Los Alamos, Trinity, and Tinian

Abstract

The most famous site of the Manhattan Project was the Los Alamos Laboratory in New Mexico, directed by Dr. J. Robert Oppenheimer. Here, physicists, chemists, engineers, and military ordnance specialists worked for over two years to design the Little Boy and Fat Man bombs. This chapter describes how the Laboratory was organized; details of the physics involved in achieving an efficient nuclear explosion; unanticipated problems which nearly rendered the plutonium bomb unworkable; the dramatic Trinity test of the plutonium bomb; the role of British scientists in the Manhattan Project; why the Little Boy and Fat Man bombs had such different designs; the training of air crews to carry out the actual bombing missions; and some of the effects of radiation.

Exercises

• 7.1 Consider a gun-type bomb with a solid cylindrical projectile piece of radius r and mass m which is fired under breech pressure P toward a mating target piece a distance x away. For simplicity, assume that the pressure P maintains its value as the projectile piece moves along the gun barrel. Using simple force and kinematic concepts [F = ma; v² = 2ax], develop an expression for the velocity that the projectile will have after traveling down the barrel. Apply your result to a projectile with r = 3 inches, P = 75,000 lb per square inch, m = 50 kg, and x = 17 feet. Be careful with conversion factors; 1 inch = 2.54 cm; 1 lb per square inch = 6895 Pa. Does you result accord approximately with the figures given in this chapter? [Ans: 1398 m/s]

• 7.2 Working from the parameter values cited in Table 7.1, verify the critical masses for U-235 and Pu-239 given in the Table; convince yourself that you understand how to solve the criticality equation (7.6). Now consider U-233: A = 233.04 gr/mol, ρ = 18.55 gr/cm³, σ_f = 1.946 bn, σ_el = 4.447 bn, v = 2.755 neutrons per fission. What is the critical mass? [Ans: 14.2 kg]

• 7.3 From the decay-rate formula and spontaneous fission data of Sect. 7.7, compute the expected number of spontaneous fissions from 20 μg of Pu-239 over the course of 30 days. Does your result agree with the value of ~0.36 cited in the text?

• 7.4 Suppose you have one gram of plutonium that is 99.99% Pu-239 by weight, with the remaining 0.01% being Pu-240 . Compute the hourly spontaneous fission rate of your gram of plutonium. [Ans: 199]

• 7.5 It is remarked in the text that the shock-wave pressure created by a nuclear explosion is proportional to E^2/3/d², where E is the energy liberated by the explosion and d is distance. If it had been predicted that the Trinity test would liberate energy equivalent to 20,000 tons of TNT, how many tons of TNT should have been used in the May, 1945, calibration test to produce the same pressure at ground zero if the Trinity and test shots were at elevations of 100 and 28 feet, respectively? [Ans: 439 tons]

• 7.6 From the dimensions given in Fig. 7.20, compute the masses of the aluminum and natural uranium tamper spheres in the Fat Man bomb; neglect the effect of the “trap-door” access. Take the densities of aluminum and natural uranium to be 2.699 and 18.95 gr/cm³, respectively. [Ans: aluminum 131 kg = 289 lb; uranium 101 kg = 223 lb]

• 7.7 In the study of thermodynamic properties of materials, the following simple differential equation is used to model the change in volume dV of a sample of material of volume V when it is subjected to a change in pressure dP:

  $$\frac{dV}{dP} = - \frac{V}{B}.$$

  B is the bulk modulus of the material, a measure of its compressibility; a material of higher-B is more difficult to compress than one of lower B. The bulk modulus of plutonium is about 30 GPa (assume constant). If an implosion bomb subjects a plutonium core to a pressure increase of one million atmospheres (1 atm ~ 10⁵ Pa), integrate the differential equation to estimate the ratio of the final volume of the plutonium to its initial volume. [Ans: V_final /V_initial ~ 0.036]

• 7.8 To minimize fallout created by an air-burst nuclear weapon, the weapon should be detonated at a height such that the fireball, at its maximum size, does not touch the ground. An approximate expression for the maximum radius in miles of the fireball created by an air-burst weapon of yield Y kilotons is R ~ 0.041Y ^0.4. If a 200-kiloton weapon is detonated at this height, to what distance from ground zero will the maximum overpressure exceed 5 psi? Use (7.19) for the maximum overpressure . [Ans: 2.48 miles]

• 7.9 The purpose of this problem is to make a very crude estimate of the radioactivity produced by a fission weapon. Suppose that fission of ²³⁵U happens exclusively by the reaction

  $${}_{92}^{235} {\text{U}} + {}_{0}^{1} {\text{n}} \to {}_{56}^{141} {\text{Ba}} + {}_{36}^{92} {\text{Kr}} + 3\left( {{}_{0}^{1} {\text{n}}} \right)$$

  Assume that 1 kg of ²³⁵U is fissioned in this way. ¹⁴¹Ba and ⁹²Kr then both subsequently decay by beta-decay with half-lives of 18 min and 1.8 s, respectively. Use the decay-rate expression of this chapter to estimate the “immediate” beta-radioactivity so generated; for simplicity, ignore the neutrons released in the reaction. If this radioactivity falls out over an area of 10 square miles, what will be the resulting immediate radioactivity in Curies per square meter? [Ans: Appx. 2.7 × 10¹³ Ci; 1.0 × 10⁶ Ci/m²]

• 7.10 Due to a reactor accident, it is predicted that a city of population 200,000 will be exposed to radiation doses averaging 3 rems per person. You are the civic official responsible for deciding whether or not to evacuate the city. The Chief of Police tells you that the chaos to be expected in an evacuation will probably result in about 300 deaths due to traffic accidents, heart attacks, and other such causes. Compare the number of expected radiation-induced excess cancer deaths to the number of deaths expected to be caused by the evacuation. What would you do? [Ans: Appx. 240 excess deaths ]

• 7.11 A typical household smoke detector contains an amount of ²⁴¹Am corresponding to a radioactivity of 1 μCi (Sect. 2.1.2). To what mass does this correspond? How many detectors would you have to collect to give one critical mass, estimated to be ~60 kg? [Ans: About 0.29 μg; ~200 billion detectors]