A corollary of this analysis concerns what happens if you begin with a subcritical core of material of normal density. If you could crush it to a small enough radius, you may be able to achieve ρR greater than the critical value (because ), and so make a nuclear weapon with less than one bare critical mass worth of material. Achieving such an implosion is very difficult, but, as was remarked in section 2.7, was crucial to making a working plutonium bomb during the Manhattan Project. This issue is explored further in chapter 5.
As described in section 2.4, a core need expand only a centimeter or two before criticality is lost. From the ρR analysis above, you might wish to try proving that (assuming a normal density core to begin with) the distance that a core expands from its initial radius to its shutdown radius Rshut is given by
For a core of two critical masses of U-235, = 1.26Rbare = 10.54 cm, and this gives an expansion distance Rshut − Rcore = 1.3 cm.
3.3 Tamped criticality
In the introduction to this chapter, the idea of surrounding a bomb core with a neutron-reflecting tamper was raised. Since a tamper reflects neutrons back into the core, it effectively increases the number of secondary neutrons per fission, which has the effect of lowering the mass necessary to achieve a chain reaction below the ‘bare’ value calculated in section 3.1. The mathematics of ‘tamped criticality’ is somewhat complicated; the analysis depends on the properties of both the core and tamper materials and on the outer radii of each. Figure 3.1 shows a schematic illustration of a spherical core/tamper configuration.
Figure 3.1. Schematic illustration of a tamped bomb core.
The end result of a criticality analysis for neutron diffusion within a tamped core is the following equation:
In this expression, Rcore is the outer radius of the core, and Rtamp that of the tamper. d has the same meaning as in equation (3.7). The transport mean free path of the core, λtrans of equation (3.2), is now written as , while that for the tamper is . This latter quantity is calculated with equation (3.2) with the transport cross-section and nuclear number density of the core replaced with those for the tamper. It is assumed that the tamper has no cross-sections for fission or neutron capture. Equation (3.10) corresponds to what a nuclear weaponeer would define as ‘tamped threshold criticality’.
There are two ways to use equation (3.10). These are: (i) specify the core radius Rcore in advance, with Rcore < Rbare for the desired fissile material. Then equation (3.10) can be solved (numerically) for Rtamp, the tamper outer radius which will just render the core critical. (ii) Conversely, choose a value for Rtamp, in which case the equation can be solved for the core radius Rcore which will just be critical for the chosen tamper size. This latter method can be handy if the outside dimensions of the core/tamper assembly are constrained in some way, such as having to be fitted inside a narrow missile housing. More elaborate analyses deal with tamped supercritical cores, that is, cores which are more than ‘just critical’ for the mass of tamper involved.
Figure 3.2 shows the results of a hybrid approach to solving equation (3.10). This is based on re-casting this equation as follows. First, define the dimensionless quantity ε, which depends only on the characteristics of the core:
Figure 3.2. Curves of Rcore/d versus Rtamp/Rcore for values of λ = 1.75, 1.5, … 0.5 (top to bottom), for ε = 0.3408 (U-235).
With this, equation (3.10) can be written as
where
Once the core material is chosen, the value of ε is fixed; values for U-235 and Pu-239 are given in table 3.1. For a chosen value of ε, equation (3.13) can be solved numerically for Rcore/d for a given value of Rtamp/Rcore, which by definition must be ⩾1. (Conversely, one could decide on a value of Rcore in advance, and use the graph to determine Rtamp for a given value of λ.) Figure 3.2 shows such solutions for λ = 1.75, 1.25, 1.0, … 0.5 for ε corresponding to U-235; the curves converge to Rcore/d = 2.379 at Rtamp/Rcore = 1, which corresponds to a bare core.
Table 3.2 lists values of for some common materials that might be used as tampers. Tungsten-carbide (WC) is a form of steel; beryllium oxide is a desirable tamper material on account of its very small neutron-capture cross-section.
Table 3.2. Transport mean free paths for selected tamper materials [2].
Material (cm)
Aluminum 5.595
Beryllium oxide 2.549
Lead 5.426
Tungsten-carbide 3.159
Depleted uranium (U-238) 4.342
An example of using figure 3.2: the Hiroshima U-235 bomb used a WC tamper, for which λ = 0.878. Suppose that Rtamp/Rcore = 3. Interpolating from the graph, Rcore/d ∼ 1.7, which, for the value of d listed in table 3.1, gives a critical radius of about 5.98 cm, corresponding to a mass of ∼16.8 kg—almost one-third of the ‘bare’ value. The tamper outer radius is then ∼17.94 cm, making its mass ∼364 kg.
The Hiroshima Little Boy bomb core comprised about 64 kg of U-235 in a cylindrical shape (see chapter 5) surrounded by a cylindrical WC tamper of mass about 310 kg (density of WC = 14.8 g cm−3). If this were a spherical arrangement, the core radius would be about 9.35 cm and the outer radius of the tamper would be about 18.0 cm. These dimensions give Rtamp/Rcore ∼ 1.9, which from figure 3.2 corresponds to Rcore ∼ 6.6 cm, or a mass of ∼23 kg. We can conclude that Little Boy utilized approximately (64/23) ∼ 2.8 ‘tamped critical masses’: this was a supercritical weapon.
Incidentally, a weapon that incorporates both implosion and a tamper will make even more efficient use of one’s precious fissile material—precisely as was done in the Nagasaki Fat Man bomb.
Exercise
Using the density data in table 3.1 and this section, verify the ‘spherical’ core and tamper radii for Little Boy quoted above.
Exercise
You have enough U-235 to make a core of radius 5.28 cm (∼11.5 kg). If your tamper material has λ = 0.5, what value of Rtamp/Rcore will be necessary to achieve threshold criticality?
Answer
∼2.
3.4 Efficiency
This section presents a more refined analysis of the efficiency of an untamped fission weapon than the very rough estimate developed in section 2.4. This is approached by estimating the energy released as the detonating bomb core expands from its initial radius Rcore to its criticality-shutdown radius Rshut of equation (3.9), and comparing the result to what would be expected if the entire core undergoes fission. The final result of this analysis, equation (3.24), is not at all obvious, so a brief derivation is presented.
One of the key results of the diffusion theory analysis behind the study of criticality in section 3.1 is an expression for the number density of neutrons N in the core as a function of time. This is
Recall that number density means number per cubic meter. In this expression, N0 is the number density of neutrons at ‘time zero’, the moment when the chain-reaction begins. The value of N0 would be set by whatever neutron-generating ‘initiator’ is used to trigger the weapon; we will see that the efficiency does not depend on this value. τ is again the neutron travel time between fissions (equation (3.5)). α is a dimensionless parameter that depends on the number of neutrons per fission v and the mass of the core. An approximate expression for α is
α is known as the ‘effective number of neutrons liberated per fission’. To appreciate this terminology, take the time-derivative of equation (3.14):
Now, in words, the rate of change of N can be expressed as
On dimensional grounds, you should be able to convince yourself by comparing equations (3.16) and (3.17) that (N/τ) plays the role of the rate of fissions per cubic meter, and α the effective number of neutrons per fission.
The efficiency calculation now proceeds through an energy argument. If (N/τ) is the volume density of rate of fissions, then it follows that if each fission liberates energy Ef, the rate of change of the density of energy being liberated at time t must be Ef (N/τ). The usual physics symbol for energy density is U:
The total energy density liberated
to time t is obtained by integrating from t = 0 to t:
This result is written as an approximation because α changes as the core expands (the argument that leads to equation (3.15) involves the radius of the core), and also because the contribution from the lower limit of integration has been discarded. This discarding is based on the rationale that exponential functions grow very quickly.
Let us now assume that all of the liberated energy goes into causing the core to expand. U(t) is an energy density; to get the total energy we must multiply by the volume V of the core at any moment. If the core has density ρ at some moment, its mass will be Vρ, and its kinetic energy can be written as (Vρ)v2/2, were v is the expansion speed of the core. For simplicity, I assume that the expansion is uniform throughout the core. The factors of V can be canceled, and the result rearranged to give an expression for v(t):
Now, v must be the rate of change of the radius of the core. To determine the time at which shutdown occurs, we can integrate equation (3.20) from Rcore to Rshut, and then back-substitute the time into equation (3.19) to determine the energy released. However, there is a problem with this, which is that the density ρ in equation (3.20) will be a function of time. For the sake of getting to some numerical estimates, I will model the density as being constant, and equal to the normal density of the core material. We then have:
or, on again dropping the lower limit of integration,
Solve equation (3.22) for tshut, and back-substitute the result into equation (3.19) to give the energy density at criticality shutdown. Abbreviating Rshut − Rcore as ΔR, the result is
If full efficiency is achieved, then the energy density would be the product of the nuclear number density n and the energy per fission, that is, nEf. Efficiency can then be defined as the ratio of the energy density at shutdown to this maximum possible energy density:
As claimed above, this result is independent of the number density of neutrons at the start of the chain reaction. ρ and n are related by equation (3.4); their ratio could be simplified.
We can now get to some numbers. Suppose that we have a U-235 core of three critical masses. With the value of v from table 3.1, equation (3.15) gives α ∼ 0.85. For the bare critical radius listed in table 3.1, ΔR ∼ 2.42 cm, or a little less than one inch. Adopting other values as needed from the table and assuming Ef = 170 MeV = 2.723 × 10−11 Joules, you should find eff ∼0.010, or about 1%. Four critical masses gives ΔR ∼ 3.45 cm and eff ∼2.8%. These values are close to the known efficiency of the Hiroshima bomb, but the apparent agreement is illusory as this development has taken no account of any tamper. A lucky case of canceling approximations?
Exercise
Consider an untamped Pu-239 core of three critical masses. Using the data of table 3.1, what does equation (3.24) predict for the efficiency?
Answer
∼1.5%.
Exercise
Thermodynamic theory shows that at extremely high temperatures, pressure is equal to 1/3 of energy density. For the U-235 core of three critical masses analyzed above, estimate the pressure at criticality shutdown.
Answer
About 4.4 × 1015 Pascals. This is equivalent to about 44 billion atmospheres.
References
[1] Reed B C 2015 The Physics of the Manhattan Project 3rd edn (Berlin: Springer) ch 2
[2] Reed B C 2015 Note on the minimum critical mass for a tamped fission bomb core Am. J. Phys. 83 969–71
IOP Concise Physics
The Manhattan Project
A very brief introduction to the physics of nuclear weapons
B Cameron Reed
* * *
Chapter 4
Obtaining fissile material
All of the experiments, theorizing, and calculations behind the Manhattan Project would have been for nothing had it not been possible to acquire, in a reasonable time, kilograms of U-235 and Pu-239. As described in chapter 1, U-235 had to be separated essentially atom-by-atom from a mass of natural-abundance uranium, and Pu-239 had to be synthesized in nuclear reactors. This chapter explores the physics of two methods that were used to isolate U-235 at Oak Ridge, Tennessee, and of how long it takes to produce a given amount of plutonium in a reactor.
4.1 U-235: the electromagnetic method
The electromagnetic method of isotope separation is based on the principle of mass spectroscopy, which is commonly employed in physics and chemistry laboratories. The origins of this technique date to the discovery of the electron by J J Thomson in 1897.
The principle of mass spectroscopy is sketched in figure 4.1. The sample of material to be analyzed is heated and vaporized in an oven inside a vacuum chamber. The heating will ionize the atoms, some of which will escape through a narrow slit. The ionized atoms are then accelerated by an electric field (recall , with m being an ion’s mass and q its net charge), and directed into a region of space where a magnetic field is present. If Vacc is the accelerating voltage corresponding to the electric field, the ions will emerge from the electrodes with speed v given by
Figure 4.1. Principle of mass spectroscopy. Positively-charged ions are accelerated by an electric field and then directed into a magnetic field which emerges perpendicularly from the page. Ions of different masses will follow different circular trajectories; those of greater mass will move in trajectories of larger radii than those of lesser mass. Reproduced from [1].
In the sketch, the magnetic field is pointing out of the page toward you, that is, perpendicular to the plane of travel of the positively-charged ions; the electrical coils or magnet poles for creating the field would lie on either side of the page and are not sketched. The magnetic field gives rise to an effect known as the Lorentz Force Law, which is written as , where is the velocity acquired by the ions from the accelerating voltage. This velocity will be vertically upwards in the sketch when the ions leave the electrodes, and be of the magnitude of equation (4.1). If you know how to compute the direction of a vector cross product, verify that the force on the ions will be to the right at the moment they leave the electrodes and enter into the magnetic field. This force causes the ions to move in circular trajectories whose radii depend on their mass, their net charge, and the magnitudes of and . If the ion stream comprises atoms of two different masses, the result will be two streams that travel in orbits of different radii, with heavier ions moving in orbits of larger radii. The streams will be maximally separated after one-half of an orbit, and this permits the collection of, in this case, the desired lighter-isotope stream.
The orbital diameter D is given by
The separation between the ion streams at the collector is the difference of the diameters:
Various magnet designs were employed at Oak Ridge as the work there evolved, but in the so-called Alpha-I design (see below) the magnetic field was of magnitude 0.34 Tesla, and the accelerating potential was 35 000 V. If the molecules were singly ionized, the prefactor in equations (4.2) and (4.3) evaluates to
In practice at Oak Ridge, these separators used uranium tetrachloride, so the ion streams were molecules of 235UCl4 and 238UCl4. The molecular weights of 235UCl4 and 238UCl4 are 375 and 378 mass units, respectively. Either of these masses, when substituted into equation (4.2), gives a beam diameter of
The separation is tiny in comparison to the diameter:
To achieve good beam separation, accelerating potentials and magnetic fields had to be kept very steady.
The facility at Oak Ridge for carrying out this work was code-named Y-12. The Y-12 isotope separators were known as ‘calutrons’, a name derived from that of a particle accelerator of related design called a ‘cyclotron’ which had been invented by physicist Ernest Lawrence of the University of California: ‘California University Cyclotron’. To produce their magnetic fields, which were about 7000 times as strong as the Earth’s natural magnetic field, calutrons used square-shaped coils about three meters on a side which were operated at currents ranging from 4500 to 7500 amperes. The calutrons were enormously consumptive of el
ectricity: by July, 1945, the Oak Ridge complex was consuming about 1% of the electricity being generated in all of the United States. The scale of the Y-12 complex was staggering; at $478 million to construct and operate, it would become the second-most expensive facility of the entire Manhattan Project, just behind the $512 million gaseous diffusion plant described in the following section. By May 1945 Y-12 employed nearly 22 500 workers in over 200 buildings.
The reason for the enormity of Y-12 traced to a physical effect that severely limits the operating capacity of any one calutron vacuum tank, the so-called ‘space-charge problem’. As the like-charged ion beams travel through a vacuum tank, they repel each other and become disrupted from their ideal circular paths. This sets a limit on the rate of processing of material, which engineers usually express as an equivalent beam current of the desired light-isotope material. In the Y-12 calutrons, this limit was about 500 microamperes. For a 24-hour operating period, this corresponds to isolating about 0.1 g of U-235 per day per vacuum tank. To collect 50 kg from one tank (not even a single bare critical mass) would require over 1300 years of operation. Manhattan Project organizers appreciated from the outset that only by building hundreds of tanks might it be possible to isolate enough material for a bomb in a year or two; Y-12 would eventually have 1152 vacuum tanks, some of which contained multiple ion-beam sources.
The Manhattan Project Page 5
