Answer
225 g day−1.
References
[1] Reed B C 2015 Atomic Bomb: The story of the Manhattan Project (San Rafael, Ca: Morgan & Claypool) figure 2.11
[2] Reed B C 2014 The History and Science of the Manhattan Project (Berlin: Springer) figure 5.11
[3] http://www.y12.doe.gov/about/history/getimages
[4] http://commons.wikimedia.org/wiki/File:Y12_Calutron_Operators.jpg
[5] Reed B C 2014 The Manhattan Project Physica Scripta 89 108003 figure 15
[6] http://commons.wikimedia.org/wiki/File:K-25_Aerial.jpg
[7] http://commons.wikimedia.org/wiki/File:Hanford_B_site_40s.jpg
[8] http://wcpeace.org/history/Hanford/HAER_WA-164_B-Reactor.pdf
IOP Concise Physics
The Manhattan Project
A very brief introduction to the physics of nuclear weapons
B Cameron Reed
* * *
Chapter 5
Los Alamos, Little Boy, Fat Man, Trinity, Hiroshima and Nagasaki
The third major site of the Manhattan Engineer District was the Los Alamos Laboratory, located on a remote but ruggedly beautiful desert mesa in northern New Mexico. As measured by number of employees, Los Alamos paled in comparison to Oak Ridge and Hanford, but was certainly the intellectual center of the Manhattan Project.
The idea of a centralized, secure laboratory to coordinate research and bomb design was raised by scientists in the spring of 1942. When General Groves was assigned to the project in September of that year, he undertook a tour of project sites and met University of California theoretical physicist Robert Oppenheimer on October 8, at which time they discussed the concept of such a facility; Oppenheimer had been involved with the Compton committee described in chapter 1. Groves approved the idea on October 19, and a survey was begun to locate a site which would be isolated, relatively inaccessible, have a climate that would permit year-round construction and operations, be large enough to accommodate a testing area, and be sufficiently inland to be secure from enemy attack. The Los Alamos site, which was then the location of the Los Alamos Ranch School, a financially-troubled wilderness school for boys, was selected in late November. Oppenheimer was formally appointed as Director of the laboratory on February 25, 1943, and construction of laboratory buildings began in March. As recorded in Oppenheimer’s appointment letter, the Laboratory’s work would involve ‘experimental studies in science, engineering, and ordnance’, and ‘large-scale experiments involving difficult ordnance procedures and the handling of highly dangerous material’. The laboratory functioned as a hybrid military−civilian organization, with civilians formally employees of the University of California, which was awarded an operating contract. All residents, civilian and military alike, were subject to military security and censorship regulations.
When Oppenheimer took on the Directorship of Los Alamos, he thought that he would require only a few dozen scientists, technicians, and engineers to carry out its mission. But almost immediately, complexities in the nature of fissile materials and the engineering of bomb mechanics, aerodynamics, and triggering mechanisms demanded expansions of the staff. By mid-1945, Los Alamos employed over 2000 people, and had a population of some 8000, many of them the young children of employees recruited as graduate students, newly-married and just starting families.
One of the first bomb designs considered at Los Alamos was the so-called ‘gun bomb’. As described briefly in section 2.6 and as sketched in figure 5.1, this involved combining two subcritical pieces of fissile material to make a supercritical mass. This was done by placing the two within the barrel of an artillery cannon and firing the ‘projectile’ piece, which was 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. The dimensions given in figure 5.1 are those for the uranium-235 Hiroshima Little Boy bomb.
Figure 5.1. Sketch (not to scale) of the Hiroshima Little Boy U-235 gun bomb. See also figure 5.2.
Various reasons played into the dimensions indicated in figure 5.1. One was the need to be able to fit a weapon into the bomb bay of a B-29 bomber. B-29s were equipped with two 150 inch long bomb bays, one forward and one aft of the wings, but they could be joined together to accommodate a longer bomb. A more important consideration, however, was the danger described in section 2.7: that of a premature detonation caused by stray neutrons, particularly those liberated in spontaneous fissions. To get an understanding of Little Boy’s dimensions and why the gun design could not be used for a plutonium bomb requires an exploration of the mathematics of pre-detonation probabilities. This is the topic of the following section, after which the discussion returns to Little Boy and Fat Man and their roles in the Hiroshima and Nagasaki bombings.
5.1 Predetonation probability
Table 5.1 lists spontaneous fission (SF) rates for relevant isotopes. Since every SF releases 2−3 neutrons, the rates listed must be multiplied by a factor of ∼ 2.5 to give rates of neutron emission.
Table 5.1. Spontaneous fission rates.
Isotope SF per kg per second
U-235 5.63 × 10−3
U-238 6.78
Pu-239 6.92
Pu-240 483 000
As the projectile piece in figure 5.1 seats with the target piece, there will come a moment when a critical mass is present in the partially-assembled system. After this moment, a chain reaction could begin at any time. Consider this moment to be ‘time zero’ for the following analysis. If the time required to complete assembly of the core after this moment is tcore, and the rate of background neutrons is neutrons s−1, probability theory gives an expression for the probability that a pre-detonation will not have occurred by time tcore. This is
The dimensionless factor α in this expression is the same as that in the analysis of efficiency in section 3.4:
where v is the number of neutrons emitted per SF, and Mbare and Mcore are again the bare critical mass of the fissile material and the mass of the core involved.
First consider the case of U-235, where the SF danger arises predominantly from any contamination of U-238. Suppose that we have a core of two untamped critical masses, that is, of mass 91.8 kg (table 3.1). If v ∼ 2.5, equation (5.2) gives α ∼ 0.56. If this core is contaminated to a level of 10% U-238 (9.18 kg), the SF rate listed in table 5.1 gives neutrons s−1.
As remarked in section 2.6, the highest projectile speeds achievable with World War II artillery pieces was about 1000 m s−1. For the ∼18 cm target/projectile lengths indicated in figure 5.1, this implies a full seating time of 180 μs. A critical mass is not likely to be present for all of this time, but I will err on the side of caution by assuming that it is. With these numbers, equation (5.2) gives Pno-pre ∼ 0.992, that is, the odds are better than 99% that no pre-detonation will occur. Even if the assembly speed is reduced to ∼300 m s−1 (the reason for this number is discussed below), that is, a seating time of 600 μs, Pno-pre ∼ 97.4%, still very good odds of having a fully-functioning bomb.
Exercise
Suppose that a group of terrorists have a uranium core exactly as described above, but instead of firing the projectile into the target, they propose to let it fall two meters under gravity to seat with the target. What will be the speed of the projectile when it arrives at the target? If the assembly is 18 cm long, what is the seating time? If the core is supercritical for 2/3 of this time, what is the probability that their device will not suffer a pre-detonation?
Answer
6.26 m s−1, 28.75 milliseconds, 43.7%.
Now consider an untamped plutonium bomb of two bare critical masses (33.4 kg; table 3.1) contaminated with 1% Pu-240. Presume again v ∼ 2.5. This gives 300 neutrons s−1, and Pno-pre ∼ 0.001% for tcore = 100 μs. This means more than a 99% chance of suffering an efficiency-robbing predetonation! Even if the time could be reduced to 10 μs, the no-predetonation probability climbs to only 33%. To achieve 90% no-predetonatio
n probability with these numbers demands a supercriticality time of about only a single microsecond.
When the high SF rate of reactor-produced plutonium was discovered at Los Alamos in the summer of 1944, it was considered an existential crisis for the laboratory. It was immediately clear that no gun-type assembly mechanism could possibly achieve a rapid-enough assembly for a plutonium bomb; a faster means needed to be found. As John Manley, one of Oppenheimer’s deputies, put it: ‘The choice was to junk the whole discovery of the chain reaction that produced plutonium, and all of the investment in time and effort of the Hanford plant, unless somebody could come up with a way of assembling the plutonium material into a weapon that would explode.’ The only realistic option appeared to be a high-speed implosion mechanism, which is described in more detail in section 5.3.
The reason for considering an assembly speed of ∼300 m s−1 for uranium is that this was what was used in the real Little Boy bomb. When a gun-type bomb was under consideration for both U-235 and Pu-239, the design was predicated on the anticipation that the pre-detonation concern with Pu-239 would be largely due to the (α, n) issue described in section 2.6. To overcome this, it was predicted that the required assembly speed would necessitate a weapon about 17 feet long, which would have required joining the B-29 bomb bays as described above; this was known as the Thin Man design. (As is described in the following section, the longer the gun barrel used, the higher will be the final assembly speed.) But when the gun bomb was abandoned for plutonium, it became clear that the length of the uranium bomb could be shortened to the extent that it could fit within a single bomb bay.
5.2 Little Boy
Figure 5.2 shows a detailed diagram of the final design of Little Boy, and figure 5.3 a photograph of the weapon itself. Overall, this bomb was 10 feet long, 28 inches in diameter, and weighed about 9700 pounds. The gun barrel was 6 feet long and weighed 1000 pounds. The target and projectile pieces each comprised a number of washer-like rings of uranium that were cast as material became available from Oak Ridge. The projectile was made up of nine rings totaling 7 inches in length, with inside and outside diameters of 4 inches and 6.25 inches. The projectile had a volume of 126.8 cubic inches, or 2078 cubic centimeters. At a density for pure U-235 of 18.71 g cm−3, the assembled projectile rings totaled 38.9 kg. The target consisted of six rings, also of 7 inches total length, but with inside and outside diameters of 1 and 4 inches for a volume of 82.4 in3 (1351 cm3) and a mass of 25.3 kg. The assembled core totaled just over 64 kg, about 60% of which resided in the projectile. The projectile piece traveled about 52 inches (∼130 cm) before meeting the target piece. By December, 1944, General Groves was confident enough of anticipated uranium production schedules that he ordered all research and development on the gun bomb to be complete by July 1, 1945. The design was frozen in February, 1945, and Little Boy was ready for combat by May, 1945, awaiting only enough U-235. Remarkably, Little Boy would be deployed in combat without having undergone a full-scale test: by mid-1945, enough U-235 was available for only one bomb.
Figure 5.2. Cross-section drawing of Little Boy showing the major components. Not shown are radar units, clock box with pullout wires, barometric switches and tubing, batteries, and electrical wiring. Numbers in parentheses indicate the quantity of identical components. The drawing is to scale. Copyright by and used with kind permission of John Coster-Mullen.
(A) Front nose elastic locknut attached to 1 inch diameter Cd-plated draw bolt
(B) 15.125 inch diameter forged steel nose nut
(C) 28 inch diameter forged steel target case
(D) Impact-absorbing anvil with shim
(E) 13 inch diameter 3-piece WC tamper liner assembly with 6.5 inch bore
(F) 6.5 inch diameter WC tamper insert base
(G) 14 inch diameter K-46 steel WC tamper liner sleeve
(H) 4 inch diameter U-235 target insert discs (6)
(I) Yagi antenna assemblies (4)
(J) Target-case to gun-tube adapter with 4 vent slots and 6.5 inch hole
(K) Lift lug
(L) Safing/arming plugs (3)
(M) 6.5 inch bore gun
(N) 0.75 inch diameter armored tubes containing priming wiring (3)
(O) 27.25 inch diameter bulkhead plate
(P) Electrical plugs (3)
(Q) Barometric ports (8)
(R) 1 inch diameter rear alignment rods (3)
(S) 6.25 inch diameter U-235 projectile rings (9)
(T) Polonium-beryllium initiators (4)
(U) Tail tube forward plate
(V) Projectile WC filler plug
(W) Projectile steel back
(X) 2 pound Cordite powder bags (4)
(Y) Gun breech with removable inner breech plug and stationary outer bushing
(Z) Tail tube aft plate
(AA) 2.25 inch long 5/8-18 socket-head tail tube bolts (4)
(BB) Mark-15 Mod 1 electric gun primers with AN-3102-20AN receptacles (3)
(CC) 15 inch diameter armored inner tail tube
(DD) Inner armor plate bolted to 15 inch diameter armored tube
(EE) Rear plate with smoke puff tubes bolted to 17 inch diameter tail tube
Figure 5.3. Little Boy in its loading pit on Tinian Island in the Pacific [1].
A calculation from undergraduate thermodynamics can be used to get a sense of some of these dimensions. This calculation concerns the work done by an adiabatically expanding gas, that is, one that expands so quickly that there is no time for any heat energy to be lost from the gas to the surroundings. This will be applied here in the sense of assuming that the conventional explosive expands adiabatically once detonated. From the work-energy theorem of Newtonian mechanics, we assume that the work done by the expanding gas goes into the kinetic energy of the projectile/tamper, which is assumed to be of mass m. If the initial pressure of the gas is P0, its initial volume V0, and its final volume is Vf, this analysis shows that the final velocity of the projectile/tamper is given by
where γ is the ‘adiabatic index’ of the gas, that is, the ratio of its specific heat at constant pressure to that at constant volume. For air, γ ∼ 1.4, which I adopt here. A longer gun barrel will mean a higher value of Vf /V0, which means a higher vfinal, as remarked above.
In various Manhattan Project documents, it is specified that the gun barrel be able to withstand a pressure of 75 000 pounds per square inch, or about 520 million Pascals, and that vfinal is of the order 1000 ft s−1, or about 300 m s−1. I take this pressure to be the value of P0. It is difficult to tell an exact value of Vf /V0 from figure 5.2, but taking Vf /V0 ∼ 5 cannot be far wrong. I take the mass of the projectile/tamper assembly to be 80 kg, twice that of the projectile alone, and I take the initial volume of the exploded gas to be that of a cylinder of diameter 6.25 inches and length 14 inches, twice the volume of the projectile alone if it were not hollow; this gives V0 ∼ 7,000 cm3. These numbers give vfinal ∼ 330 m s−1, or ∼1,080 ft s−1, very close to the quoted value.
Exercise
Verify the numbers in the above computation. Suppose that in the plutonium Thin Man design, the projectile/tamper projectile had mass 20 kg, Vf /V0 ∼ 25, P0 = 75 000 psi, V0 ∼ 7500 cm3, and γ = 1.4, what would have been the value of vfinal?
Answer
837 m s−1, or ∼2,750 ft s−1.
5.3 Implosion and Fat Man
The idea of triggering a nuclear weapon by implosion had been conceived in mid-1942. When Los Alamos was established, Oppenheimer assigned a research group to investigate the method, but it was initially given lower priority than the apparently surer and easier gun method. When the spontaneous fission crisis broke in July, 1944, implosion research was moved into high gear: Oppenheimer immediately reorganized the laboratory by establishing two new experimental divisions and an associated theoretical group. Over the next year followed an intense period of research and development; it is not an exaggeration to say that implosion was one of the trickiest engineering accomplishments in history to its day. Par
ticularly significant to its development were the contributions of a group of British scientists working at Los Alamos, who had brought with them considerable experience in developing ‘shaped charge’ explosives used to penetrate German tank armor. Implosion involved modifying the shaped-charge concept into a system of three-dimensional implosion ‘lenses’.
The idea behind an implosion lens is sketched in figure 5.4, which shows a single lens in side-view cross-section. To extend the concept to three dimensions, imagine a somewhat pyramidal-shaped five or six-sided block about a foot across and a foot and a half from end-to-end (left to right in the figure, which is not to scale). Each block is assembled from two separate castings of different explosives, with the castings fitting together very precisely. The outer casting of each block is a fast-burning explosive known as ‘Composition B’ (Comp B), while the inner lens-shaped casting is a slower-burning material known as Baratol, a mixture of barium nitrate and TNT. A detonator at the outer edge of the block of Comp B triggers an outward-expanding detonation wave, which progresses to the left in the figure. When the detonation wave hits the Baratol, it too begins exploding. If the interface between the two materials is of just the right shape, the two waves can be arranged to combine as they progress along the interface to create an inwardly-directed converging burn wave in the Baratol. In effect, this ‘binary explosive’ focuses the detonation wave as does an optical lens with light waves. Implosion not only achieves criticality of the fissile material much more rapidly than could a gun mechanism, but has the further advantage that because it crushes the fissile material to higher-than-normal density, the requisite critical mass is less than what would be required for a core of normal density (section 3.2).
The Manhattan Project Page 7