induce a fission. For both U-235 and Pu-239, scattering is in fact more likely than fission for fast neutrons. Of equal importance is the number of secondary neutrons emitted per fission. Here again, more is better: some neutrons do escape, so, phrased colloquially, the more that are ‘born’ per fission generation, the more are available to ‘reproduce’ themselves in subsequent fissions. Finally, we will see that the density of the material affects critical mass: as you might guess, higher density means a lower critical mass because neutrons are more likely to find a nucleus to react with before they escape.
We begin with the roles of the cross-sections. Recall the concept of mean free path for fission that was introduced in section 2.2. This concept and an analogous one known as the ‘transport mean free path’ express the roles of fission and scattering cross-sections in criticality calculations. These quantities are respectively symbolized as λfiss and λtrans, and are given by
and
In these expressions, σfiss is the fission cross-section for the material, and σtrans is its transport cross-section. To a reasonable approximation, the transport cross-section is given by the sum of the fission and elastic-scattering cross-sections:
In words, λfiss and λtrans mean the average distance a neutron will travel before it is consumed in causing another fission, and the average distance a neutron will travel before it is scattered or causes a fission.
The symbol n in equations (3.1) and (3.2) represents the number density of nuclei, that is, the number of nuclei per cubic meter. If the material has density ρ grams per cubic centimeter and atomic weight A grams per mole, then n in nuclei per cubic meter is given by
As in section 2.2, another important quantity is the average time that a neutron will travel before causing a fission; this is again designated by the symbol τ. If neutrons have average speed vneut and travel for an average distance λfiss before causing a fission, then it follows (as before) that
In the case of an untamped spherical bomb core, diffusion theory shows that criticality will occur when the radius of the core, Rcore, satisfies the condition:
In this expression, d has units of meters and can be thought of as a characteristic core radius for criticality. It is given by
where v is the number secondary of neutrons liberated per fission. The quantity η in equation (3.6) is dimensionless, and involves the ratio of the transport mean free path to the characteristic size:
Because of the presence of the cotangent, equation (3.6) cannot be solved analytically for Rcore; it can only be solved numerically.
As remarked above, an untamped core is also known as a bare core, and the value of Rcore obtained by solving equation (3.6) is thus known as the bare critical radius, Rbare; the corresponding bare critical mass Mbare then follows from the density. Table 3.1 shows numbers for U-235 and Pu-239. (Reminder: cross-sections are usually measured in ‘barns’; 1 bn = 10−28 m2.)
Table 3.1. Parameter values and critical radii and masses for bare criticality. Cross-sections and secondary neutron numbers represent values averaged over the energy spectrum of fission-liberated neutrons. Data from [1].
For comparison, an American softball has a radius of about five centimeters and a mass of about 180 grams; the bare critical radius of Pu-239 is only slightly larger, but corresponds to about 90 times as much mass. Forty-six kilograms is equivalent to about 101 pounds, and 16.7 kg to about 37 pounds. As in section 2.2, note that the neutron travel-times-to-fission, τ, are on the order of nanoseconds: nuclear explosions are incredibly brief phenomena.
Exercise
A hypothetical fissile material has A = 250 g mol−1, ρ = 17.4 g cm−3, σfiss = 1.55 bn, σelas = 6 bn, and v = 2.95. Determine its critical radius and mass.
Answer
6.70 cm, 21.9 kg.
3.2 Supercriticality and the radius-density effect
Another result of diffusion theory is that if you start with a core of more than one critical mass, then the rate of creation of neutrons will for a time exceed the rate of escapes. As remarked above, this condition is called supercriticality. With a supercritical core, the production of neutrons (and the energy release) will grow exponentially so long as criticality holds. But beyond the advantage of achieving more energy release, there is a very important reason why it is desirable to assemble a core of more than one critical mass in the first place. This relates to the estimate in section 2.4 that a chain-reacting core will rapidly heat up, expand, and disperse itself over a timescale on the order of microseconds.
To analyze this, look at equation (3.8) for the dimensionless parameter η. This quantity depends only on the cross-sections and v, none of which can be changed by any human intervention; once they are chosen, η is fixed. Be sure to convince yourself in particular that the density of the material cancels out when computing η. With η determined, equation (3.6) will be satisfied by some unique value of Rcore/d. But d is proportional to the inverse of the mass density ρ through the mean free paths and number density n (equations (3.1)–(3.4)), so we can equivalently say that the solution of equation (3.6) will be some unique value of ρRcore for a given value of η. The ‘criticality value’ of ρR can be determined by multiplying the bare radii listed in table 3.1 by the corresponding densities, which are the normal densities of U-235 and Pu-239. These evaluate to ρnormal Rbare = 156.5 g cm−2 for U-235, and to 99.0 g cm−2 for Pu-239. This means that any core of U-235 of some radius R and density ρ will be critical if the product ρR exceeds 156.5 g cm−2; similarly for Pu-239. If ρR is exactly equal to the critical value, the core will be equivalent to bare critical; if ρR exceeds the critical value, the core will be supercritical.
Now consider what happens as a supercritical core begins
expanding. The core will start out with a radius Rcore > Rbare, but, as its radius increases due to expansion, the density must drop. If at any time the density is ρ and the radius R, what happens to the product ρR as time advances? The mass M of the core is essentially fixed, so the density must behave as . This means that, at any time, . Consequently, ρR will eventually fall below the critical value ρnormal Rbare, at which time criticality will be lost. But if you start with only a single critical mass, this would happen as soon as the core begins expanding, which would be essentially immediately. It is to avoid this prompt shutdown that it is necessary to assemble a multiple-critical-mass core with an initial radius Rcore greater than Rbare. In technical parlance, the moment of shutdown is known as ‘second criticality’. ‘First criticality’ denotes that moment during the assembly of a bomb core when a critical mass has been achieved and a chain reaction could start.
The Manhattan Project Page 4