λfiss is known as the mean free path for fission; it will come up again in the discussion of critical mass in chapter 3. For U-235, λfiss ∼ 16.9 cm. A neutron of kinetic energy 2 MeV moves at a speed of about 1.96 × 107 m s−1, which gives τ ∼ (0.169 m)/(1.96 × 107 m s−1) ∼ 8.62 × 10−9 s, a remarkably brief interval. This implies that only ∼0.7 microseconds need elapse to achieve 80 generations. A real bomb will likely contain tens of kilograms of fissile material, but this argument indicates that the entire chain reaction will require only on the order of microseconds to occur even if all of the material undergoes fission.
Exercise
The mass of a neutron (A = 1) is 1.67 × 10−27 kg. Verify the speed given above for a kinetic energy of 2 MeV. If a fission fragment of mass number A = 100 has kinetic energy 80 MeV, what is its speed?
Answer
About 1.2 × 107 m s−1, similar to that of the neutron.
2.3 Temperature equivalent of fission fragments
From kinetic theory, temperature is equivalent to kinetic energy:
where k is Boltzmann’s constant, 1.38 × 10−23 J K−1. A fission fragment with kinetic energy 80 MeV travels at a speed equivalent to a temperature of nearly 6 × 1011 K, that is, about 600 billion Kelvins. The claim that nuclear weapons create conditions of temperature and pressure like those inside the cores of stars is entirely true.
2.4 A first glimpse of the efficiency issue
A very important issue in nuclear weapons design is the question of efficiency. The chain reaction timescale and fragment speeds derived in the preceding sections can be used to give a very rough estimate of bomb efficiency. This is a very preliminary treatment; a fuller examination appears in chapter 3.
From collisions of the fission fragments with other nuclei in a bomb core, the core will very quickly heat up and begin expanding as the chain reaction proceeds. We will see in chapter 3 that if a core expands by more than about a centimeter or two beyond its initial size, the conditions necessary to continue the chain reaction will no longer hold; this is because as the material spreads out, neutrons become less and less likely to find a nucleus with which to react before escaping. For simplicity, imagine that the core is expanding uniformly at some average speed due to the effects of fragment/nuclei collisions. If a full microsecond is to be available for the chain reaction to proceed to completion, then this average speed must be such as to keep the core expansion under a centimeter during the 1 μs. This gives a maximum tolerable average speed of ∼ (10−2 m)/(10−6 s) ∼ 104 m s−1, which is about 1000 times less than the typical fragment speed. For the average nucleus speed to reach this value, only about one nucleus in one million need fission, an efficiency of much less than 1%. This is obviously a very crude estimate, but it does indicate that it will likely be very difficult to have the entire core fission before criticality is lost. Indeed, the Little Boy U-235 bomb dropped on Hiroshima contained about 64 kg of U-235, and the resulting explosion has been estimated to have released about 13 kilotons; from these figures we can infer that only some 0.8 kilograms of the U-235 actually underwent fission, or about 1.2% of that available. The Nagasaki bomb achieved about 20% efficiency because of its implosion-based design.
2.5 Neutron pairing energy, the fission barrier, and plutonium
This section and the following two analyze why so very few isotopes are feasible for use in powering nuclear weapons.
Three main factors come into play in evaluating the suitability of an isotope for ‘weaponizability’: the energetics involved in the process of neutron capture, the long-term stability of heavy-element isotopes against various decay mechanisms, and the phenomenon of spontaneous fission.
In general, we can designate a nucleus of some isotope of mass number A of element X which has atomic number Z by the symbol . When such a nucleus captures an incoming neutron, it will become a heavier isotope of itself, a so-called ‘compound nucleus’. The compound nucleus will have a mass number which is greater by one than that of the original nucleus: . If the compound nucleus has a mass which is less than the sum of the masses of the neutron and the original nucleus (which is the case for all isotopes considered here), the ‘lost mass’ becomes energy thanks to E = mc2. This energy is called the ‘neutron pairing energy’, and it appears as agitation of the compound nucleus. As in section 2.1, these energies are designated by the symbol Q, and can be calculated via the mass excesses of the isotopes involved. Neutron pairing energies are typically on the order of a few MeV. This energy should not be confused with the ∼200 MeV liberated when a compound nucleus subsequently fissions, even though they are designated with the same symbol.
For U-235 and U-238, the neutron-capture reactions are
and
In the case of U-235, the Δ-values are (left-to-right in equation (2.7)) 8.071, 40.921, and 42.446 MeV, from which equation (2.1) gives Q = 6.546 MeV. For U-238, the Δ-values are (left-to-right in equation (2.8)) 8.071, 47.310, and 50.575 MeV, which give Q = 4.806 MeV. If the bombarding neutrons are ‘slow’, that is, if they bring very little kinetic energy into the reactions, then the nucleus of U-236 formed in the first reaction will find itself in an excited state with an internal energy of about 6.6 MeV, while the U-239 nucleus formed in the second reaction will have a like energy of about 4.8 MeV.
The importance of these neutron pairing energies emerges when they are connected to another concept. Theory indicates that any nucleus can be induced to fission if it can be given enough agitation energy. Every isotope has its own minimum fission-inducing energy requirement; this is called the ‘fission barrier energy’ for that isotope, and is designated here as Ebarr. This energy can be supplied in either, or a combination of, two ways: (i) the neutron pairing energy liberated upon neutron capture as calculated above, and/or (ii) kinetic energy carried in by the bombarding neutron. The barrier energy that is relevant for assessing weaponizability is that for the compound nucleus , as it is compound nuclei that go on to fission (or not!). If Q − Ebarr () > 0, then the isotope is said to be ‘fissile’, which means that capture of a neutron of any kinetic energy can induce fission. Conversely, an isotope for which Q − Ebarr () < 0 is said to be ‘fissionable’: it can only be fissioned if the neutron brings in enough kinetic energy to supply the difference between the pairing energy and the fission barrier.
This paragraph contains the key point of this section. For U-236, Ebarr ∼ 5.67 MeV, so for neutron bombardment of U-235, Q − Ebarr () ∼ +0.9 MeV: U-235 is fissile. In contrast, for U-239, Ebarr ∼ 6.45 MeV, so, for neutron bombardment of U-238, Q − Ebarr () ∼ −1.6 MeV. This means that U-238 can be fissioned only by neutrons whose kinetic energy exceeds ∼1.6 MeV. These results are the first indications that U-235 can be used to power a bomb, but that U-238 cannot.
That U-238 is not fissile does not necessarily render it useless as a bomb fuel (and similarly for any other isotope with a negative Q − Ebarr value). The secondary neutrons emitted in fissions emerge with a spectrum of kinetic energies, with a mean value of about 2 MeV. About half of these neutrons have kinetic energy greater than ∼1.6 MeV, the threshold fission-induction energy for U-238, so it looks as if there might yet be a chance for this isotope. However, for such energetic neutrons, the reaction cross-section for inelastic scattering against U-238 nuclei is about eight times the fission cross-section. The problem is that inelastic collisions reduce the kinetic energies of bombarding neutrons so drastically that the vast majority of them that strike U-238 nuclei are promptly slowed to energies less than the 1.6 MeV threshold. U-238 is useless as a fission-bomb fuel because of the combination of its negative Q − Ebarr value and scattering properties.
Figure 2.1 shows a graph of fission barrier energy versus mass number. The barrier is very great for the vast majority of elements, peaking at about 55 MeV for nuclei with A ∼ 100. Even for radium (Z = 88, A = 226 in its most common form), the barrier still amounts to ∼10 MeV, so there is no hope of achieving a self-sustaining chain reaction with any elements lighter than
Z ∼ 90 (thorium).
Figure 2.1. Fission barrier energy in MeV versus mass number A. This is an approximate curve; it does not reflect the effect of isotope-to-isotope variations within a given mass number.
The curve in figure 2.1 can be expressed very approximately as
This expression derives from an empirical fit to barrier energies; while it is handy for making quick estimates, it has no underlying physical rationale. For A = 235, it gives Ebarr ∼ 7.3 MeV, somewhat on the high side but not too far afield.
Exercise
Apply equation (2.9) to tin, A = 120.
Answer
Ebarr ∼ 52.4 MeV.
The rapid decline in barrier energy for heavy elements suggests that any heavy element (uranium and beyond, say) should be viable for use as a bomb material. We will see in the next section why this is not so.
This paragraph contains another very important point. Despite its non-fissility, U-238 was crucial to the success of the Manhattan Project as it can be used to create the synthetic fissile material plutonium-239 (Pu-239). The U-239 nucleus formed in equation (2.8) sheds its excess energy in a series of two beta-decays, ultimately giving rise to Pu-239:
Before reading on, do the following calculation.
Exercise
Neutron bombardment of Pu-239 gives . Here the left-to-right Δ-values are 8.071, 48.591, and 50.128 MeV. What is the Q value? The fission barrier for Pu-240 is 6.05 MeV; would you expect Pu-239 to be fissile?
Answer
Q = 6.53 MeV; yes.
Exercise
Now do the same calculation for thorium, , for which the Δ-values are 8.071, 35.452, and 38.737 MeV, and for which the fission barrier is 6.65 MeV.
Answer
Q = 4.79 MeV; not fissile.
That Pu-239 might be fissile as this exercise suggests occurred to Princeton University physicist Louis Turner in early 1940. Turner wrote up his speculation in a brief paper dated May 29 of that year, but in accordance with wartime censorship guidelines he voluntarily withheld publication until 1946. Future events fully vindicated him.
If you look at the above numbers for U, Pu, and Th, a pattern emerges: U-235 and Pu-239 are fissile, whereas U-238 and Th-232 are not. The features that these pairs have in common is that first two contain odd numbers of neutrons, while the latter two contain even numbers of neutrons. Nuclear physicists have known for many decades that the stability of nuclei depends on the evenness or oddness of the number of nucleons that they possess. In the present case, we need only worry about the number of neutrons N in a nucleus since fission is a neutron-induced phenomenon.
Empirically, nuclear forces are of short range, and neutrons seem to prefer to bind in pairs. This means that a nucleus with even N is intrinsically more mass-energetically stable than one with odd N; if you are familiar with some quantum mechanics, even-N nuclei effectively reside in a deeper potential well than do odd-N nuclei. An equivalent way of thinking about this is via the perspective of mass changes upon neutron capture. Because of the neutron pairing energy release, if a compound nucleus is of even N, then the difference between its mass and the sum of the two input masses will be greater than this difference when N is odd. The difference in mass differences between these two cases is equivalent to an energy of about 1.5 MeV, which leaves an odd-N to even-N capture (as in U-235 → U-236 and Pu 239 → Pu-240) with more agitation energy than one resulting from an even-N to odd-N capture (as with U-238 → U-239 and Th-232 → Th-233). This additional energy is often enough to endow an even-N compound nucleus with an agitation energy which exceeds its fission barrier, which means that nuclei which undergo odd-to-even N transitions are inherently more likely to be fissile than their even-to-odd N counterparts. These patterns can be summarized with the following approximate expressions:
and
where the energies are the (approximate) neutron pairing energy releases.
2.6 Decay mechanisms and the (α, n) problem
Let us now consider what heavy-element isotopes could be practical candidates for weaponizability.
All isotopes of all elements with atomic numbers Z > 83 are radioactive; they decay to isotopes of other elements by various processes, although some have very long half-lives. If a nuclear weapon is to be kept in a stockpile for many years, it will obviously not be sensible to fuel it with a short-lived isotope. For reasons that will become clear in this section and the following one, it is quite reasonable to adopt 1000 years as an acceptable lower limit of half-life for considering weaponizability.
For an isotope with a decay mechanism of half-life t1/2 seconds, the rate of decays per second per gram of material is given by
where NA is Avogadro’s number and A the atomic weight of the isotope in grams per mole. For t1/2 = 1000 years (= 3.154 × 1010 s) and A = 240 (typical of heavy elements), R ∼ 5.5 × 1010 g−1 s−1. If you have a bomb core of mass 10 kg, this means a decay rate of R ∼ 5.5 × 1014 per second. The practical implication of this large number involves the nature of the dominant mode of decay for fissile materials, which is alpha decay, and how a fission bomb is triggered.
In the Hiroshima uranium bomb, two subcritical pieces of fissile material were combined to make a supercritical mass; this was achieved by firing a ‘projectile’ piece initially located in the tail of the bomb toward a mating ‘target’ piece located in the nose by igniting a conventional gunpowder charge to propel the projectile piece forward (see chapter 5, particularly figures 5.1 and 5.2). For pieces on the order of 10 cm in size and the maximum speed that could be achieved (about 1000 m s−1), about 100 μs would be required for the pieces to seat fully together after the leading edge of the projectile piece encounters the target piece. This is a very brief interval, but for a decay rate of ∼1014 s−1, some 1010 alphas (10 billion!) will be emitted during the seating time. It is this large number that is the issue.
If the chain reaction should begin before the fissile pieces are fully seated, the result will be an explosion of less efficiency than that which the bomb was designed to achieve. The 10 billion alpha-particles are themselves not directly the problem; they are not energetic enough to closely approach the nuclei of the fissile material against the Coulomb repulsion that the nearly 100 protons contained within the latter will exert on them. However, there is a problem known as the (α, n) effect. When struck by alpha-particles, nuclei of light elements such as aluminum, and particularly beryllium, emit neutrons. Some impurities will inevitably be present due to chemical processing of the fissile material; if the level of such impurities is too great, a ‘predetonation’ (also known as a ‘preignition’) is virtually guaranteed during the seating time due to alpha-particles emitted by the fissile material striking nuclei of the impurity material. Since even only a single stray neutron will have some probability of initiating a chain reaction, it is very important for bomb engineers to keep the neutron background as low as possible—and hence the presence of any contaminants. The mathematics of pre-detonation probabilities is explored in more detail in chapter 5.
Fortunately, the (α, n) effect is mitigated by the fact that atoms are mostly empty space; most of the alphas will not strike an impurity nucleus, especially if they are few in number to begin with. Nuclear physicists quantify this in terms of the ‘neutron yield’ y of a reaction: the number of neutrons generated per second divided by the number of alpha particles emitted per second in a well-mixed sample of an alpha-emitter and impurity material. Experimentally, (α, n) neutron yields are typically y ∼ 10−4. Based on theoretical calculations of how particles travel through bulk matter, there is a semi-empirical expression available for estimating the rate of neutrons created per second, Rn, if the rate of alpha emission per second is Rα. This is
In this expression, (nimpurity/nfissile) is the ratio of the number densities of the impurity and fissile nuclei (number density means the number of nuclei per cubic meter), and the A’s are their molecular weights. Good-quality chemical purification techniques should achieve (nimpurity/nfissile
) ∼ 1/20 000. For the alpha rate computed above for a 10 kg core, assuming Aimpurity = 10 (about that of beryllium) and setting Afissile = 240,
For a 100 μs seating time this implies a few tens of neutrons, which would virtually guarantee a predetonation. To cut this down to one neutron over 100 μs would demand increasing the alpha-decay half-life to tens of thousands of years, so a lower-limit cutoff of 1000 years will not exclude any fissile-material alpha-emitter from practical weaponization consideration.
Exercise
U-235 has an alpha-decay half-life of 705 million years. Compute the (α, n) rate for U-235 for a 60 kg bomb core (the Hiroshima bomb), assuming the impurity density and atomic weight as above. You should be able to convince yourself that (α, n) predetonation was not a serious probability for the Hiroshima bomb. Then repeat the calculation for a 10 kg Pu-239 core, which has a half-life of 24 100 years. The purity requirements for the fissile material of the Nagasaki bomb were much more demanding than for its Hiroshima counterpart, although not impossible to achieve. But we will see in the following section that plutonium suffered from a much more serious complication.
Answer
4.95 neutrons s−1; 23 500 neutrons s−1.
Another comment on neutron emission is appropriate here. Stray neutrons are a hazard while a bomb core is being assembled, but then you will want to have some available to initiate the chain reaction once the core has been assembled. In the Manhattan Project, this was done by fabricating neutron ‘initiators’. These devices, approximately the size of a golf ball, contained alpha-emitting polonium and inert beryllium which were initially separated inside the initiator. When the initiator was crushed by the assembling core, the two elements would mix, and alphas from the polonium would strike beryllium nuclei and release a small torrent of neutrons to initiate the explosion. Ironically, the very (α, n) process which bomb designers sought to avoid during core assembly was used to trigger a bomb when desired!
The Manhattan Project Page 3