Simple calculation of the critical mass for highly enriched uranium and plutonium-239 Christopher F. Chyba, and Caroline R. Milne Citation: American Journal of Physics 82, 977 (2014); doi: 10.1119/1.4885379 View online: https://doi.org/10.1119/1.4885379 View Table of Contents: https://aapt.scitation.org/toc/ajp/82/10 Published by the American Association of Physics Teachers ARTICLES YOU MAY BE INTERESTED IN Heisenberg and the critical mass American Journal of Physics 70, 911 (2002); https://doi.org/10.1119/1.1495409 Note on the minimum critical mass for a tamped fission bomb core American Journal of Physics 83, 969 (2015); https://doi.org/10.1119/1.4931721 Arthur Compton’s 1941 Report on explosive fission of U-235: A look at the physics American Journal of Physics 75, 1065 (2007); https://doi.org/10.1119/1.2785193 A brief primer on tamped fission-bomb cores American Journal of Physics 77, 730 (2009); https://doi.org/10.1119/1.3125008 fissioncore: A desktop-computer simulation of a fission-bomb core American Journal of Physics 82, 972 (2014); https://doi.org/10.1119/1.4894165 Estimate of the critical mass of a fissionable isotope American Journal of Physics 58, 363 (1990); https://doi.org/10.1119/1.16172 Simple calculation of the critical mass for highly enriched uranium and plutonium-239 Christopher F. Chybaa) Department of Astrophysical Sciences and Woodrow Wilson School of Public and International Affairs, Princeton University, Princeton, New Jersey 08544 Caroline R. Milneb) Woodrow Wilson School of Public and International Affairs, Princeton University, Princeton, New Jersey 08544 (Received 10 December 2013; accepted 13 June 2014) The correct calculation of and values for the critical mass of uranium or plutonium necessary for a nuclear fission weapon have long been understood and publicly available. The calculation requires solving the radial component in spherical coordinates of a diffusion equation with a source term, so is beyond the reach of most public policy and many first-year college physics students. Yet it is important for the basic physical ideas behind the calculation to be understood by those without calculus who are nonetheless interested in international security, arms control, or nuclear non-proliferation. This article estimates the critical mass in an intuitive way that requires only algebra. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4885379] I. INTRODUCTION with a source term. This or simplified versions may also be found in a number of journal articles.5–9 Fissile materials are materials that can sustain an explo- 1 The ready availability of the derivation and the value of the sive fission chain reaction. A “threshold bare critical mass” critical mass for HEU and Pu assures that further publications in Mc (henceforth just “critical mass”) is the minimum mass of this realm cannot provide any significant information to states or fissile material that is needed for that chain reaction. At the terrorists pursuing nuclear weapons. However, a derivation ac- threshold critical mass, neutron production by fission within cessible to those without calculus could help ensure that the next a volume just balances neutron loss through the volume’s generation of students interested in international security, arms surface, and the number density of neutrons is constant in control, or nuclear non-proliferation has an intuitive quantitative time. A sphere gives the smallest Mc because a sphere mini- understanding of the critical mass needed for a nuclear explo- mizes the ratio of surface area to volume for a solid. “Bare” sion. We provide such a derivation here, in the hopes of captur- critical mass indicates that there is no neutron reflector; such ing the key physical ideas without doing too much damage to a component reflects neutrons that would otherwise escape the underlying physics through various simplifications. from the sphere back into its interior. The “critical radius” Rc is the radius of the sphere of fissile material corresponding to Mc. Real nuclear weapons may be II. A SIMPLE MODEL expected to employ neutron reflectors and implosive spheri- Hafemeister10 suggests following the simple physical pic- cal compression, both of which serve to reduce the amount 3 of fissile material below that of the bare critical mass.1 ture of Serber to derive Rc by equating neutron production rate PN due to nuclear fission within a spherical volume to Nevertheless, the value of Mc is a useful benchmark for the material requirements for a nuclear weapons program—for neutron loss rate LN through the boundary of that volume. example, the rapidity with which a potential nuclear prolifer- Hafemeister provides a straightforward estimate of PN within ator could produce a bomb’s worth of highly enriched ura- asphereofradiusR. Take the typical velocity v of a neutron nium (HEU) or plutonium-239 (Pu) from gas centrifuges or produced by a nuclear fission to be that corresponding to its nuclear reactors, respectively. (Weapons grade plutonium kinetic energy ( 2 MeV). This neutron has a mean free path produced in a plutonium production reactor is preferable for between fission events kf ¼ 1/nrf,whererf is the fission cross section and n is the number density of fissile nuclei.11 The nu- military purposes to power-reactor plutonium, but the latter À8 may also be used to produce a weapon, albeit with greater clear generation lifetime s ¼ kf/v 10 s is the correspond- difficulty.2) The value of the critical mass provides the fun- ing time between fissions. In each fission, an average number damental context for arms control efforts to constrain or of neutrons is produced. Let N bethefreeneutronnumber density in the sphere. (In reality, of course, N is a function of reduce fissile material stocks below a certain number of 3,4 equivalent warheads. A basic understanding of the derivation radial distance within the sphere, but it is interesting to see of the critical mass is therefore an important underpinning to how far one can go with a much simpler assumption.) Then PN is given by the total number of neutrons in the spherical key aspects of contemporary international security. 3 The standard derivation of the critical mass involves solv- sample, N(4/3)pR , times the net number of neutrons pro- ing a time-dependent diffusion equation with a source term, duced per neutron in a nuclear generation lifetime, ( À 1)/s, and may be found in Serber’s long-declassified Los Alamos where there term À 1 takes into account the loss of the ini- Primer3 and in Reed’s The Physics of the Manhattan tiating neutron in each fission event. We therefore have Project.4 At the threshold critical mass, the time dependence 4 3 À 1 disappears, and one is left with having to solve the radial PN ¼ N pR : (1) component in spherical coordinates of a diffusion equation 3 s 977 Am. J. Phys. 82 (10), October 2014 http://aapt.org/ajp VC 2014 American Association of Physics Teachers 977 The neutron loss rate LN is the trickier part of the estimate. We now describe the simple picture underlying our model, (Even Serber’s Los Alamos lectures had trouble setting up building on Hafemeister’s approach. We divide our fissile the condition at the loss boundary.3) For his simple model spherical mass into concentric layers. The interior layer is a Hafemeister puts sphere of radius R À x. The outer layer is a spherical shell of 2 thickness x with neutron number density N. In a time s, half LN ¼ 4pR Nv; (2) of these outer-layer neutrons—those whose net motion is radially outward—escape the shell and cause no fissions. saying that this represents the “neutron loss rate through an The other half, whose net motion is radially inward, contrib- area” but does not explain further. However, this expression ute to maintaining the constant number density N of the inner may be shown to result from an assumption that in a given nu- sphere. Each neutron present within this inner sphere causes clear generation lifetime all neutrons within a distance k of f a fission event in time s. the outer boundary of the sphere of fissile material escape the We therefore have, for neutron production, sphere, in an approximation where kf R. That is, the num- ber of neutrons in the outermost spherical shell of thickness kf 4 À 1 3 3 2 3 is just ½ð4=3ÞpR ð4=3ÞpðR À k Þ N 4pR k N,withthe PN ¼ N pðR À xÞ : (5) f f 3 s last approximation holding provided kf/R 1. If these neu- trons escape in time s,Eq.(2) follows because kf/s ¼ v.The Meanwhile, neutron loss from the outer layer during this critical radius R ¼ Rc then results from putting PN ¼ LN: same time period is given by 3vs 3 3 3 Rc ¼ ¼ kf : (3) N 4 R ðR À xÞ ð À 1Þ ð À 1Þ L ¼ p : (6) N 2 3 s There are several problems with this simple model. Clearly, Equating neutron production and loss (at which point not all neutrons within kf of the edge will escape, since most 3 3 will not have a purely radial velocity. Moreover, as Reed6 R Rc) gives Rc ¼ð2 À 1ÞðRc À xÞ ,or has emphasized, the cross section for neutron scattering in 1=3 both HEU and Pu is larger than the cross section for fission, 1 ð2 À 1Þ Rc ¼ pffiffiffiffiffi kf ; (7) so that scattering cannot be neglected even in a lowest-order 3g ð2 À 1Þ1=3 À 1 estimate. 12 Solving for numerical values emphasizes the problems.