Physics of Nuclear Explosives by Barrosso

Ref: Dalton Barrosso (2009). Physics of Nuclear Explosives.

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Summary

• Despite the greater availability of information and the diffusion of knowledge in related areas, as far as I have been concerned there is no publication that addresses explicitly and at the desired technical level the theory of nuclear explosives. This book is an attempt to fill part of this gap by specifically addressing subjects critical to understanding the physics of nuclear explosives. Elaborated on the basis of strictly academic scientific activity and with technical rigor consistent with limitations in data availability, it is intended for those who wish to know more deeply the theory and physical processes involved in nuclear explosions.

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Ch 1 Introduction

• The atomic nucleus of the elements is composed of protons and neutrons (the nucleons). These particles are bound by the strongest force of nature: the nuclear force (‘strong interaction’). In spite of its short range (restricted to nucleus dimensions, 10-13 cm), this attractive force between nucleons overcomes the repulsive coulombian force between protons and holds them tied, together with neutrons, in a complex and violent internal dynamic. The nuclear force is responsible for the strong binding energy of the particles in the nucleus. While in the chemical bindings and in atomic transitions (governed by the EM forces) the energy is measured in electronvolt (eV) (1 eV = 1.602x10-12 ergs) or in kilo-electronvolt (keV), the binding energy of the protons and neutrons in nucleus is measured in millions of electronvolt (MeV). Therefore, the energy released in nuclear reactions or transitions is orders of magnitude greater than the energy released in chemical reactions or in atomic transitions.

1.1 Nuclear fission and fusion

• Fission, discovered in 1938, is the division of an excited atomic nucleus into two other nuclei of smaller masses, which are distributed around half the mass of the original nucleus.

• In the fission of the nucleus of the fissile Isotopes, ~200 MeV (3.210 ergs) are released. To get an idea of what this energy represents, the fission of 1 kg of U or Pu generates approximately 8x1020 ergs, or 19 kt, more than the energy released in the explosion of the Hiroshima atomic bomb.

• The absorption of a neutron by the nucleus of a fissile element can lead it to a complicated process of oscillation or vibration until it is completely destabilized, culminating in its separation into preferentially two intermediate mass nuclei, called fission fragments (or fission products). These carry, in the form of kinetic energy, 80% of the energy released; the remaining 20% is distributed by emission of x-rays (4%), neutrons (3%) and neutrinos (5%), plus a posterior portion of 8% due to beta decay of fission products and the non-fission capture of neutrons.

• Fusion is induced by a thermal process, that is to say, for two nuclei come together to form a third it is necessary to impart them with sufficient kinetic energy so that they overcome the coulombian repulsion force between them (due to protons) and enter into the working distance of the nuclear attractive force. For example, the easiest fusion is obtained with the heavy H isotopes- DT and tritium.

1.2 Fission chain reaction and the criticality concept

• On average, 2.5 to 3 neutrons are released in a fission reaction, with an average energy of about 2 MeV. Immediately the idea arose that these neutrons could be used to promote new fissions, in a continuous and self-sustained process called chain-reaction.

• In order to visualize the progression of the fission chain reaction in a supercritical medium, consider that each fission emits on average 3 neutrons (as is the case in the fission of the Pu-239 isotope), and that 2 of these neutrons always cause fission of new fissile nuclei present in the medium; (it may be assumed that the third neutron is lost by leakage or by non-fission capture in any present isotope). Assuming that the chain reaction starts from a single fission, the number of fissions increases geometrically with multiplicative ratio k=2: Number of fissions = 1, k, k2, k3, ..,kn, where n represents the n-generation of neutrons. Each generation follows the previous one with an average time interval defined between them.

• Fission neutron velocity is of the order of 109 cm/s.

• Taking as an example the 8th generation, that is, after a time of approximately 10-7 s from the beginning of the fission chain reaction with multiplicative factor k=2, the number of fissions multiplies to the astonishing value of 280 = 1.2x1024 fissions, with release of about 9 kt of energy only in this last generation.

1.3 Fissile materials

• The principal fissile isotopes are U-235, U-233, and Pu-239. These isotopes are classified as fissile because they have high probabilities of being fissioned by neutron absorption for all energy range of the incident neutrons, particularly for the energy band of thermal neutrons, defined around 0.025 eV (range, however, unimportant for nuclear explosives, which work only with fast neutrons). They are distinguished from the other elements by having a "critical mass', which is the minimum mass in which the development of a self-sustained fission chain reaction is possible. Other isotopes, such as U-238 and Th-232, have relatively high probabilities of being fissioned, but only by very fast neutrons, with high kinetic energies; (for U238 , the incident neutron energy must be greater than about 1 MeV). Of the fissile isotopes, only U235 is found in nature, but its proportion is only 7 nuclei per 1000 existent in natural U, composed mostly of U-238. The use of U in nuclear explosives requires that it be enriched in U-235 from the natural content of 0.7% up to an optimum weight fraction of about 93%. The main processes of isotopic enrichment are well known: gas diffusion, ultracentrifugation and, more recently, by laser beams. The other fissile isotopes, the U-233 and the Pu-239 , are artificially produced inside nuclear reactors. The U-233 is the product of the radioactive decay chain that originates from Th-232 , after the capture of a neutron by this isotope, while the Pu-239 comes from the radioactive decay chain that begins with the capture of a neutron by the U-238 nucleus. The isotopes that, by radiative capture of neutrons, give rise to the formation of fissile isotopes are called fertile isotopes.

• In relation to Pu and U (U highly enriched in U-235), both have favorable and unfavorable characteristics which make their uses dependent on the type of application the nuclear explosives are intended to. Pu has a small critical mass (~10 kg), but is composed of a series of isotopes some of them with a high spontaneous fissions rate whose neutron production can cause the pre-detonation of the nuclear explosive, compromising its efficiency. In U, the pre-detonation problem is minor, but its high critical mass (~48 kg for pure U-235), compared to that of Pu, constitute a serious disadvantage in face of constraints in nuclear weapons design. The critical mass of the fissile elements can be substantially reduced with the addition of a cover material that serves not only as a tamper but also as a neutron reflector. The reflection of neutrons in the cover decreases their leakage, increasing their availability within the fissile mass and, consequently, the criticality. Recommended materials with suitable properties to perform this function, in nuclear explosives, are natural-U, tungsten and Be.

1.4 Supercritical assembly of fissile mass

• In U explosives, in addition to the implosion method, the so-called "cannonball" method is used, in which two subcritical fissile masses are quickly united by means of ballistic projection of one against the other, forming a single supercritical mass. The time for the insertion of criticality in this process is limited to the range of 10-3-10-4 s.

• In a nuclear explosion, the energy released depends on a competition between the propagation rate of the fission chain reaction in the supercritical fissile mass (which is greater the faster the neutrons in the system) and the velocity with which the hydrodynamic expansion of the fissile mass occurs in the direction of subcriticality.

1.5 Extreme conditions of the fissile mass in nuclear explosion

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Ch 2 Neutronic & Criticality

• Chapter 2 covers the neutronic formalism to numerically simulate the fission chain reaction that develops in a nuclear explosion. The main purpose is to determine the time evolution of neutron flux and the energy deposition in the fissile mass during the propagation of this fission chain reaction.

2.1 Neutron interaction with matter. Cross sections

• The main nuclear reactions of neutrons with matter which are relevant for the calculation of neutron population in a fissile medium (in nuclear reactors or nuclear explosives) are: fission, radiative capture and scattering (elastic and inelastic). Other secondary reactions, such as the (n,2n) reaction (absorption of a neutron with emission of two, are commonly taken into account, but constitute a refinement (in fact, in very fast neutron spectra, this reaction is very important).

• Nuclear Fission: The division of the nucleus destabilized by a neutron absorption. The nucleus is split into two fragments of intermediate masses, releasing in the process on average 2.5 to 3 neutrons (depending on the fissile element) with average energy of 2 MeV. These neutrons are responsible for the feasibility of the fission chain reaction. The total energy released by fission is about 200 MeV, 80% of which is carried by the fission fragments, which normally deposit their energy locally in the medium.

• Radiative Capture: Consists of the simple absorption of a neutron by the nucleus, which returns from the excited state to the ground state emitting gamma rays. The atomic mass of the nucleus is then increased by one. It is also termed parasite absorption, since it removes neutrons from the system and degrades the criticality. It is responsible, however, for the isotopes production such as those of Pu within nuclear reactors.

• Elastic Scattering: The incident neutron is scattered by the nucleus with no variation of internal energy state of the nucleus. In this case, the kinetic energy of the neutron-nucleus system is conserved.

• Inelastic Scattering: In this case, the nucleus with which the neutron collides is-left in an excited state, so there is no conservation of the kinetic energy of the system.

• Each one of these reactions has a certain probability of occurring depending on the energy of the incident neutron and the characteristics of nuclei with which it interacts. The probability is defined in terms of cross section concept.

• Microscopic Cross Sections: The effective target area that a particular nucleus exposes to an incident neutron for a particular reaction to occur. It is given in barns (10-24 cm2) and is represented by the Greek letter sigma (σ). Microscopic cross sections are an intrinsic feature of atomic nuclei. The probability of reaction also depends, of course, on the density of these nuclei in the medium. 

• Macroscopic Cross Section: A measure of the probability of nuclear reaction per cm traveled by the neutron in the medium. It is given by: ∑ = N*σ (cm-1); N = atomic density of the material (atoms/cm3).

• Subscripts f, a, s, are used to represent the interactions of fission, absorption (fission + radiative capture), scattering.

• Neutron Flux: The density of neutrons population in a given fissile system multiplied by the velocity of the neutrons; it has units of neutrons/cm2/s. This abstract scalar quantity, represented by Ø, has its mathematical convenience justified by the calculation of the reaction rates/cm3 of a particular interaction, given by: 

• Reaction rate = N*v*∑ = Ø *∑ (in cm-3s-1); N = density of neutrons, v = velocity of neutrons, ∑ = macroscopic cross section.

2.2 Neutron transport equation. The ⍺ and Keff eigenvalues

• Neutron Transport Equation: The fundamental equation to be solved in neutronic problems. Its solution determines the distribution of neutron flux inside the fissile system and its evolution over time. 

• Neutron Flux (Φ): A measure of the intensity of neutron radiation; the total number of neutrons passing through a unit area in a unit time. It is given by: Φ = n*v (n = number of neutrons per cm3, v = neutron velocity.

• Typical neutrons velocities inside the fissile core in nuclear explosions are of the order of 4,400 km/s (0.1 MeV) to 44,000 km/s (10 MeV), while the expansion velocity of the fissile core acquires hundreds or thousands of kilometers per second only in final stages of the explosion, when the system is probably subcritical and the fission power decays to negligible values.

• ⍺ eigenvalues: The criticality analysis of a fissile system in terms of the dominant alpha eigenvalue: 

• ⍺ < 0: The system is subcritical (decreasing flux).

• ⍺ > 0: The system is supercritical (increasing flux).

• ⍺ = 0: The system is critical (stationary flux).

• Keff (k-effective) eigenvalues: The criticality of a fissile system is more commonly analyzed through the Keff; the greatest eigenvalue in magnitude corresponding to an everywhere positive eigenfunction.

• Keff < 1: The system is subcritical (since it is necessary to increase the values of v).

• Keff > 1: The system is supercritical (there is an excess of neutron production).

• Keff = 1: the system is exactly critical.

• This eigenvalue is called the effective multiplication factor. In the fission chain reaction it gives the multiplication of the number of neutrons after each generation of these neutrons in the reaction.

• A simple relationship between the ⍺ and Keff eigenvalues is given by: ⍺ = (keff – 1)/l (l = avg lifetime of prompt neutrons- the time between the birth of the neutron and its disappearance by leakage or absorption).

• In nuclear explosives, the higher the keff value introduced by the supercritical assembly of the fissile mass, the greater will be the nuclear explosion power resulting from the fission chain reaction, since greater will be the multiplication of neutrons and the number of fissions after each generation of these neutrons in the reaction.

2.3 Solution of transport equation. Transport codes

2.4 Criticality in Uranium and Plutonium metallic balls

2.5 Time evolution of neutron flux through the quasi-static approximation. AX-1 code scheme

2.6 Burn of fissile material. Nuclear explosive efficiency

• The efficiency of nuclear explosions, defined as the ratio between the fissile mass consumed (fissioned) during the explosion and the initial fissile mass, is generally low. In U explosives it is around 1% to 2% for medium-power explosions (10-20 kt); in Pu explosives it is higher: around 10% to 20% for the same range of explosion.

2.7 Numerical solution of the time-dependent neutron transport equation

• Tamper (Neutron Reflector): Resides around the fissile mass and increases the average neutron lifetime and, consequently, the time transients to asymptotic distributions.

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Ch 3 Hydrodynamic & Thermodynamic at High Temperatures & Densities

• In this chapter, we study the hydrodynamic calculation, that is, the calculation of the hydrodynamic response of the fissile mass and its cover (tamper or neutron reflector) to the energy released by the fission chain reaction. 

• It is mainly by hydrodynamic expansion that the fissile mass becomes subcritical.

3.1 Dense and heavy plasmas. Hydrodynamic Characteristics

• At high temperatures and because of the high atomic number and the strong ionization, the fissile mass can be treated as a dense fluid (plasma) of free electrons and multiple ionized atoms. Since there is no charge separation on the hydrodynamic scale (ne(r) ~ Zeni (r)), the plasma is electrically neutral, and its motion is governed only by the hydrodynamic equations, without the need to include Maxwell's equations. 

• Average Degree of Ionization (Ze): The number of free electrons present in the plasma per original atom.

• The fission fragments, which are deposited locally in the fissile mass, transfer almost all their energy directly to electrons, which, by successive collisions, rapidly lose any excess of energy.

• It is imperative to solve the radiation transport equation simultaneously with the hydrodynamics equations.

• The hydrodynamic simulation of a fission nuclear explosion can be based on the following calculation scheme:

• a) Equations of hydrodynamics at one-temperature model, which presupposes complete thermodynamic equilibrium between electrons and thermal radiation.

• b) Mean ion approximation to determine the average degree of ionization considering complete equilibrium between ionization and electron capture processes.

• c) Equation of state and specific internal energy of the ideal gas model, with incorporation of ionization and thermal radiation energies.

• d) Assuming local thermodynamic equilibrium, energy and pressure of thermal radiation varying with the fourth power of the temperature (blackbody): transfer of this radiation using the conduction approximation to the radiation transport equations. Such an approximation is admissible because of the high fissile mass opacity, even at extremely high temperatures.

3.2 Hydrodynamic equations. Numerical solution scheme

3.3 Radiation transport equation. The conduction approximation

• In a nuclear explosion, the energy transfer from the fissile mass to the tamper, via radiation, is comparable to or even greater than the energy transfer by the hydrodynamic shock wave generated by the fissile mass expansion.

3.4 Opacity and Rosseland mean free path

• The calculation of opacity, that is, of the radiation mean free path in heavy elements at high temperatures is one of the most difficult tasks related to the study of thermal radiation transfer in these elements. The main reasons are the high number of electrons (both free and bound to ions in the most diverse quantum states) and the multiplicity of possible transitions related to the radiation absorption in complex atoms or ions. These transitions can be free-free (inverse bremsstrahlung absorption), bound-free (photoionization) and bound-bound (spectral absorption). The first two transitions result in continuous absorptions, while the latter transitions are discrete and result in spectral lines.

3.5 Average degree of ionization

• The degree of ionization in gases (plasmas) at high temperatures relates to the number of free electrons present in the plasma per original atom.

• The electron ionization and recombination processes in atoms or ions of dominant importance for the determination of the degree of ionization are the following: 

• Photoionization: The electron is ejected by a quanta of radiation.

• Collision with Electrons: The impact of a free electron with an ion causes the ejection of a bound electron.

• Radiative Recombination: A free electron is captured by an ion, emitting a quanta of radiation.

• Three-Body Recombination: One free electron is captured by an ion and the excess of energy.

3.6 Equation of state and specific internal energy. The Thomas-Fermi model

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Ch 4 Numerical Simulation of Nuclear Explosions

• The objective of this chapter is to determine the moves of the fissile mass and its cover coupled to the neutronic calculations at successive time intervals. 

4.1 Coupled neutronic-radiation-hydrodynamics equations. The RIC1 code

4.2 Nuclear explosion of a plutonium sphere with natural-uranium tamper

• Consider a compressed Pu sphere with initial density of 35 g/cm3 (1.8 times the ⍺-phase Pu density), with external radius of 3.793 cm (8 kg) and with 3 cm of natural-U tamper at normal solid density of 18.9 g/cm3 (20.5 kg). Isotopic composition of typical weapon-grade Pu was used: Pu239 (93.8%) + Pu240 (5.8%) + Pu241 (0.35%) + Pu242 (0.05%). The natural-U has its normal composition: 99.3% of U238 + 0.7% of U235. In this initial configuration, keff=1.385 and ⍺ = 19.6 x 107 s.

• Total internal energy = kinetic energy of ions and free electrons + ionization energy + energy of thermal radiation.

• The efficiency of the explosion was 11.6% with 8.4% of the fissions occurring in the U tamper. Practically all the energy of the explosion is released in a time interval of about 10-7 s, with about 30% of the energy being generated after the fissile mass becomes subcritical (Keff < 1).

• A considerable amount of energy escaping from the fissile core to outside medium occurs in the form of thermal X-ray; 20% of the total energy released.

• X-rays play a key role in thermonuclear explosives, since they seem to be the only means by which the energy of the fission explosion can be conveyed to the thermonuclear module of the explosive under the conditions required for a thermonuclear detonation. The use of a more optically transparent material as the Be (with 3 cm thick as well) around the Pu rises the thermal X-rays energy escaping to outside to about 56% of the total energy of the explosion!

• The energy release process in a nuclear explosion depends only on the fissile mass and the material used as neutron reflector, being completely independent of the external components of the nuclear explosive and of the external environment, which have only influence on the later spread of nuclear explosion energy.

• The nuclear explosion of HEU has a behavior similar to that of nuclear explosion of Pu, with the difference that, due to the lower propagation rate of the fission chain reaction in U, the energy is released in a comparatively longer time.

4.3 Analysis of sensitivity

4.4 Explosion versus Uranium criticality

• The excess of criticality is the main parameter on which the nuclear explosion power depends. These results refer to the explosion calculations of 93%-U235 enriched U sphere, with an initial density of 18.5 g/cm3 and with 2 cm of natural-U tamper, as a function of the initial excess of criticality.

• An advantage of tungsten (W) is the absence of spontaneous fissions, which could greatly increase the probability of pre-ignition in the ballistic artifact if natural-U were used.

4.5 Analysis of the influence of the tamper (neutron reflector)

• The cover, neutron reflector or the so-called tamper around the fissile mass of nuclear explosives has a significant effect on the efficiency of nuclear explosion and on the economy of fissile material. This influence, which is both neutronic and hydrodynamic, depends on the type of material(s) that compose it: whether it is light or heavy, fissionable or not, its ability to reflect neutrons and the speed with which the reflection process occurs (less moderation is better).

• From the viewpoint of the static neutron transport calculation, the influence of the tamper on the criticality is maximum when its thickness corresponds to the so-called infinite reflection, which for natural-U corresponds to about 10 cm (adding more material does not change significatively the criticality). However, from the viewpoint of kinetics, the neutronic influence of the tamper may cease to be effective long before that value. The reason has is that neutrons that go too far into the tamper shell can take a very long time in relation to the time when most of the energy from the nuclear explosion is released to be reflected back to the fissile mass; thus, the outermost regions of very thick tamper shells (mainly if they are composed of light materials) may contribute little or nothing to the increase of the fission chain reaction that develops in the internal fissile mass. [This fact is reflected in the ⍺-eigenvalue, which, unlike the keff, depends on the average neutron lifetime, which increases with the tamper thickening.]

• The results show that the neutronic influence of the tamper, that is, the increase in the criticality due to neutron reflection, is the predominant factor contributing to the energy yield increase of a nuclear explosion.

• The definition of the tamper is linked to other factors that determine the design of nuclear explosives, especially with respect to size and weight limitations. In the supercritical assembly of the Pu, for example, it is necessary to minimize the overall mass to be compressed by the HE in order to reduce the criticality insertion time and the probability of nuclear pre-ignition. In this aspect, the Be presents greater advantage over the natural-U, due to the great difference of mass density between the two materials.

4.6 Comparison with quasi-static approximation (alpha-eigen value)

4.7 Hiroshima bomb

• It is reported that the final configuration of the bomb (‘Little Boy’), after the shot of a subcritical mass against another subcritical target mass, was more or less a square cylinder containing a supercritical U mass of 64 kg, with an average enriched of 80% in U235 and surrounded by a tamper of W-carbide with about 8.25 cm thickness (sufficient to an infinite reflection).
  

4.8 Fission micro-explosions

4.9 Post-explosion phenomena

• The main and most spectacularly visible phenomenon characteristic of the nuclear explosion, in the moments after the release of the total energy of the explosion, occurs in the atmosphere, with the splendor of the so-called fireball, formed by the rapid transfer of radiation and its absorption in the atmospheric environment neighboring to the point of explosion. The movement of the shock wave generated by the explosion, which strongly heats the medium in which it propagates, is given as follows: R = [E/(β * ɣ * ⍴0)]^(1/5 * t2/5). R = radius and V = velocity of the shock wave expansion at time t; ⍴0 =density of the medium in which it propagates (in air, ⍴0 = .012 g/cm3); E = total energy released by the explosion; and β is a constant that depends on the ratio between the specific heats (ɣ = cp / cv).

• The pressure exerted by the shock wave can be estimated by means of the limit formula for strong shock waves, which is given by P ~ [2/ (ɣ +1)] * ⍴0 * V2.

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Ch 5 Hydrodynamic Theory of Chemical High-Explosive Detonation

• It is through the detonation of powerful chemical explosives that the supercritical assembly of the fissile mass takes place, in the speed and precision required. The need for rapid introduction of criticality is due to the high probability that the nuclear explosive pre-detonates with low efficiency - that is, the probability that an early generation of a fission chain reaction occurs before inserting and reaching the maximum planned criticality - when this introduction is too slow.

5.1 Hydrodynamic theory of chemical detonations

• Von Neumann Spike: The point on a pressure/time graph that represents the overpressure which results from the initial compression of the unreacted explosive, which gives rise to the detonation process.

• Properties of HE

• TNT: C7H5(NO2)3; ⍴ = 1.64 g/cm3; D = 6.95 km/s; P = 190 kbar.

• RDX: C3H6N3(NO2)3; ⍴ = 1.80 g/cm3; D = 8.75 km/s; P = 347 kbar.

• HMX: C4H8 N4(NO2)3; ⍴ = 1.90 g/cm3; D = 9.10 km/s; P = 393 kbar.

• Composite B: 64% RDX/36% TNT; ⍴ = 1.70 g/cm3; D = 8.00 km/s; P = 292 kbar.

• Baratol: 60% Ba (NO3)2/40% TNT; ⍴ = 2.60 g/cm3; D = 5.20 km/s; P = 140 kbar.

• PBX: 94% HMX/3% NC/3% CEF; ⍴ = 1.84 g/cm3; D = 8.80 km/s; P = 370 kbar.

5.2 Hydrodynamics and numerical scheme of solution. The LUTI code

• The detonation wave is nothing more than a shock wave followed by a chemical reaction process.

5.3 Plane detonation waves

5.4 Acceleration of projectiles by high explosives

5.5 Convergent detonation waves

• While in the plane detonation wave its propagation velocity is constant and the conditions on detonation front remain stationary (in the reference system moving with the wave front), in the cylindrical or spherically convergent detonation waves the hydrodynamic convergence of the detonation products produces cumulative effects and the conditions in the detonation front vary continuously as it approaches to the axis (cylinder) or to the center (sphere) of convergence. In these cases, we have the so-called forced detonations, where D > Dcj & P1 > Pcj.

5.6 Interaction of the detonation wave with inert materials

5.7 Explosive "lenses"

• Explosive Lens(es): Devices that consist in the use of explosives with different detonation velocities and arranged in a certain geometry in order to produce detonation waves in a desired geometric form.

• Spherically convergent detonations with one or two detonation points utilize an outer layer containing an explosive with a high detonation velocity and with only one point of ignition, involving a second explosive whose main property is to have the lowest detonation velocity possible; and finally, a third spherical explosive with high detonation power, since it will be responsible for the implosion of the internal solid mass. The objective of the strip-shaped external explosive 1 is to detonate the internal explosive 2 at successive points on the interface so that, by means of a suitable geometrical shape, a spherically convergent detonation wave is generated in the innermost explosive 3 in contact with the solid. This geometric form is called a "logarithmic spiral", an exponential function

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Ch 6 Dynamic Compression of Solids

• It is possible to greatly increase the criticality of the fissile mass by compressing it to densities well above its normal solid density. This criticality increase results from the reduction of neutrons leakage due to reduction of their mean free path in the compressed fissile mass.

• In the vast majority of nuclear explosives, the excess of criticality is obtained by compressing the fissile mass using powerful chemical HE. In the implosion method, this compression benefits from cumulative hydrodynamic effects, characteristic of convergent geometries, through which it is possible to reach densities 2 or more times the normal densities of U and Pu in the convergence central region.

6.1 Shock waves. Rankine-Hugoniot relations. Numerical simulation

• The shock wave formation process results essentially from the increase in the velocity of sound propagation (or of a disturbance) in the material with the increase of its density. Thus, under the effect of strong impulsive force, disturbances caused in sequence during the rapid transient of acceleration and compression of the material (whose density increases continuously) overlap and "collapse" forming a shock wave front. In the absence of viscosity, this shock wave would constitute an unphysical hydrodynamic discontinuity, propagating in the material with supersonic speed (in relation to the undisturbed material). In the real shock wave, the effect of viscosity and other mechanisms of energy dissipation interfere in the formation process of the wave front, whose width is extremely narrow (of the order of a few molecular or particle mean free paths), but finite.

• ⍴ = density, V = specific volume (1/⍴), u = particle or fluid velocity, P = pressure, E = specific internal energy, us = propagation velocity of the shock wave.

6.2 Experimental determination of dynamic compression of solids. Hugoniot curves

6.3 Forces resisting to compression. Equations of state

• In dynamic compression caused by the propagation of strong shock waves, the main forces that resist to compression in the solid are of elastic and thermal origin. Dynamic compression approximates the atoms in the crystalline lattice, shifting them from their equilibrium position towards the Coulomb repulsion, while increasing the vibration energy of these atoms, leading to an increase of temperature and kinetic pressure. The Coulomb components of pressure and internal energy in the shock wave depend only on the specific volume. They are called elastic or "cold" pressure and elastic or "cold" internal energy (Pc and Ec, respectively), since they represent the variation of pressure and internal energy with the density at 0 Kelvin (isothermal at 0 Kelvin).

6.4 Allotropic variations in materials under compression

6.5 Dynamic compression of porous materials

6.6 Implosion of plutonium masses: hydrodynamic analysis and criticality insertion rate

• Assuming that the nuclear explosion occurs at the point of maximum criticality, the RIC1 code calculated the following released energies and efficiencies: almost solid Pu: 1.42 kt (E=1.5%); hollow plutonium: 3.3 kilotons (&=2.9%); and porous plutonium: 2.7 kilotons (8=2.4%).

6.7 Multiple shock waves and isentropic compression

• In the case of isentropic compression of materials by high-explosives, the strong shock wave generated in the material by the impact of the detonation wave must, by some means, be replaced by a sequence of weaker shock waves properly modeled in time. The practical way to achieve such multiple compressions probably consists in the alternating and successive use of layers containing high and low dynamic impedances materials, 61,02 inserted between the explosive and the material to be compressed. The partitioning of the strong initial shock wave into a sequence of weaker shock waves would be achieved by the effect of the multiple reflections, back and forth, of the shock waves in the layers of high dynamic impedance, between which are interposed the layers of low dynamic impedance. To achieve the desired effect, the thicknesses of the layers must vary in a ratio which depends on the propagation time of the waves in the materials and the distances to be traveled. Such a scheme also allows a cumulative shock pressure amplification.

6.8 Extreme shock waves. Influence of thermal radiation

• At extremely high shock pressures (in the range of hundreds to thousands of megabars), all materials, particularly the lightest ones, tend to behave as an ideal gas (plasma). This occurs when the temperature acquired by the material in terms of kT is much higher than the Coulomb interaction energy between the particles and the ionization energy of the atomic electrons. In heavy materials, where the bonding energy of the internal electrons is in the keV range, multiple ionizations continue to play an important role even at extreme temperatures, as in the case of uranium, analyzed in Chapters 3 and 4. On the other hand, light materials become fully ionized and behave as an ideal plasma at relatively lower temperatures.

• With the increase of shock pressure and temperature, the thermal radiation becomes increasingly important, and there may be a situation where much of the energy supplied by the shock wave is transformed into radiation. The radiation affects the characteristics of the shock wave basically in two ways: increases the degrees of freedom of internal energy partition (thereby increasing total heat capacity and serves as a mechanism for the rapid transfer of energy from the shock wave region to the region ahead of the shock wave, preheating that region. The increase in the heat capacity (as occurs also in the case of ionization) and the energy leakage through radiation cause the increase of the maximum ratio between the final and initial densities of the material hit by the shock wave, which for an ideal monatomic gas is p/po-4 (see Appendix F).

• The results show that the influence of thermal radiation becomes more significant only from a shock pressure of the order of 103 Mbar, when temperatures of several million degrees are reached in the LiDT. At these temperatures, intense radiation wave propagates downstream of the shock wave, the difference between the propagation velocities of the two waves being dependent on the radiation intensity and transparency of the LiDT at the temperatures reached.

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Ch 7 The Pre-Ignition Problem

• The basic question involved in the problem of pre-ignition or pre-denotation (early detonation) in nuclear explosives can be formulated as follows: If a certain neutron source of intensity S (neutrons/s) is present in an initially subcritical medium, what is the probability that a given excess of reactivity, ⍴ = (keff-1)/keff , be introduced in this medium, without any persistent fission chain reaction established during the time, t, necessary for the introduction of this reactivity? There is another question: Once the persistent fission chain reaction (ignition) is established, how long does it take for the energy generated in the system put a break on the reactivity insertion process, that is, to provide a negative reactivity feedback? In other words: how much additional reactivity can be inserted after the premature establishment of the persistent fission chain reaction? In nuclear explosives, during the supercritical assembly of the fissile mass, the presence of random neutrons from external sources or from the fissile material itself is inevitable. These neutrons can cause the fission chain reaction as soon as the system becomes supercritical. This reaction, as we have seen, propagates extremely fast, because of the excess of reactivity involved and the short average generation time of prompt neutrons (in the order of a few nanoseconds). Considering the velocities with which the required excesses of reactivity can be introduced in practice, the time available for introducing additional reactivity after early nuclear ignition is, as will be seen in this chapter, extremely short, due to the destruction of the supercritical configuration caused by the rapid release of fission energy. The nuclear explosive is likely to explode prematurely and inefficiently. Therefore, it is essential that the supercritical assembly of the fissile mass be carried out with a minimum chance of nuclear ignition before reactivity values close to the maximum planned value are reached.

7.1 Statistical neutronic

• The issue of nuclear pre-ignition is pertinent to the area of statistical neutronics, which studies the fluctuations in the neutron population when the population is very low and, consequently, when the statistical nature of nuclear reactions has great influence on the behavior of the population. The deterministic treatment, implicit in the use of the transport equation (2.3), is valid only in the application to a large number of neutrons, when fluctuations can be neglected and the cross sections are treated as average probabilities of nuclear reactions.

7.2 Spontaneous fissions and the plutonium isotopic vector

• The problem of pre-ignition is crucial in Pu explosives. Hence the importance of having low contents of Pu240 and Pu242 in the Pu masses used in nuclear explosives (weapon-grade Pu).

• The Pu for bombs is produced in reactors specially designed for this purpose, although part of them has a hybrid purpose of also producing electricity. Examples are the Hanford's water-cooled graphite-moderated reactors and Savanna River's heavy-water reactors in the USA; similar reactors in Chelyabinsk and Dadonovo in Russia;" in England, the Calder Hall-type reactors; G1 (air-cooled) and G2 and G3 (CO2-cooled) graphite-moderated reactors in France. These reactors (most of them already out of service) present two important points in common: the use of natural U as their main fuel and the charge and discharge of the fuel are performed with reactor in operation, because frequent shutdowns would be costly in operational and economic terms.

7.3 Probability of no pre-ignition during an arbitrary insertion of reactivity

• For the calculation of probability of no pre-ignition (no persistent fission chain reaction) in an initially subcritical medium, where an excess of reactivity is inserted, and in presence of a neutron source, S, two basic assumptions are firstly admitted: 1) All neutrons behave identically, i.e., they have the same probability, p, of inducing fission (p=k/v, where k=keff is the multiplication factor and V is the average number of neutrons emitted per fission) and each fission has the same probability P(v) of emitting v neutrons; 2) the neutron source is of the "weak" type, that is, S 𝛕 < <1, where 𝛕 is the mean lifetime of prompt neutrons.

• The first hypothesis neglects the spatial and energetic dependence of neutrons, considering, for example, that all neutrons emitted from the source have the same "importance". This fault could be partially corrected by assuming effective source S weighted in the adjunct transport flux.

• The second hypothesis implies that the neutron population is predominantly determined by the first persistent chain reaction in the system, and that the probability that it occurs is practically null for k < 1. In nuclear explosives, S240 ~ 105-106 and t ~ 10-9 , which fully satisfies the above inequality.

7.4 Pre-ignition probability in plutonium explosives

7.5 Ignition by artificial neutron source

The source commonly used in nuclear explosives is Po-Be, which emits neutrons through (⍺,n) reaction and has no associated ɣ radiation. In the inactive state, Po and Be are separated by a thin layer of inert material so that this layer fully absorbs the a particles emitted by the Po and no reaction takes place. The activation of the source occurs by the rupture of this layer impacted by the implosion shock wave and having a special format) and the consequent mixture between the two elements. [The absence of ɣ emission is important because it would be impossible to avoid the (ɣ, n) reaction.]

• In the implosion method this source is made in spherical form and placed under suspension or accommodated in the center of the fissile mass. Thus, the mixing of Po with Be occurs naturally in moments close to the maximum reactivity: in hollow spheres after compaction, and in solid spheres after the shock wave convergence to the center. (It seems that in modern nuclear explosives, the source is placed outside the fissile core and consists of a miniaturized neutron generator. Although time synchronization is a problem, this is highly recommended because a higher compression ratio is attained in the central region of the fissile mass if no neutron source is there).

7.6 Nuclear explosives and plutonium from nuclear power reactors

• The reliability of Pu nuclear explosives is directly linked to the amount of Pu240 present in the mass of Pu used and to the speed with which its supercritical assembly is performed. Because of the statistical nature of the pre-ignition problem, two identical nuclear explosives, under conditions of equal implosion performance, can produce completely different nuclear explosions.

• There are two additional difficult-to-circumvent circumstances related to Pu with high content of even isotopes: its higher radioactivity, which would make difficult to handle it (probably requiring a remote handling and under some sort of shielding), and the problem of self-heating caused by the ⍺-radiation emitted by these isotopes. A realistic analysis of this issue, involving the use of Pu238 in the Pu isotopic composition to "denature" it even more, that is, to render it virtually useless to be used as a nuclear explosive, was recently undertaken by Kessler. Pu238 is a fissile isotope (in fast energy range), but its high ⍺-decay activity (with a half-life of 87.7 yrs) yields 570 W/kg, serving well to this purpose. [It is also an excellent source of energy for space vehicles sent to distant space.]

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Ch 8 Implosion-Type Nuclear Explosives

• In the implosion-type nuclear explosives the fissile mass is made supercritical by mean of its strong spherical compression, which is obtained by impacting on the fissile core a spherically convergent detonation wave generated in a HE placed around it in the nuclear explosive.

• Once the problem of pre-ignition in Pu explosives became apparent at the time, when the large number of spontaneous fissions of its Pu240 isotope was discovered, it became clear that the faster and already known process of implosion was absolutely indispensable to achieve supercriticality if using Pu masses.

• In this chapter, we analyze in more details the implosion method and its performance in both Pu and HEU explosives. For this purpose, we make intensive use of computer simulation. Pu is initially admitted at its normal ⍺-phase and only the simplest configurations of the explosive are considered: a fissile mass, a cover (tamper or neutron reflector) element and the HE.

8.1 Advantages of the implosion method

• Three parameters are basic and fundamental to the implosion design:

• 1) The degree of supercriticality to be inserted: The fissile mass-tamper system must achieve certain excess of reactivity (criticality) so that the nuclear explosion releases the desired amount of energy. It depends on the mass and type of fissile material (Pu239, U235 or U233), the tamper (neutron reflector), the assembly scheme and the amount of chemical HE employed.

• 2) The criticality insertion rate: Should be as high as possible in order to minimize the pre-ignition problem. In addition to the factors listed in 1, it is conditioned, in the case of Pu, by the presence of even isotopes in its isotopic vector and by the reliability to be assigned to the explosive.

• 3) The initial subcriticality: All nuclear explosives, consisting primarily of the fissile mass, the tamper and the chemical HE, must be in a safe initial subcriticality state prior to supercritical assembly.

• Main advantages of Implosion:

• Much faster approach to the supercritical configuration due to geometry of the implosion process and the possibility of generating higher impulse pressures through spherically converging detonation waves, when cumulative hydrodynamic effects occur in the HE.

• Because the fissile mass is compressed to densities well above the normal solid density, high criticalities can be inserted using fissile masses smaller than the critical mass of the fissile elements.

• Flexibility in subcritical arrangement of the pre-assembly configuration. One of the greatest difficulties faced by nuclear explosives designers is in establishing the initial distribution of the fissile mass in the explosive so as to have with absolute assurance the subcriticality of the entire system prior to supercritical assembly.

• The main disadvantages of the implosion method are the greater difficulty of its technological domain, especially regarding to the process of convergent and uniform detonation of the spherical HE.

8.2 The cover (tamper or neutron reflector) definition

• The best tamper or neutron reflector tend to be comprised of natural-U and Be (a heavy material and a lightweight one, respectively). During the implosion phase, the cover serves as a tamper to contain the internal fissile mass after its maximum compression, preventing that the rarefaction wave coming from the pressure drop in the HE reaches the interior of the fissile mass before nuclear ignition. During the nuclear explosion phase, its primary function is to reflect neutrons back to the fissile mass and, if composed of heavy material, to resist to the expansion of the fissile mass in order to increase its supercritical confinement time.

8.3 Influence of the chemical high-explosive on the criticality

• The chemical HE, which surrounds the entire central fissile core and is made up of light neutron-moderating elements (H, C, O, N), and has great influence on the criticality of the whole system. As this influence is undesirable, since most of the detonated HE in no way or very little contributes to the propagation of the fission chain reaction that develops in the internal fissile mass, during the nuclear explosion, it is desirable as much as possible to decouple them neutronically. The objective is to decrease the influence of the HE on the initial criticality of the fissile mass (which must be subcritical) and on the probability of pre-ignition. (It is important to reiterate that the reflection time of neutrons moderated by the light elements of the HE is too long compared to the time interval in which most of the fission chain reaction develops. These neutrons play role analogous to that of the delayed neutrons.)

• The natural way to neutronically uncouple the HE from the inner fissile core is to place highly neutron-absorber elements between them, so that those neutrons that are moderated and reflected by the HE be absorbed in these elements.

• The HE has a great influence on the criticality of the system, especially for small thicknesses of natural-U in the tamper. For thickness of 2 cm, the increase in keff (in absolute value) was ~10%, while for thicknesses beyond 5 cm, the contribution of the HE to the criticality decreases significantly, due to the smaller leakage of neutrons and their higher absorption in natural-U when reflected from RDX.

• The optimum combination of elements to be used around the fissile mass, under the neutronic viewpoint, would be one that had a high ability to reflect fast neutrons coming from the fissile mass but also high epithermal and thermal capture cross sections of neutrons coming from the moderation in the chemical HE.

8.4 Computer analysis of the implosion method. Plutonium explosives

• Model: Weapon-grade isotope composition of the Pu is used: Pu239 (93.8%), Pu240 (5.8%), Pu241 (0.35%) and Pu242 (0.05%).

• The adoption of a fissile mass inner cavity is probably one of the main expedients used to make the nuclear explosive subcritical in the pre-assembly configuration (by increasing neutron leakage due to increase of the ratio between the outer surface of the fissile mass, through which the neutron escape, and its volume).

• 100 kt of energy requires the fission of approximately 5.5 kg of fissile material; assuming a burn efficiency of 40% for the Pu and 10% for the U-235, it would be necessary to have at least 13.7 kg and 55 kg of these elements, respectively.

8.5 Numerical simulation of the Fat Man atomic bomb

• It is believed that the purposes of the Al are to minimize the effect of the rarefaction wave coming from the pressure drop in the HE after the impact of the detonation wave, to increase the effect of shock wave convergence and also to minimize the effects of the Rayleigh-Taylor instabilities in the explosive-solid interface. As for the borated acrylic (the B10 is highly neutron absorber) its probable function is to absorb neutrons moderated and reflected from the Al and the HE, improving the conditions for the initial subcritical configuration and decreasing the probability of pre-ignition.

• Fat Man’s initial value was Keff =- .97 and its max value was Keff ~ 1.51, resulting in ~18-24 kt yield.

• A Po-Be source type, whose purpose is to produce a neutron pulse (with an estimated value of 107 n/s, responsible for generating a persistent fission chain reaction in the fissile mass.

8.6 HEU explosives

• In U explosives, the main characteristics are the higher fissile mass involved due to the higher critical mass of enriched U and the less serious problem of pre-ignition due to the low background of neutrons coming from the lower number of spontaneous fission of the U isotopes.

• The largest pure fission explosive to date was produced in a U explosive and released about 500 kt of energy.

8.7 Mixed uranium and plutonium explosives

• Pu and U explosives have both advantages and disadvantages. In Pu, there is a much more reduced critical mass, but the problem of pre-ignition caused by the high background of neutrons from spontaneous fissions or even isotopes of the Pu is crucial; in U, despite the minor problem of pre-ignition, the higher critical mass of HEU implies the use of higher amounts of HE so that the implosion produce the required excesses of criticality…The idea of a mixed explosive is precisely to minimize these difficulties by jointly employing the two fissile elements in the fissile core of the nuclear explosive. 

• Pu should be preferably placed in the central region because of its higher reactivity.

8.8 The ballistic or "cannonball" method

• The contact time of the masses, i.e., the time during which the system remains at maximum criticality (without geometric deformations due to impact), can be conservatively estimated by the shock wave propagation time over the distance separating the impact surface and the outer surface of the fissile mass. Using the linear relation (6.5) for U, and knowing that the velocity u is equal to Vo/2 (shock of identical materials), where Vo is the impact velocity, the value of us is easily obtained and, therefore, the value of the time referred above. The nuclear ignition (by an artificial neutron source created at the moment of impact) must occur within this time interval.

8.9 Exotic concept of a device with multiple nuclear explosions

• The thermal radiation (X-rays) from the explosion of a primary atomic bomb (shown above) diffuses rapidly through a light semitransparent material called radiation diffuser (in whose interior are inserted subcritical fissile masses, which are opaque to radiation), heating it to temperatures of several million degrees equally in all its regions. The resulting thermal and radiation pressures in this region- added to the pressure resulting from the shock reaction caused by the violent ablation of the outer region of the fissile mass tampers impacted by the incident X-rays (ablation pressure) - evenly distributed, cause the perfectly spherical implosion of the subcritical fissile masses, making them highly supercritical. Pressures of the order of thousands of Mb can be easily reached in the X-ray-heated radiation diffuser region. Pressure equalization at all points around the secondary fissile masses (regardless of their geometric shapes) is ensured by the rapid diffusion of radiation trapped in a cavity surrounded by an opaque external tamper.

• The maximum theoretical value of criticality is Keff ~ 2.8.

• The average neutron lifetime at maximum criticality is l = 3x10-10 s, resulting in an extremely fast fission chain reaction compared to the expansion time of the compressed fissile mass.

8.10 General aspects and nuclear explosive safety

• Nuclear explosives are a complex arrangement of basic internal components (neutron initiator, fissile mass, neutron reflector, high-explosive) and complementary external components (explosive lens, detonators and associated electronic circuits, structural material, safety devices etc.).

• The nuclear explosive must be proof of significative nuclear detonation if the chemical HE is accidentally detonated at one point ("one point safe"). Such a situation may occur due to external projectile action, accidental fall or sudden shocks capable of sensitizing the internal HE.

• To make them even more safe with regard to accidental detonation, some are using highly detonation-insensitive chemical HE (IHE).

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Ch 9 The “Boosted” Bombs

• Boosted bombs are a more complex type of fission nuclear explosive that incorporates into its fissile core a small amount of fusion fuel material (DT or LiDT) in order to increase the efficiency of the nuclear fission explosion.

9.1 Physical principle

• The physical principle of the boosted bombs working is simple: the fusion of small amount of DT placed in the center of the fissile mass, where temperatures of tens of millions of degrees are reached during the nuclear explosion, releases a relatively negligible sum of energy, but a significant number of neutrons, which will boost the fission chain reaction, resulting in a greater number of fissions and fissile mass consumption. It is worth noting that the number of neutrons released by DT fusions is 16x higher than those released by Pu fissions, by weight of material.

9.2 Numerical simulation of boosted bombs.

9.3 Analysis of the boosted bomb with the Plutonium in its normal solid density

• For a Pu mass with an external radius of 5.2 cm, there is an increase in nuclear explosion energy from 0.45 to 15 kt (33.3x higher). This increase results, from the effect, on the fission chain reaction, of the 4.5x1023 neutrons released by burning of 44% of the DT.

• The fusion occurs only with the increase of the Pu temperature and the consequent intensification of the thermal radiation transfer from the Pu to the DT, as well as with the continuous increases of DT bulk density due to compression caused by the Pu internal expansion.

• The Pu confines the DT to high densities and provides energy via neutron emission and radiation transfer.

• In the case of Pu with an external radius of 5.2 cm, the preheating was basically due to fission neutrons, since during the initial shock compression phase of the DT, the temperatures in the Pu were too low to cause a strong radiation emission. The neutron energy deposition was sufficient, however, to rise the temperature of the central region of the DT (before it is hit by the shock wave) from the initial value of 5 eV to a temperature of about 50 eV.

• In the case of Pu with an external radius of 6.0 cm, the preheating is much higher due to the much more intense and early emission of thermal radiation from the fissile mass. This is explained by the following. Because of the much higher reactivity implicated in this case, the energy released by the fission chain reaction increases much faster, and the internal energy and temperature inside the plutonium plasma are also much higher up to the point where the DT begins to be compressed. As a result, there is a greater transfer of thermal radiation from the fissile mass to the DT during the compression of the latter. As the thermal radiation wave propagates to the DT with a velocity much faster than the shock wave propagation velocity, it causes a strong preheat of the DT region that is ahead of the shock wave, degrading the final compression of the DT much more intensely than that caused by preheating due to fission neutrons. As can be seen in Figure 9-5, the temperature of the central region of the DT rises to 1 keV before the impact of the shock wave, at t = 16.3 x 10-8 s.

• An example of slow volume fusion is one that occurs in the Sun, where the fusion energy is in equilibrium with the gravitational attraction energy. In the explosion of Supernovas, one of the processes to explain this catastrophic event supposes the existence of a thermonuclear detonation, induced by a shock wave generated by internal instabilities, after a certain burning cycle of the star.

• The higher the initial density of the DT (which, therefore, leads to a higher final density after its compression caused by the Pu internal expansion), the higher the fusion burn rate achieved in the DT.

• LiDT: Li0.5 D0.25 T0.25 (Li-natural)

• The great advantage of using LiDT, which is solid at room temperature and has a density under normal conditions of 0.85 g/cm3, is to avoid cryogenic problems with the use of DT unless the DT is injected as a gas). The disadvantage, as discussed in Ch. 10, is the much less efficient propagation of thermonuclear reaction due to the presence of Li. However, as can be seen in Table 9-4, the DT fusion in the mixture is also sufficient to cause a very significant increase in nuclear explosion energy, especially for high initial densities of the mixture.

9.4 Boosted bomb simulation with the implosion of the fissile mass and DT by high-explosives

• In the boosted bomb, the neutron population reached after the relatively rapid release of fusion neutrons appears to have a far greater importance to the explosion efficiency than the neutron multiplication factor in the supercritical fissile mass; that is, the magnitude of the external source S0 formed by the fusion neutrons, and the quickness with which these neutrons are released, have a far greater weight in increasing explosion efficiency than the keff value inserted by the supercritical fissile mass assembly and on which the neutron multiplication depends, as can be seen from the analysis of the results in Table 9-6. These results show that the reactivity must only have a value that is sufficient to cause an explosion of fissile mass which leads to a significant fusion of DT in the center. Moreover, if this fusion is self-sustained, the final burn and explosion efficiency will be greater than that obtained by volume fusion (not much greater, however, as we have seen).

• This extremely important feature of the boosted bombs, that is, their capacity of producing reasonably powerful nuclear explosions with low fissile mass reactivity, makes them an ideal instrument for the "miniaturization" of nuclear explosives, which is nothing more than the reduction of their size and weight without effective loss of nuclear explosion power. Naturally, the lower the reactivity to be inserted, the lower the amount of chemical HE required for implosion.

9.5 Uranium boosted bombs

• Note that while the fission chain reaction develops more slowly in U (due to the longer mean lifetime of prompt-neutrons), its displacement (hydrodynamic expansion) towards subcriticality occurs, in turn, more slowly too, because of its greater mass and inertia. Thus, the neutron multiplication, although slower, extends over much longer periods of time than those existing in the chain reaction in the Pu, particularly after the jump in the neutron population caused by the injection of fusion neutrons at still early stages of fissile mass displacement. It should also be noted that in U explosives much greater fissile mass is available for burning, which allows the release of more energy and a lower criticality fall due to this burning.

9.6 Fusion fuel placed outside the fissile mass

• In this nuclear explosive configuration, the goal is that the fusion induced by an internal fission explosion makes a direct and significant contribution to the total energy yield, although the fusion contribution (via neutrons release) to boost the fission explosion remains dominant, even if the fissile mass is subcritical at the time when considerable fusion occurs. (As shown in Table 9-9, this boost effect raises the energy yield from 11 to 129 kt in the HEU mass.) The results also show a considerable fission in the external natural-U tamper, due to the 14.1 MeV fusion neutrons, energy that is well above the threshold energy for fission reactions in U238 (~ 1 MeV). Another interesting effect is that the LiDT is strong and doubly compressed in two opposite directions: by the expansion of the internal fissile core and by the expansion of the internal region of the outer tamper ablated by the thermal X-rays coming from the fissile core.

9.7 Boosted micro-explosions

9.8 Final considerations

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Ch 10 Thermonuclear Detonations

• Thermonuclear reaction waves differ from chemical reaction waves in several ways: 

• 1) The medium is completely ionized, forming a plasma, due to the higher temperatures involved (108 - 109 K).

• 2) The energy of the alpha, neutron, and other particles are deposited in regions outside where the reaction occurred due to the long mean free path of the reaction products.

• 3) The reaction occurs in the absence of thermodynamic equilibrium between the medium constituents (ions and electrons of the plasma.

• 4) The mechanism of thermal energy conduction results from the high mobility of electrons which dissipate energy and lead to the preheating of the region ahead of the wave front

• 5) There is significant energy loss by radiation (bremsstrahlung) due to the high temperature and transparency of the material. 

• Self-sustained thermonuclear reactions will occur only if the energy deposited locally by the fusion outweighs the energy losses by hydrodynamic expansion, conduction and radiation, so that Lawson's criterion is met locally, in the reaction zone.

• In a thermonuclear explosion, the fission nuclear explosive (called the primary module) ignites the thermonuclear section (called the secondary module).

10.1 Inertial confinement fusion

• The main nuclear fusion reactions involved in thermonuclear detonations occur with the heavy isotopes of H, DT and T, and are:

• D + T = ⍺ (3.5 MeV) + n (14.1 MeV); ⍺ = He42 nucleus.

• D + D = He32 (.82 MeV) + n (14.1 MeV) & D + D = T (1.01 MeV) + p (3.02 MeV)

• During the thermalization, the He32 and T products have much higher probabilities of reacting with the DT. With the He32, we have the third relevant reaction: D + He32 = ⍺ (3.6 MeV) + p (14.7 MeV).

• The hydrodynamic equations describing the thermonuclear detonation must be solved at two-temperatures, one for the ions and other for the electrons in the plasma. (In cases where radiation transport becomes important, there is a three-temperature model, when it is also unlikely to exist thermodynamic equilibrium between electrons and radiation).

10.2 Thermonuclear detonation waves

• The most important physical effects: thermonuclear reaction, hydrodynamics, heat conduction, energy exchange between ions and electrons and bremsstrahlung.

• The direct method (rocket target), with emphasis on the direct use of laser beams for the DT microsphere implosion, as well as the so-called indirect method (cannonball target or hohlraum target), which implies the conversion into X-ray of the energy deposited by particle or laser beams and the use of more complex structured targets (Fig. 10-2). The objective of both processes is to generate, in a small central region of the microsphere, the density and temperature conditions for thermonuclear fusion ignition and its propagation to the rest of the microsphere. In simplified terms, the rapid energy deposition of laser, particle or X-ray beams in the outer region of the microsphere causes its violent vaporization (ablation), generating a reaction implosion pulse (convergent shock wave) necessary for high compression and heating of the central region. Densities thousands of times that of solid DT density and temperatures of some keV are required.

• In thermonuclear explosives, fusion explosion is induced by thermal X-rays coming from a fission nuclear explosion, and for practical, economic and military reasons the emphasis is on the use of solid thermonuclear fuels at room temperature involving mixtures of Li with D and T.

• Energy Transfer Processes in the thermonuclear reaction wave:

• 1) Hydrodynamic Transport: Because of the sudden release of energy in the thermonuclear reaction zone, it expands and generates a shock wave that strongly compresses the cold, intact neighboring region of the thermonuclear fuel. The thermonuclear detonation wave will predominantly propagate by a mechanism analogous to that of the chemical detonation wave. This propagation mechanism is favored in high density plasmas.

• 2) ⍺-Particle Transport: Local deposition of energy or part of the energy of the ⍺ particles generated by the fusion reactions. After ignition, the temperature rises and the plasma expands, increasing the mean free path of these particles in the plasma medium. As a result, they tend to move more easily from one region to another of the thermonuclear fuel. If this transport is highly supersonic, such a propagation mechanism may predominate over hydrodynamic transport. The mean free path of the 3.5 MeV ⍺ particles in DT plasma at temperatures up to 100 keV is given approximately by the expression: λ⍺ = [ 1.9 / (⍴ * (1 +122/T5/4))] in cm; T = temperature in keV. At the solid density of DT and temperature of 10 keV, λ⍺ = 6.5 cm.

• 3) Neutron Transport: Neutrons carry most of the energy released in the DT fusion (14.1 MeV). Because of their long mean free path (22 cm in DT plasma at solid density, it is common in many situations to consider free escape of neutrons produced by the fusion reaction. In DT, the mean free path of 14.1 MeV neutrons as a function of density is given by: λn = 4.6 / ⍴ in cm.

• 4) Energy transport by thermal conduction due to electrons: Energy transport by thermal conduction is mainly due to the high mobility of electrons and the high temperature gradients in the plasma. 

• 5) Radiation Transport: Normally, in non-extremely dense plasmas, the radiation (bremsstrahlung) escapes freely from the plasma medium without significantly interacting with it. In fact, thermonuclear ignition is mainly conditioned by the competition between local energy deposition rate by fusion reaction and bremsstrahlung energy loss rate. Only when the former overcomes the latter, a self-sustained thermonuclear reaction can be established. In DT plasmas (Z = 1, A = 2.5), the Planck radiation mean free path, due to absorption by inverse bremsstrahlung, is given by: λp = 14 * T7/2 / ⍴2 in cm.

• It is important to note that in plasmas confined by inert materials (tampers), the radiation escaping from the plasma is strongly absorbed and reemitted by the inert material, increasing the radiation influence on the system.

• In thermonuclear explosives, probably most of the fuel (or all the fuel) is made up of LiD to avoid excessive inventory of the difficult-to-be-made and expensive tritium, whose radioactive decay would require constant replacement to prevent explosive degradation over time. The LiDT (if it is actually used) would be confined to an ignition region, where the onset of a self-sustained thermonuclear reaction would be more easily established for further propagation to an adjacent LiD fuel.

• The density of LiD (⍴ = 0.8 g/cm3) is slightly less than that of LiDT and A = 4 (for a mixture with Li). The fusion reaction rate per unit volume is: R = (⍴*N)2 < vσDD > /8, where N = 1.5 x 1023 /g is the specific atomic density of the mixture. This rate is much lower than that in LiDT, given the much lower < vσDD > parameter as a function of temperature for DD reactions. 

• In the process of shock wave plasma heating, the energy of the shock wave is initially transferred to the ions. Only after a finite time of successive collisions do they equipart energy with electrons, which in turn transfer energy to the radiation field (via bremsstrahlung) after a finite time of successive collisions as well.

10.3 The three-temperature hydrodynamics model and thermonuclear burning. Numerical solution scheme

• In the so-called "three temperature model", the equations of hydrodynamics are solved by considering different equilibrium temperatures for ions, electrons and radiation. 

• Charged particles produced by thermonuclear reactions include ⍺, p, T, and He3.

10.4 Thermonuclear explosives. Basic components

• The following facts regarding thermonuclear fuel must first be taken into account:

• 1) For practical reasons already explained, it should be composed mostly of LiD, but incorporating an ignition region containing LiDT. Both are solid at room temperature. (The use of pure LiD, without initial T, requires a special scheme, with the placement of a fissile mass in the center of thermonuclear fuel.

• 2) For the thermonuclear reaction to propagate in a self-sustained and more efficient manner, the thermonuclear fuel must be strongly compressed. This compression must be sufficient to: a) increase the thermonuclear reaction rate, so that the reaction zone have a length compatible with the characteristic dimensions of the fuel (remembering that the fusion reaction rate varies with the square of density; b) provide a complete reabsorption of the a particles energy in the thermonuclear reaction zone; c) allow that a significant part of the neutron energy be deposited in the fuel, so that it have an effective participation in the propagation of thermonuclear reaction; and d minimize the energy loss due to radiation leakage.

• 3) The compression and transmission of energy must heat the LiDT to thermonuclear ignition temperature.

• The solution of this enigma goes first, of course, through the analysis of the energy released by the fission explosion. We see that a large part of this energy is released to the external environment (out of the fissile core) in the form of thermal X-rays. As these X-rays propagate at the speed of light (or at a considerable fraction of this in light materials), they precede in time the hydrodynamic energy carried by the expanding bomb material. Thus, the use of these thermal X-rays for the thermonuclear detonation process seems evident, since they can transport a large amount of energy almost instantly to the module containing the fusion material, before it is destroyed by the expanding plasma of the fission bomb…Thermal X-rays emitted from the fission bomb explosion propagate rapidly to a region of cylindrical ring containing material with low density and low atomic number (called radiation diffuser), contained externally by a heavy material external tamper or casing). The rapid diffusion of these X-rays by the semitransparent material of the radiation diffuser (trapped between two optically opaque materials) heats it to temperatures of several million degrees equally in all its regions, establishing a strong pressure gradient between that region and the cold internal region of the thermonuclear fuel (not penetrated by the thermal X-rays, blocked by an internal tamper). This pressure gradient, added to the pressure of the shock reaction ("rocket effect") caused by the violent ablation of the external region of the internal tamper penetrated by X-rays, are responsible for generation of a convergent shock wave and the consequent implosion of the innermost region of the internal tamper and the thermonuclear fuel. The implosion of the fuel makes it possible, then, to concentrate energy in the cylinder axis (via convergence of the shock wave) and to obtain, finally, in that region, the density and temperature necessary to ignite the LiDT fuel. The internal tamper around the thermonuclear fuel is assumed to be composed of natural-U (or lightly enriched-U), with the aim of interacting with the fusion neutrons through fast fissions in the U238 (and all fissions in U235), multiplying the neutrons and increasing the overall efficiency of the explosion.

• The radiation diffuser material must have low density and low atomic number to allow a rapid diffusion of radiation. It can be composed, for example, of Li or a plastic material (for example, polystyrene), in solid form or, more appropriately, in the form of "foam" (Styrofoam seam to be a good material) to increase their transparency to radiation. If through this process it is possible to reach ignition temperatures of around 3 keV in the central region of LiDT and extreme compressions such that the fuel is optically thick to radiation and neutrons, then possibly the thermonuclear reaction, initiated and propagated into LiDT, will extend axially (or radially, depending on fuels arrangement) to the adjacent pre-compressed LiD fuel. The natural-U in the radiation shielding will then serve, in this phase, not only as a thermonuclear fuel tamper, thus preventing energy loss by hydrodynamic expansion, but also as neutron and radiation reflector, having fundamental importance for the increase of thermonuclear burning fraction. The possibility of significant pre-production of T should also be analyzed if the fuel is LiD, due to multiplication of neutrons in natural-U and its reflection and absorption by the Li6.

• The fission bomb must meet special characteristics. At first glance, it should be of ballistic type, since the chemical HE cannot interfere with the process of transmitting energy to the fusion module. However, it is believed that the transparency of the HE (composed by light elements) is sufficient to allow a satisfactory conduction of the radiation to the fusion module (a cylindrical implosion process of the fissile mass could also be used. The lateral tamper (neutron reflector) of the fissile mass in front of the fusion module must be composed of light transparent material, in order to maximize the thermal radiation emission from the fissile plasma (Be is the recommended material). Its power should probably be in the range of 10 to 20 kt. For compact thermonuclear explosive and to minimize the HE content, this fission bomb has to be of boosted type.

10.5 Numerical simulation of cylindrical and spherical implosions of LiDT with U cover (tamper). Thermonuclear explosion calculations

• Initially, generated by the external pressure of 103 Mbar, a shock wave propagates through the U, compressing it to a density of about 4x its initial density. When this shock wave reaches the internal surface of the U, a shock wave is transmitted to the LiDT, compressing it in a similar way (in the external region) to a density of about 4x its initial density; at the same time, due to the large difference between the dynamic impedances of U and LiDT, a rarefaction wave propagates back through the U mass. When this rarefaction wave reaches the outer surface of the U, a new shock wave is generated (although much weaker) if there is a persistent external pressure responsible for the implosion…This process of U acceleration -by means of a sequence of shock and rarefaction waves - tends to continue as long as the external pressure on it persists and also the formation of rarefaction waves at the interface with LiDT. An important consequence of this process (which is much more pronounced the smaller the thickness of the U) is the generation of successive compressions on the surface of LiDT, increasing its final density. In the cases analyzed here, however, this process had little significance due to the relatively wide thickness of U. Because of the lower dynamic impedance of the LiDT, the pressure in its external region reached by the shock wave, before convergence and reflection in the center, is about 5x less than the external pressure applied on the U. Its temperature reaches about 73 eV, while the temperature in the U shock wave is about 100 eV (in the model considered). After a time of approximately t = 0.72 µs, the spherically converging shock wave in the LiDT is reflected in the center, greatly increasing the density and temperature of this region at this moment. This reflected shock wave then collides with the external mass of LiDT still in motion of implosion and is reflected again, now on the internal surface of U. At t = 0.92 µs, a new convergence and reflection of the shock wave occurs in the center, this time generating densities and temperatures much higher than those generated in the previous reflection. At this moment, not only the global mass of LiDT, but also that of U are in extremely high densities, with the system tending towards stagnation and maximum compression (maximum internal energy at the expense of kinetic energy). The U kinetic and internal energy curves start to separate one from the other (i.e., from the 1/2 fraction of the total energy when the U shock wave reaches its internal surface. From this moment on, the U velocity increases due to decompression, and the kinetic energy grows continuously, at the expense of the decrease of internal energy. Subsequently, with the system stagnation and maximum compression, the distributions are reversed and the internal energy far exceeds the kinetic energy in the U. In the LiDT, the curves of kinetic and internal energies start to deviate from 1/2 fractions with the convergence of the shock wave to the center, the internal energy reaching its maximum value with the subsequent compression (mainly isentropic) of the material. The total energy of the system, at the time of maximum central compression of the LiDT, adds up to 0.16 kt.

• There is a greater burning of D in cases where LiDT is surrounded by a covering of heavy material (tamper), such as U. This can basically be attributed to the higher ⍴R values reached in the implosion of the LiDT in these cases and to the fuel containment effect provided by the U tamper. As already pointed out, the higher the ⍴R value, the lower the radiation leakage and the greater the local energy deposition by the fusion neutrons (arbitrarily fixed by parameter fn), improving the conditions for thermonuclear reaction propagation throughout the fuel. Due to the high opacity of U, it also serves to confine radiation inside the LiDT, preventing energy leakage to the outside environment.

10.6 Progression of thermonuclear reaction through the main LiD fuel

• The main fuel of thermonuclear explosives must be composed mostly of LiD in order to avoid excessive inventory of T, which decays with a half-life of 12 yrs and whose production is difficult and costly economically. The disadvantage, in relation to LiDT fuel, is the much less efficient propagation of thermonuclear fusion reaction.

10.7 Compact thermonuclear explosives with pure LiD fuel. Warhead W-87

• There are two essential aspects of the thermonuclear explosive in the W-87: the use of thermal X-rays emitted from the primary fission bomb to promote the implosion of the secondary module, through the process called radiation implosion, and the use of a fissile mass inside the thermonuclear fuel. It also uses spherical symmetry in the secondary module.

• The purpose of a fissile mass inside the thermonuclear fuel (also called a "spark plug") is to provide the additional energy needed to ignite a pure LiD fuel (without initial T) by means of a nuclear fission explosion when the fissile mass becomes highly supercritical with the implosion of the system. In addition, the existence of a fissile mass inside the thermonuclear fuel allows the total energy released by the explosive to be greatly increased, due to the symbiotic process of fusion and fission that is established between the two components - for example, the huge amount of neutrons emitted from the fusion causes the fission of the remaining fissile mass not burned in the internal fission explosion preceding the fusion; on the other hand, the fission neutrons play an important role in the initial transmutation of Li (one of the Li isotopes) into T, enhancing the thermonuclear ignition process.

• Although there is initially no T in the fuel, in all the cases treated here the reaction D-T is dominant, contributing about 82-84% to the total fusion reactions. Hence the importance of the T production by the neutron-Li reaction to enhance the thermonuclear ignition and explosion. The contribution of only about 16 - 18% of the D-D reactions is due to their much lower fusion cross sections in the range of the temperatures here obtained, despite the much higher number density of DT. (The contribution of the D-He3 reaction is less than 0.08%). One advantage of having a predominance of the D-T reaction is that this reaction produces 17.6 MeV of energy, whereas the D-D reactions produce only, on average, 3.65 MeV.

• For a "clean thermonuclear explosive" - for peaceful purposes, with minimum radioactivity production — we can replace the external U tamper by an inert material and use a minimum Pu internal mass as spark plug.

• The propagation speed of thermal X-rays in polystyrene is of the order of 11,000 km/s. Observing the diametral dimension of the secondary module, the difference between the arrival times of the X-rays to the anterior (opposite to the primary module) and posterior (opposite side) regions of the U tamper around the LiD is of the order of 10-8 s. Although extremely small, differences of this order of magnitude in the time of arrival of the thermal X-rays at the U surface may be sufficient to cause an asymmetric ablation of that surface and therefore an asymmetric implosion of the system, impairing the performance of the explosive. A possible solution to this issue is shown in Figure 10-41. Glued to the external U tamper, a polystyrene region is defined with greater opacity than that of the polystyrene in the rest of the cavity. Its geometric shape, as you can see in the figure, has the objective of producing the "lens" effect necessary for the perfect synchrony in the arrival of X-rays to the external surface of U. As for the opacity of polystyrene, it can be controlled by varying its density or composition; in the latter case, the material is generally doped with small amounts of heavy material.

10.8 Final considerations

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APPENDIX A: Simple derivation of the neutron transport equation

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APPENDIX B: Algorithm for numerical solution of the energy equation by Gauss elimination method

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APPENDIX C: Details of opacity calculation

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APPENDIX D: General equations of variation of the atomic state

• The processes normally considered for ionization and electron recombination in ions are 1) photoionization (when radiation is significant and plasma is optically non-transparent), 2) collision ionization (with electrons),  and 3) three-body recombination and radiative recombination. In some cases, 4) dielectronic recombination is also considered, which consists of capturing a free electron with the excess energy being used to excite a bound electron of the ion.

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APPENDIX E: Radiation transport: Eddington variable method

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APPENDIX F: Rankine-Hugoniot relations. Shock waves in ideal gases.

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APPENDIX G: Equation of state of a degenerate free electron gas.

Degeneracy and stellar evolution.

• White dwarfs are formed when, during the star collapse process, the force of gravitational attraction is balanced by the repulsion force arising from the degeneracy of electrons, which becomes stronger and stronger when stellar matter density increase with the contraction. White dwarfs are therefore "frozen" stars in a similar situation. Over time, they slowly cool (emitting radiation) and their brightness becomes increasingly weak and changing, leaving the trail of obscure or totally invisible bodies (the so-called "black dwarfs").

• Neutron stars are formed when, at extreme densities, the Fermi energy of the (almost) fully degenerate electron gas (which can reach the order of several MeV) is sufficient to make energetically possible the interaction of electrons with protons with the consequent production of neutrons and neutrinos (this process is called a β-inverse reaction). The β-reaction (that is, the spontaneous neutron decay in electrons, protons and antineutrinos) would be inhibited by the fact that the electrons produced in this decay would have energy below the Fermi energy of the electron gas of the star, so there would be no quantum space available for them. It is believed that this imbalance between the two opposite reactions is the factor responsible for the neutronization of the star (see more details in the book of Kourganoff). Hydrostatic stability is now achieved by balancing the force due to degenerate neutron gas (which are also fermion particles, with the complication of interference of the strong interaction between them and the force of gravitational attraction. The density of the neutron star is of the order of 1014 g/cm3 (comparable to that of the atomic nucleus), its mass is comparable to that of the Sun and its external radius extends for no more than about 10 km. (It is interesting to note that, for both white dwarfs and neutron stars, the larger their masses, the smaller their radius will be.) Like white dwarfs, neutron stars are unstable for masses greater than a certain limit mass, most recently estimated at about 2.5x the solar mass.

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APPENDIX H: Criticality benchmarks

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PHYSICAL CONSTANTS AND CONVERSION FACTORS

• Boltzmann constant, k, = 1.380 x 10-16 erg/K.

• Planck constant, h, = 6.625 x 10-27 erg-s.

• Stefan-Boltzmann constant, σ, = 5.67 x 10-5 erg-K-4-cm-2-s-1

• Light speed, c, = 2.998 x 1010 cm/s.

• Gas constant, R, = 8.317 x 107 erg/ (K-mol)

• Avogadro number, N0, = 6.023 x 1023/mol

• Electron number, me, = 9.108 x 10-28 g

• Proton mass, mp, = 1.673 x 10-24 g

• Atomic mass unit (AMU), m0, = 1.660 x 10-24 g

• Electron charge, e, = 4.803 x 10-10 esu

• 1 eV = 1.602 x 10-12 erg = 1.160 x 104 K

• 1 keV = 103 ev = 1.160 x 107 K

• 1 kt = 103 tons TNT = 4.18 x 1019 ergs

• 1 erg = 10-7 joules

• 1 Mbar = 1012  dynes/cm2 = 106 atm = 105 Mpa

• Typical pressure of chemical HE: .2-.4 Mbar

• Typical pressure at Earth center: ~4 Mbar

• Typical pressure at Sun Center: ~105 Mbar

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Terminology

• Adamantine: Unbreakable. 

• Binding Energy: The energy that holds the protons and neutrons together in the nucleus of an atom. Measured in millions of electronvolt (MeV).

• Critical Mass: The minimum mass in which the development of a self-sustained fission chain reaction is possible.

• Fertile Isotopes: Isotopes that give rise to the formation of fissile isotopes by radiative capture of neutrons.

• Fission: The division of an excited atomic nucleus into two other nuclei of smaller masses, which are distributed around half the mass of the original nucleus.

• Nuclear Force (‘Strong Interaction’): The fundamental force that holds nucleons in an atom, thereby overcoming the repulsive coulombian force between protons; restricted to nucleus dimensions, 10-13 cm.

• US Nuclear Weapons

• W53: A US Thermonuclear weapon with a power of 9 Mt and weight of ~4 tons.

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