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Elements of Fission Weapon Design

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Elements of Fission Weapon Design archived as http://www.stealthskater.com/Documents/Nuke_041.doc [pdf] more on nuclear weapons at http://www.stealthskater.com/Nuke.htm note: because important websites are frequently "here today but gone tomorrow", the following was archived from http://nuclearweaponarchive.org/Nwfaq/Nfaq4-1.html#Nfaq4.1 on October 20, 2003 . This is NOT an attempt to divert readers from the aforementioned website. Indeed, the reader should only read this back-up copy if it cannot be found at the original author's site. COPYRIGHT CAREY SUBLETTE This material may be excerpted, quoted, or distributed freely provided that attribution to the author (Carey Sublette) and document name (Nuclear Weapons Frequently Asked Questions) is clearly preserved. I would prefer that the user also include the URL of the source. Only authorized host sites may make this document publicly available on the Internet through the World Wide Web, anonymous FTP, or other means. Unauthorized host sites are expressly forbidden. If you wish to host this FAQ, in whole or in part, please contact me at: [email protected] This restriction is placed to allow me to maintain version control. The current authorized host sites for this FAQ are the High Energy Weapons Archive hosted/mirrored at http://nuketesting.enviroweb.org/hew/ and "mirrored" at http://nuclearweaponarchive.org/ , http://gawain.membrane.com/hew/ and Rand Afrikaans University Engineering hosted at http://www-ing.rau.ac.za/ 4.1 - Elements of Fission Weapon Design 4.1.1 Dimensional and Temporal Scale Factors In Section 2, the properties of fission chain reactions were described using 2 simplified mathematical model: (1) the discrete step chain reaction and (2) the more accurate continuous chain reaction model. A more detailed discussion of fission weapon design is aided by introducing more carefully defined means of quantifying the dimensions and time scales involved in fission explosions. These scale factors make it easier to analyze time-dependent neutron multiplication in systems of varying composition and geometry. These scale factors are based on an elaboration of the continuous chain reaction model. It uses the concept of the "average neutron collision" which combines the scattering, fission, and absorption cross- sections with the total number of neutrons emitted per fission to create a single figure of merit which can be used for comparing different assemblies. The basic idea is this: When a neutron interacts with an atom, we can think of it as consisting of 2 steps: 1. the neutron is "absorbed" by the collision; and 2. zero-or-more neutrons are emitted. If the interaction is ordinary neutron capture, then no neutron is emitted from the collision. If the interaction is a scattering event, then one neutron is emitted. If the interaction is a fission event, then the average number of neutrons produced per fission is emitted (this average number is often designated by υ). By combining these, we get the average number of neutrons produced per collision (also called the number of secondaries) designated by c: 1 c = (cross scatter + cross fission avg_n_per_fission) / cross total Eq. (4.1.1-1) The total cross-section cross total is equal to: cross total = cross scatter + cross fission + cross absorb Eq. (4.1.1-2) The total neutron mean free path (the average distance a neutron will travel before undergoing a collision) is given by: MFP = 1 / ( cross total * N) Eq. (4.1.1-3) where N is the number of atoms per unit volume, determined by the density. In computing the effective reactivity of a system, we must also take into account the rate at which neutrons are lost by escape from the system. This rate is measured by the number of neutrons lost per collision. For a given geometry, the rate is determined by the size of the system in MFPs. Put another way, for a given geometry and degree of reactivity, the size of the system as measured in MFPs is determined only by the parameter c. The higher the value of c, the smaller the assembly can be. An indication of the effect of c on the size of a critical assembly can be gained by the following table of critical radii (in MFPs) for bare (unreflected) spheres: Table 4.1.1-1. Critical Radius rC versus Number of Secondaries c c value rC (crit. Radius in MFP) 1.00 infinite 1.02 12.027 1.05 7.277 1.10 4.873 1.20 3.172 1.40 1.985 1.60 1.476 If the composition, geometry, and reactivity of a system are specified, then the size of a system in MFPs is fixed. From Eq. (4.1.1-3), we can see that the physical size or scale of the system (measured in centimeters, say) is inversely proportional to its density. Since the mass of the system is equal to volume*density and volume varies with the cube of the radius, we can immediately derive the following scaling law: 2 2 mcritc = mcrit0 / (ρ/ ρ0) = mcrit0 / C Eq. (4.1.1-4) That is, the critical mass of a system is inversely proportional to the square of the density. C is the degree of compression (density ratio). This scaling law applies to bare cores. It also applies cores with a surrounding reflector if the reflector is density has an identical degree of compression. This is usually not the case in real weapon designs, a higher degree of compression generally being achieved in the core than in the reflector. An approximate relationship for this is: 2 1.2 0.8 mcritc = mcrit0 / (Cc * Cr ) Eq. (4.1.1-5) where Cc is the compression of the core and Cr is the compression of the reflector. Note that when Cc = Cr, then this is identical to Eq. (4.1.1-4). For most implosion weapon designs (since Cc > Cr), we can use the approximate relationship: 1.7 mcritc = mcrit0 / Cc Eq. (4.1.1-6) These same considerations are also valid for any other specified degree of reactivity, not just critical cores. Fission explosives depend on a very rapid release of energy. We are thus very interested in measuring the rate of the fission reaction. This is done using a quantity called the effective multiplication rate or "alpha α". The neutron population at time t is given by: (α*t) Nt = N0 e Eq. (4.1.1-7) α thus has units of 1/t, and the neutron population will increase by a factor of e (2.71...) in a time interval equal to 1/ α. This interval is known as the "time constant" (or "e-folding time") of the system, tC. The more familiar concept of "doubling time" is related to α and the time constant simply by: doubling time = (ln 2)/ α = (ln 2) tc Eq. (4.1.1-8) α is often more convenient than tc or doubling times since its value is bounded and continuous: zero at criticality; positive for supercritical systems; and negative for subcritical systems. The time constant goes to infinity at criticality. The term "time constant" seems unsatisfactory for this discussion though since it is hardly constant, tc continually changes during reactivity insertion and disassembly. Therefore I will henceforth refer to the quantity 1/α as the "multiplication interval". α is determined by the reactivity (c and the probability of escape) and the length of time it takes an average neutron (for a suitably defined average) to traverse an MFP. If we assume no losses from the system, then α can be calculated by: α = (1/τ) (c - 1) = (vn / totalMFP) (c - 1) Eq. (4.1.1-9) where τ is the average neutron lifetime between collisions and vn is the average neutron velocity (which is 2.0x109 cm/sec for a 2 MeV neutron, the average fission spectrum energy). The "no losses" assumption is an idealization. It provides an upper bound for reaction rates and a good indication of the relative reaction rates in different materials. For very large assemblies consisting of many critical masses, neutron losses may actually become negligible and approach the α's given below. The factor c-1 used above is the "neutron number". It represents the average neutron excess per collision. In real systems, there is always some leakage. When this leakage is taken in account, we get the "effective neutron number" which is always less than c-1. When the effective neutron number is zero, the system is exactly critical. 4.1.2 Nuclear Properties of Fissile Materials The actual value of alpha at a given density is the result of many interacting factors: the relative neutron density and cross-sections values as a function of neutron energy, weighted by neutron velocity 3 which in turn is determined by the fission neutron energy spectrum modified by the effects of both moderation and inelastic scattering. Ideally the value of α should be determined by "integral experiments". That is, measured directly in the fissile material where all of these effects will occur naturally. Calculating τ and α from differential cross-section measurements, adjusted neutron spectrums, etc. is fraught with potential error. In the table below, I give some illustrative values of c, total cross-section, total mean free path lengths for the principal fissionable materials (at 1 MeV), and the α's at maximum uncompressed densities. Compression to above normal density (achievable factors range up to 3 or so in weapons) reduce the MFPs, α's(and the physical dimensions of the system) proportionately. Table 4.1.2-1. Fissile Material Properties isotope c crosstotal totalMFP density α tdouble (barns) (cm) (1/μsec) (nano-sec) U233 1.43 6.5 3.15 18.9 273 2.54 U235 1.27 6.8 3.04 18.9 178 3.90 Pu239 1.40 7.9 2.54 19.8 315 2.20 Values of c and totalMFP can be easily calculated for mixtures of materials as well.
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