|
|
|
4.1.1 Dimensional and Temporal Scale Factors
NOTE: MILNET has redacted two tables which are of some
value to actual weapons designers. The remaining material is only
somewhat useful to actual weapons designers. Key to this is the absence
of critical masses for various types of bomb making materials.
Knowledge of early experiments and critical mass experimentation, we
believe gives the fledgling weapons designer a "true" leg up in their
design and MILNET does not wish to provide that leg up. As a
result, we do not include those tables or discussion paragraphs which
deal with actual or experimentally/computationally derived critical
masses. Having said that, the well known published figure for a
typical critical mass of a certain material is around 10kg, not all
that very large (10 kg of a very massive material like uranium is not
all that large), making a fission primary for a thermonuclear weapon
quite small, thus able to fit in a trunk or an artillery shell (gun
design core assembly). The ZIP file has been disable until this
new version of section 4.1 is replaced in the zip file.
In Section 2 the properties of fission chain reactions were described using two simplified mathematical models: the discrete step chain reaction, and 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 two steps:
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 nu). By combining these we get the average number of neutrons produced per collision (also called the number of secondaries), designated by c:
Eq. 4.1.1-1
c = (cross_scatter + cross_fission*avg_n_per_fission)/cross_total
the total cross section, cross_total, is equal to:
Eq. 4.1.1-2
cross_total = cross_scatter + cross_fission + cross_absorb
The total neutron mean free path, the average distance a neutron will travel before undergoing a collision, is given by:
Eq. 4.1.1-3where N is the number of atoms per unit volume, determined by the density.
MFP = 1/(cross_total * N)
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.
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:
Eq. 4.1.1-4That 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.
mcrit_c = mcrit_0/(rho/rho_0)^2 = mcrit_0/C^2
An approximate relationship for this is:
Eq. 4.1.1-5where C_c is the compression of the core, and C_r is the compression of the reflector. Note that when C_c = C_r, then this is identical to Eq. 4.1.1-4. For most implosion weapon designs (since C_c > C_r) we can use the approximate relationship:
mcrit_c = mcrit_0/(C_c^1.2 * C_r^0.8)
Eq. 4.1.1-6
mcrit_c = mcrit_0/C_c^1.7
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". The neutron population at time t is given by:
Eq. 4.1.1-7
N_t = N_0*e^(alpha*t)
Alpha 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/alpha. This interval is known as the "time constant" (or "e-folding time") of the system, t_c. The more familiar concept of "doubling time" is related to alpha and the time constant simply by:
Eq. 4.1.1-8
doubling_time = (ln 2)/alpha = (ln 2)*t_c
Alpha is often more convenient than t_c 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, t_c continually changes during reactivity insertion and disassembly. Therefore I will henceforth refer to the quantity 1/alpha as the "multiplication interval".
Alpha 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 alpha can be calculated by:
Eq. 4.1.1-9
alpha = (1/tau)*(c - 1) = (v_n/total_MFP)*(c - 1)
where tau is the average neutron lifetime between collisions; and v_n is the average neutron velocity (which is 2.0x10^9 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 provides 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 alphas 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 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 alpha should be determined by "integral experiments", that is, measured directly in the fissile material where all of these effects will occur naturally. Calculating tau and alpha from differential cross section measurements, adjusted neutron spectrums, etc. is fraught with potential error.
All nations interested in nuclear weapons technology have performed integral experiments to measure alpha, but published data is sparse and in general is limited to the immediate region of criticality. Collecting data for systems at high densities requires extremely difficult high explosive experiments, and data for high alpha systems can only be done in actual nuclear weapon tests.
Some integral alpha data is available for systems near prompt critical. The most convenient measurements are of the negative alpha value for fast neutron chain reactions at delayed criticality. Since at prompt critical alpha is exactly zero, the ratio of the magnitude of this delayed critical measurement to the fraction of fission neutrons that are delayed allows the alpha value to be calculated. These were the only sort of alpha measurements available to the Manhattan Project for the design of the first atomic bombs.
<>The most informative values are from the Godiva and Jezebel unreflected reactor experiments. These two systems used bare metal weapon grade cores, so the properties of weapons material was being measured directly.Eq. 4.1.2-1
alpha_eff = alpha_max*[1 - (r_c/R)^2]
= alpha_max*[1 - (M_crit/m)^(2/3)]
4.1.3 Distribution of Neutron Flux and Energy in the Core
Since neutron leakage occurs at the surface of a critical or supercritical core, the strength of the neutron flux is not constant throughout the core. Since the rate of energy release at any point in the core is proportional to the flux at that point, this also affects the energy density throughout the core. This is a matter of some significance, since it influences weapon efficiency and the course of events in terminating the divergent fission chain reaction.
4.1.3.1 Flux Distribution in the Core
For a bare (unreflected) critical spherical system, the flux distribution is given by:
Eq. 4.1.3.1-1
flux(r) = max_flux * Sin(Pi*r/(r + 0.71*MFP))/(Pi*r/(r + 0.71*MFP))
(using the diffusion approximation) where Sin takes radians as an argument.
If we measure r in MFPs, then by referring to Table 4.1.1-1 we can relate the flux distribution to the parameter c. Computing the ratio between the flux at the surface of the critical system, and the maximum flux (in the center) we find:
Table 4.1.3-1 Relative Flux at Surface
c value flux(r_c)
1.0 0.0 (at the limit)
1.02 0.0587
1.05 0.0963
1.10 0.1419
1.20 0.2117
1.40 0.3182
1.60 0.4018
This shows that as c increases, the flux distribution becomes flatter with less drop in the flux near the surface.
The flux distribution function above applies only to bare critical systems. If the system is supercritical, then the flux distribution becomes flatter, since neutron production over-balances loss. The greater the value of alpha for the system, the flatter it becomes. The addition of a neutron reflector also flattens the distribution, even for the same degree of reactivity. The flux distribution function is useful though, since the maximum rate of fission occurs at the moment when the core passes through second criticality (on the way to disassembling, see below).
4.1.3.2 Energy Distribution in the Core
As long as the geometry doesn't change, the relative flux distribution remains the same throughout the fission process. The fission reaction rate at any point in the core is proportional to the flux. The net burnup of fissile material (and total energy release) is determined by the reaction rate integrated over time.
This indicates that the degree of burnup (the efficiency of utilization) varies throughout the core. The outer layers of material will be fissioned less efficiently than the material near the center. The steeper the drop off in flux the greater this effect will be. We can thus expect less efficient utilization of fissile material in small cores, and in materials with low values of c. From the relatively low value of c for U-235 compared to U-233 and Pu-239, we can expect that U-235 will be used less efficiently. This is observed in pure fission tests, the difference being about 15% in nominal yield (20 kt) pure fission designs.
The energy density (energy content per unit volume) in any region of the core is determined not only by the total energy produced in that region, but also by the flow of heat in to and out from the region.
The energy present in the core rises by a factor of e (2.71...) every multiplication interval (neglecting any losses from the surface). Nearly all of the energy present has thus been produced in the last one or two multiplication intervals, which in a high alpha system is a very short period of time (10 nanoseconds or less). There is not much time for heat flow to significantly alter this energy distribution.
Close to the end point of the fission process, the energy density in the core is so high that significant flow can occur. Since most of the energy is present as a photon gas the dominant mechanism is radiation (photon) heat transport, although electron kinetic heat transport may be significant as well. This heat flow can be modelled by the diffusion approximation just like neutron transport, but in this case estimating the photon mean free path (the opacity of the material) is quite difficult. A rough magnitude estimate for the photon MFP is a few millimeters.
The major of effect of energy flow is the loss of energy from a layer about 1 photon mean free path thick (referred to as one optical thickness) at the surface of the core. In a bare core this cooling can be quite dramatic, but the presence of a high-Z tamper (which absorbs and re-emits energy) greatly reduces this cooling. Losses also occur deeper in the core, but below a few photon MFPs it becomes negligible. Otherwise, there is a significant shift in energy out of the center of the core that tends to flatten the energy distribution.
The energy density determines the temperature and pressure in the core, so there is also a variation in these parameters. Since the temperature in radiation dominated matter varies with the fourth power of the energy density, the temperature distribution is rather flat (except near the surface perhaps). The pressure is proportional to the energy density, so it varies in similar degree.
4.1.4 History of a Fission Explosion
To clarify the issues governing fission weapon design it is very helpful to understand the sequence of events that occurs in every fission explosion. The final event in the process - disassembly - is especially important since it terminates the fission energy release and thus determines the efficiency of the bomb.
4.1.4.1 Sequence of Events
Several distinct physical states can be identified during the detonation of a fission bomb. In each of these states a different set of physical processes dominates.
4.1.4.1.1 Initial State
Before the process that leads to a fission explosion is initiated, the fissile material is in a subcritical configuration. Reactivity insertion begins by increasing the average density of the configuration in some way.
4.1.4.1.2 Delayed Criticality
When the density has increased just to the point that a neutron population in the mass is self-sustaining, the state of delayed criticality has been achieved. Although nearly all neutrons produced by fission are emitted as soon as the atom splits (within 10^-14 sec or so), a very small proportion of neutrons (0.65% for U-235, 0.25% for Pu-239) are emitted by fission fragments with delays of up to a few minutes. In delayed criticality these neutrons are required to maintain the chain reaction. These long delays mean that power level changes can only occur slowly. All nuclear reactors operate in a state of delayed criticality. Due to the slowness of neutron multiplication in this state it is of no significance in nuclear explosions, although it is important for weapon safety considerations.
4.1.4.1.3 Prompt Criticality
When reactivity increases to the point that prompt neutrons alone are sufficient to maintain the chain reaction then the state of prompt criticality has been reached. Rapid multiplication can occur after this point. In bomb design the term "criticality" usually is intended to mean "prompt criticality". For our purposes we can take the value of alpha as being zero at this point. The reactivity change required to move from delayed to prompt criticality is quite small (for plutonium the prompt and delayed critical mass difference is only 0.80%, for U-235 it is 2.4%), so in practice the distinction is unimportant. Passage through prompt criticality into the supercritical state is also termed "first criticality".
4.1.4.1.4 Supercritical Reactivity Insertion
The insertion time of a supercritical system is measured from the point of prompt criticality, when the divergent chain reaction begins. During this phase the reactivity climbs, along with the value of alpha, as the density of the core continues to increase. Any insertion system will have some maximum degree of reactivity which marks the end of the insertion phase. This phase may be terminated by reaching a plateau value, by passing the point of maximum reactivity and beginning to spontaneously deinsert, or by undergoing explosive disassembly.
4.1.4.1.5 Exponential Multiplication
This phase may overlap supercritical insertion to any degree. Any neutrons introduced into the core after prompt criticality will initiate a rapid divergent chain reaction that increases in power exponentially with time, the rate being determined by alpha. If exponential multiplication begins before maximum reactivity, and insertion is sufficiently fast, there may be significant increases in alpha during the course of the chain reaction. Throughout the exponential multiplication phase the cumulative energy released remains too small to disrupt the supercritical geometry on the time scale of the reaction. Exponential multiplication is always terminated by explosive disassembly. The elapsed time from neutron injection in the supercritical state to the beginning of explosive disassembly is called the "incubation time".
4.1.4.1.6 Explosive Disassembly
The bomb core is disassembled by a combination of internal expansion that accelerates all portions of the core outward, and the "blow-off" or escape of material from the surface, which generates a rarefaction wave propagating inward from the surface. The drop in density throughout the core, and the more rapid loss of material at the surface, cause the neutron leakage in the core to increase and the effective value of alpha to decline.
The speed of both the internal expansion and surface escape processes is proportional to the local speed of sound in the core. Thus disassembly occurs when the time it takes sound to traverse a significant fraction of the core radius becomes comparable to the time constant of the chain reaction. Since the speed of sound is determined by the energy density in the core, there is a direct relationship between the value of alpha at the time of disassembly and the amount of energy released. The faster is the chain reaction, the more efficient is the explosion.
As long as the value of alpha is positive (the core is supercritical) the fission rate continues to increase. Thus the peak power (energy production rate) occurs at the point where the core drops back to criticality (this point is called "second criticality"). Although this terminates the divergent chain reaction, and exponential increase in energy output, this does not mean that significant power output has ended. A convergent chain reaction continues the release of energy at a significant, though rapidly declining, rate for a short time afterward. 30% or more of the total energy release typically occurs after the core has become sub-critical.
4.1.4.2 The Disassembly Process
The internal expansion of the core is caused by the existence of an internal pressure gradient. The escape of material from the surface is caused by an abrupt drop in pressure near the surface, allowing material to expand outward very rapidly. Both of these features are present in every fission bomb, but the degree to which each contributes to disassembly varies.
Consider a spherical core with internal pressure declining from the center towards the surface. At any radius r within the core the pressure gradient is dP/dR. Now consider a shell of material centered at r, that is sufficiently thin so that the slope of the pressure gradient does not change appreciably across it. The mass of the shell is determined by its area, density, and thickness:
m = thickness * area * densityThe outward force exerted on the shell is determined by the pressure difference across the shell and the shell area:
F = dP/dR * thickness * areaFrom Newton's second law of motion we know that acceleration is related to force and mass by:
a = F/m
so:
a = (dP/dR * thickness * area)/(thickness * area * density)
= (dP/dR)/density
If density is constant in the core, then the outward acceleration at any point is proportional to the pressure gradient; the steeper the gradient, the greater the acceleration. The kinetic energy acquired comes at the expense of the internal energy of the expanding material.
The limiting case of a steep pressure gradient is a sudden drop to zero. In this case the acceleration is infinite, the internal energy of the material is completely converted to kinetic energy instantaneously and it expands outwards at constant velocity (escape velocity). The edge of the pressure drop propagates back into the material as a rarefaction wave at the local speed of sound. The pressure at the leading edge of the expanding material (moving in the opposite direction at escape velocity) is zero. The pressure discontinuity thus immediately changes into a continuous pressure change of steadily diminishing slope. See Section 3.6.1.1 Release Waves for more discussion of this process.
In a bare core, thermal radiation from the surface causes a large energy loss in a surface layer about one optical thickness deep. Since energy lost from the core by thermal radiation cannot contribute to expansion, this has the effect of delaying disassembly. It does create a very steep pressure gradient in the layer however, and a correspondingly high outward acceleration. Deeper in the core, the pressure gradient is much flatter and the acceleration is lower. After the surface layer has expanded outward by a few times its original thickness, it has acquired considerable velocity, and the surface pressure drop rarefaction has propagated a significant distance back into the core. At this point the pressure and density profile of the core closely resembles the early stages of expansion from an instantaneous pressure drop, the development of the profile having been delayed slightly by the time it took the surface to accelerate to near escape velocity.
A bomb core will typically be surrounded by a high-Z tamper. A layer of tamper (about one optical thickness deep) absorbs the thermal radiation emitted by the core and is heated by it. As its temperature increases, this layer begins to radiate energy back to the core, reducing the core's energy loss. In addition, the heating also generates considerable pressure in the tamper layer. The combined effect of reduced core surface cooling, and this external pressure is to create a much more gradual pressure drop in the outer layer of the core and a correspondingly reduced acceleration.
The expanding core and heated tamper layer creates a shock wave in the rest of the tamper. This has important consequences for the disassembly process. The rarefaction wave velocity is not affected by the presence of the tamper, but the rate at which the density drops after arrival of the rarefaction wave is strongly affected. The rate of density drop is determined by the limiting outward expansion velocity, this is in turn determined by the shock velocity in the tamper. The denser the tamper the slower the shock, and the slower the density decrease behind the rarefaction wave. In any case the shock velocity in the tamper is much slower than the escape velocity of expansion into a vacuum. The disassembly of a tamped core thus more closely resembles one dominated by internal expansion rather than surface escape.
4.1.4.3 Post Disassembly Expansion
The expanding core creates a radiation dominated shock wave in the tamper that compresses it by at least a factor of 7, and perhaps as high as 16 due to ionization effects. This pileup of high density material at the shock front is called the "snow plow" effect. By the time this shock has moved a few centimeters into the tamper, the rarefaction wave will have reached the center of the core and the entire core will be expanding outward uniformly.
The basic structure of the early fireball has now developed, consisting of a thin highly compressed shell just behind the shock front containing nearly all of the mass that has been shocked and heated so far. This shell travels outward at nearly the same velocity as the shock front. The volume inside this shell is a region of very low density. Temperature and pressure behind the shock front is essentially uniform though since nearly all of the energy present is contained in the radiation field (i.e. it exists as a photon gas). Since the shock wave is radiation dominated, the front does not contain an abrupt pressure jump. Instead there is a transition zone with a thickness about equal to the radiation mean free path in the high-Z tamper material (typically a few millimeters). In this zone the temperature and pressure climb steadily to their final value.
This overall explosion structure remains the same as the shock expands outward until it reaches a layer of low-Z material (a beryllium reflector, or the high explosive).
The transition zone marking the shock front remains thin as long as the shock is travelling through opaque high-Z material. Low-Z material becomes completely ionized as it is heated, and once it is completely ionized it is nearly transparent to radiation and is no longer efficiently heated. When the shock front emerges at the boundary of the high-Z tamper and the low-Z material, it spits into two regions. A radiation driven shock front moves quickly away from the high-Z surface, bleaching the low-Z material to transparency. This faster shock front only creates a partial transition to the final temperature and pressure. The transition is completed by a second shock, this one a classical mechanical shock, driven by the opaque material.
4.1.5 Fission Weapon Efficiency
Fundamental to analyzing the design of fission bombs is understanding the factors that influence the efficiency of the explosion - the percentage of fissile material actually fissioned. The efficiency and the amount of fissile material present determine the amount of energy released by the explosion - the bomb's yield.
I have organized my discussion of design principles around the issue of efficiency since it is the most important design characteristic of any fission device. Any weapon designer must have a firm grasp on the expected efficiency in order to make successful yield predictions, and a firm grasp on the factors affecting efficiency is required to make design tradeoffs.
In the discussion below (and in later subsections as well) I assume that the system under discussion is spherically symmetric, and of homogenous density, unless otherwise stated. Spherical symmetry is the simplest geometry to analyze, and also happens to be the preferred geometry for efficient nuclear weapons.
4.1.5.1 Efficiency Equations
It is intrinsically difficult to accurately predict the performance of a particular design from fundamental physical principles alone. To make good predictions on this basis requires sophisticated computer simulations that include hydrodynamic, radiation, and neutronic effects. Even here it is very valuable to have actual test data to use for calibrating these simulation models.
Nuclear weapon programs have historically relied heavily on extrapolating tested baseline designs using scaling laws like the efficiency equations I discuss below, especially in the early years of development. These equations are derived from idealized models of bomb core behavior and consequently have serious limitations in making absolute efficiency estimates. The predictions of the Theoretical Section at Los Alamos underestimated the yield of the first atomic bomb by a factor of three; an attempts a few years later to recompute the bomb efficiency using the best models, physical data, and computers available at the time led to a yield overestimate by a factor of two.
From the description of core disassembly given above we can see that two possible idealizations are possible for deriving convenient efficiency equations:
The basic approach is to model how quickly the core expands to the point of second criticality. To within a constant scaling factor, this fixes the efficiency of the explosion.
In the first modelling approach, the state of second criticality is based on the average density of the entire core. In the second approach, second criticality is based on the surface loss of excess critical masses from a residual core which remains at constant initial density.
The first efficiency equation to be developed was the Bethe-Feynmann equation, prepared by Hans Bethe and Richard Feynmann at Berkeley in 1942 based on the uniform expansion model. A somewhat different efficiency equation was presented by Robert Serber in early 1943 at Los Alamos, which was also based on uniform expansion but also explicitly included the exponential growth in energy release (which the Bethe-Feynmann equation did not). A problem with these derivations is that to keep the resultant formulas relatively simple, they assume that the expanding core remains at essentially constant density during deinsertion, which is only true (even approximately) when the degree of supercriticality is small.
For the purposes of this FAQ I have taken the second approach for deriving an efficiency equation, using the surface escape model. This model has the advantage that the residual core remains at constant density regardless of the degree of supercriticality. Comparing it to the other efficiency equations provides some insight into the sensitivity of the assumptions in the various models.
4.1.5.1.1 The Serber Efficiency Equation Revisited
Let us first consider the factors that affect the efficiency of a homogenous untamped supercritical mass. In this system, disassembly begins as fissile material expands off the core's surface into a vacuum. We make the following simplifying assumptions:
If r is the initial outer radius, and r_c is the critical radius, then the reaction halts when:
Eq. 4.1.5.1.1-1
Integral[c_s(t) dt] = r - r_c
where c_s(t) is the speed of sound at time t.
If kinetic pressure is negligible compared to radiation pressure (this is true in all but extremely low yield explosions), then:
Eq. 4.1.5.1.1-2
c_s(t) = [(E(t)*gamma)/(3*V*rho)]^0.5
where E(t) is the cumulative energy produced by the reaction, V is the volume of the core, and rho is its density.
We also have:
Eq. 4.1.5.1.1-3
E(t) = (E1/(c - 1)) * e^(alpha*t)
where E1 is a constant that gives the energy yield per fission (E1 = 2.88 x 10^-4 erg/fission). Thus:
Eq. 4.1.5.1.1-4
Eff(t) = E(t)/E_total = (E1/((c - 1)*E_total)) * e^(alpha*t)
where Eff(t) is the efficiency at time t, and E_total is the energy yield at 100% efficiency.
Thus:
Eq. 4.1.5.1.1-5
r - r_c = Integral[(E(t)*gamma/(3*V*rho))^0.5 dt]
= (gamma*E1/(3*M*(c-1)))^0.5 * Integral[e^(alpha*t/2)dt]
= (gamma*E1/(3*M*(c-1)))^0.5 * 2/alpha * e^(alpha*t/2)
where M is the fissile mass.
Rearranging and squaring we get:
Eq. 4.1.5.1.1-6
e^(alpha*t) = (r - r_c)^2 * ((3M*(c-1))/(gamma*E1)) * (alpha^2)/4
Substituting into the efficiency equation:
Eq. 4.1.5.1.1-7
Eff(t) = [3*alpha^2 * M * (r - r_c)^2]/(4*gamma*E_total)
If E2 is a constant equal to fission energy/gram in ergs (7.25 x 10^17 erg/g for Pu-239), and gamma is equal to 4/3 for a photon gas, then:
Eq. 4.1.5.1.1-8
Eff(t) = [9*alpha^2 * (r - r_c)^2]/(16*E2)
We can observe at this point that efficiency is determined by the actual value of alpha and the difference between the actual radius of the assembly, and the radius of the mass just sufficient to keep the chain reaction going. Note that it is the values of these parameters WHEN DISASSEMBLY ACTUALLY OCCURS that are relevant.
Now using r = r_c(1 + delta) so that (r - r_c) = delta*r_c, we get:
Eq. 4.1.5.1.1-9
Eff(t) = [9*alpha^2 * delta^2 * r_c^2]/(16*E2)
If we let tau = (total_MFP/v_n) then:
Eq. 4.1.5.1.1-10
alpha_max = (v_n/total_MFP)*(c - 1) = (c - 1)/tau
and
Eq. 4.1.5.1.1-11
alpha_eff = ((c - 1)/tau)*[1 - (1/(1 + delta)^2)]
Now:
Eq. 4.1.5.1.1-12
Eff(t) = ((c-1)/tau)^2 * 9/(16*E2) * r_c^2 * delta^2 *[1-(1/(1+ delta)^2)]^2
= ((c-1)/tau)^2 * 9/(16*E2) * r_c^2 *[delta - (delta/(1+ delta)^2)]^2
In the range of 0 < delta < 1 (up to 8 critical masses), the expression
[delta - (delta/(1+ delta)^2)]^2
is very close to 0.6*delta^3, giving us:
Eq. 4.1.5.1.1-13
Eff(t) = 0.338*((c-1)/tau)^2 * r_c^2/E2 * delta^3
= 0.338/E2 * alpha_max^2 * r_c^2 * delta^3
This last equation is identical with the equation derived by Robert Serber in the spring of 1943 and published in The Los Alamos Primer, except that his constant is 0.667 (i.e. gives efficiencies 1.98 times higher). Serber derived his efficiency equation from rough dynamical considerations without using a hydrodynamic model of disassembly and admits that his result is 2-4 time higher than the true value. This is consistent with the above derivation.
Both the equation given above and Serber's equation differ significantly from the Bethe-Feynmann equation however, which gives an efficiency relationship of:
Eq. 4.1.5.1.1-14
Eff = (1/(gamma - 1)E2) * alpha_max^2 * r_c^2 *
(delta*(1 + 3*delta/2)^2)/(1 + delta)
after reformulating to equivalent terms. This is a much more linear relationship between delta and efficiency, than the cubic relationship of Serber. Due to the crudeness of all of these derivations, the significance of this difference cannot be assessed at present.
Equation 4.1.5.1.1-13 shows that efficiency is proportional to the square of the maximum multiplication rate of the material, and the critical radius (also due to material properties), and is the cube of the excess critical radius excess delta.
Extending to larger values, we can approximate it in the range 1 < delta < 3 (up to 64 critical masses), with the expression:
Eq. 4.1.5.1.1-15
Eff(t) = 0.338/E2 * alpha_max^2 * r_c^2 * delta^(7/3)
4.1.5.1.2 The Density Dependent Efficiency Equation
The efficiency equations given above leave something to be desired for evaluating fission weapon designs. I have included it to assist in making comparisons with the available literature, but I will give it a different form below.
The choice of fissile materials available to a weapon designer is quite limited, and the nuclear and physical properties of these materials are fixed. It is desirable then to separate these factors from the factors that a designer can influence - namely, the mass of material present, and the density achieved. The density is of particular interest since it is the only factor that changes in a given design during insertion. Understanding how efficiency changes with density is essential to understanding the problem of predetonation for example.
Returning to equation Eq. 4.1.5.1.1-8:
Eff(t) = [9*alpha^2 * (r - r_c)^2]/(16*E2)
we want to reformulate it so that it consists of two parts, one that does not depend on density, and one that depends only on density.
Let the composition and mass of the system be fixed. We will normalize the radius and density so that they are expressed relative to the system's critical state. If rho_crit and r_crit are the values for density and radius of the critical state, and rho_rel and r_rel are the values of the system that we want to evaluate:
Eq. 4.1.5.1.2-1
rho_rel = rho_actual/rho_crit
and
Eq. 4.1.5.1.2-2
r_rel = r_actual/r_crit
When the system is exactly critical, rho_rel = 1 and r_rel = 1. Of course we are interested in states where rho_rel > 1, and r_rel < 1. We can relate r_rel to rho_rel:
Eq. 4.1.5.1.2-3
r_rel = (1/rho_rel)^(1/3) * r_crit
Using this notation, and letting alpha_max_c be the value of alpha_max at the critical state density, we can write:
alpha = alpha_max_c * rho_rel * (1 - (r_c/r_rel)^2)
In this case r_c refers to the effective critical radius at density rho_rel not rho_crit; that is, r_c IS NOT r_crit. Instead it is equal to r_crit/rho_rel. Using this, and the relation for r_rel above, we can eliminate r_crit:
Eq. 4.1.5.1.2-4
alpha = alpha_max_c * rho_rel * (1 - ((1/rho_rel)/(1/rho_rel)^(1/3))^2)
= alpha_max_c * rho_rel * (1 - (rho_rel)^(-4/3))
Substituting into the efficiency equation:
Eq. 4.1.5.1.2-5
Eff = (9/16*E2) * alpha^2 * (r_rel - r_c)^2
we get:
Eq. 4.1.5.1.2-6
Eff = (9/(16*E2))*(alpha_max_c*rho_rel*(1 - (rho_rel)^(-4/3)))^2 *
(r_rel - r_c)^2
Splitting constant and density dependent factors between two lines:
Eq. 4.1.5.1.2-7
Eff = (9/(16*E2)) * alpha_max_c^2 *
rho_rel^2 * (1-(rho_rel)^(-4/3))^2 * (r_rel - r_c)^2
We can eliminate r_rel and r_c, replacing them with expressions of rho_rel and r_crit:
Eq. 4.1.5.1.2-8
r_rel - r_c = (1/rho_rel)^(1/3) * r_crit) - (r_crit/rho_rel)
= ((1/rho_rel)^(1/3) - (1/rho_rel)) * r_crit
Substituting again:
Eq. 4.1.5.1.2-9
Eff = (9/(16*E2)) * alpha_max_c^2 * r_crit^2 *
rho_rel^2 * (1-(rho_rel)^(-4/3))^2 * ((1/rho_rel)^(1/3)-(1/rho_rel))^2
Recall that the rho_rel, the relative density, is not generally the compression ratio compared to normal density. This is true only if amount of fissile material in the system is exactly one critical mass at normal density (as was approximately true in the Fat Man bomb). For "sub-crit" systems, rho_rel is smaller than the actual compression of the material since compressive work is required to raise the initial sub-critical system to the critical state. For a system consisting of more than one critical mass (at normal density), rho_rel is higher than the actual compression.
By looking in turn at each of the density dependent terms we can gain insight into the significance of the efficiency equation. First note that alpha_max_c is a fundamental property of the fissile material and does not change, even though it is system dependent (being normalized to the critical density of the system).
The term (rho_rel^2) is introduced by the reduction of the MFP with increasing density and contributes to enhanced efficiency at all values of rho_rel.
The term (1-(rho_rel)^(-4/3)))^2 represents the effect of neutron leakage. At rho_rel=1 the value is 0. It has a limiting value of 1 when rho_rel is high, i.e. no leakage occurs. As this term approaches one, and leakage becomes insignificant, it ceases to be a significant contributor to further efficiency enhancement.
The term ((1/rho_rel)^(1/3)-(1/rho_rel))^2 describes the distance the rarefaction wave must travel to shut down the reaction. At rho_rel=1 it is 0. It initially increases rapidly, but soon slows down at reaches a maximum at about rho_rel = 5.196. Thereafter it declines slowly. This signifies that fact that once the critical radius of the system at rho_rel is small compared to the physical radius no further efficiency gain is obtained from this source. Instead further increases in density simply reduce the scale of the system, allowing faster disassembly.
We can provide some approximations for the efficiency equation to make the overall effect of density more apparent.
In the range of 1 < rho_rel < 2 it is approximately:
Eq. 4.1.5.1.2-10
Eff = (9/(16*E2)) * alpha_max_c^2 * r_crit^2 * ((rho_rel - 1)^3)/8
In the range of 2 < rho_rel < 4.5 it is approximately:
Eq. 4.1.5.1.2-11
Eff = ((9/(16*E2)) * alpha_max_c^2 * r_crit^2 * ((rho_rel - 1)^(2.333))/8
In the range of 4 < rho_rel < 8 it is approximately:
Eq. 4.1.5.1.2-12
Eff = (9/(16*E2)) * alpha_max_c^2 * r_crit^2 * ((rho_rel - 1)^(1.8))/5
4.1.5.1.3 The Mass and Density Dependent Efficiency Equation
The maximum degree of compression above normal density that is achievable is limited by technology. It is of interest then to consider how the amount of material present affects efficiency at a given level of compression, since it is the other major parameter that a designer can manipulate.
To examine this we would like to reintroduce an explicit term for mass. To do this we renormalize the equation to a fixed standard density rho_0 (the uncompressed density of the fissile material), and use rho_0 and the corresponding value of the critical mass M_c to replace the scale parameter r_crit. Thus:
Eqs. 4.1.5.1.3-1 through 4.1.5.1.3-5
alpha_max_crit = alpha_max_0 * (rho_crit/rho_0)
m_rel = m/M_c
rho_crit = rho_0/m_rel^(1/2)
rho_rel = rho/rho_crit = (rho/rho_0)*m_rel^(1/2)
r_crit = ((m/rho_crit)*(3/(2Pi)))^(1/3)
= (m*m_rel^(1/2)/rho_0)^(1/3) * (3/2Pi)^(1/3)
= (m^(3/2)/(M_c^(1/2) * rho_0))^(1/3) * (3/2Pi)^(1/3)
= m^(1/2) * (M_c^(1/2) * rho_0)^(-1/3) * (3/2Pi)^(1/3)
Assuming the density rho >= rho_crit, we get:
Eq. 4.1.5.1.3-6
Eff = (9/(16*E2))*(3/(2Pi))^(2/3) * alpha_max_0^2 *
(rho_crit/rho_0)^2 * (rho/rho_crit)^2 *
(m^(1/2) * (M_c^(1/2) * rho_0)^(-1/3))^2 *
(1-((rho_0/rho)^(4/3) * m_rel^(-2/3)))^2 *
(((rho_0/rho)^(1/3) * m_rel^(-1/6)) - ((rho_0/rho) * m_rel^(-1/2)))^2
Simplifying:
Eq. 4.1.5.1.3-7
Eff = (9/(16*E2))*(3/(2Pi))^(2/3) * alpha_max_0^2 *
(rho/rho_0)^2 * m/(M_c^(1/3) * rho_0^(2/3)) *
(1-((rho_0/rho)^(4/3) * m_rel^(-2/3)))^2 *
m_rel^(-1) * (((rho_0 * m_rel)/rho)^(1/3) - (rho_0/rho))^2
Then:
Eq. 4.1.5.1.3-8
Eff = (9/(16*E2))*(3/(2Pi))^(2/3) * alpha_max_0^2 * m/(M_c^(1/3)) * (M_c/m)
(rho^2)/(rho_0^(8/3)) * (1 - ((rho_0/rho)^(4/3) * m_rel^(-2/3)))^2 *
(((rho_0 * m_rel)/rho)^(1/3) - (rho_0/rho))^2
And finally:
Eq. 4.1.5.1.3-9
Eff = (9/(16*E2))*(3/(2Pi))^(2/3) * alpha_max_0^2 * M_c^(2/3) *
(rho/(rho_0^(4/3)))^2 * (1 - ((rho_0/rho)^(4/3) * m_rel^(-2/3)))^2 *
(((rho_0 * m_rel)/rho)^(1/3) - (rho_0/rho))^2
The first line of this equation consists entirely of constants, some of them fixed by the choice of material and reference density. From the next two lines it is clear that the density dependency is the same. The effect of increasing the mass of the system is to modestly reduce leakage and retard disassembly.
4.1.5.1.4 The Mass Dependent Efficiency Equation
It is useful to also have an equation that considers only the effect of mass. Including this as the only variable allows presenting a simplified form that makes the effect of varying the mass in a particular design easier to visualize. Also in gun-type designs no compression occurs, so the chief method of manipulating yield is by varying the mass of fissile material present.
Taking the mass and density dependent equation, we can set the density to a fixed nominal value, rho, and then simplify. Let rho = rho_0:
Eq. 4.1.5.1.4-1
Eff = (9/16*E2)*(3/2Pi)^(2/3) * alpha_max_0^2 * M_c^(2/3) *
(rho_0/(rho_0^(4/3))^2 *(1 - ((rho_0/rho_0)^(4/3) * m_rel^(-2/3)))^2 *
(((rho_0 * m_rel)/rho_0)^(1/3) - (rho_0/rho_0))^2
= (9/16*E2)*(3/2Pi)^(2/3) * alpha_max_0^2 * M_c^(2/3) *
rho_0^(-2/3) * (1 - m_rel^(-2/3))^2 * ((m_rel)^(1/3) - 1)^2
Since M_c/rho_0 is the volume of a critical assembly (m_rel = 1):
Eq. 4.1.5.1.4-2
Eff = (9/16*E2)*(3/2Pi)^(2/3) * alpha_max_0^2 * vol_crit^(2/3) *
(1 - m_rel^(-2/3))^2 * ((m_rel)^(1/3) - 1)^2
And finally:
Eq. 4.1.5.1.4-3
Eff = (9/16*E2)*(2^(2/3)) * alpha_max_0^2 * r_crit^2 *
(1 - m_rel^(-2/3))^2 * ((m_rel)^(1/3) - 1)^2
Again the top line consists of numeric and material constants, the second of mass dependent terms. This equation shows that efficiency is zero when m_rel = 1, as expected. Efficiency is negligible when m_rel < 1.05, similar to the power of conventional explosives. It climbs very quickly however, increasing by a factor of 400 or so between 1.05 and 1.5, where efficiency becomes significant. The Little Boy bomb had m_rel = 2.4. If its fissile content had been increased by a mere 16%, its yield would have increased by 75% (whether this could be done while maintaining a safe criticality margin is a different matter).
4.1.5.1.5 Limitations of the Efficiency Equations
These formulas provide good scaling laws, and a rough means to calculate efficiency. But we should return to the simplifying assumptions made earlier to understand their limitations.
It is obvious that alpha is not constant during disassembly. As material blows off, the size of the core and the value of alpha both decrease, which has a negative effect on efficiency. This is the most important factor not accounted for, and results in a lower effective coefficient in the efficiency equation.
The assumption about uniform temperature, and no energy loss is also not really true. The energy production rate in any region of the core is proportional to the neutron flux density. This density is highest in the center and lowest at the surface (although not dramatically so). Furthermore, the high radiation energy density in the core corresponds to a high radiation loss rate from the surface. Based on the Stefan-Boltzmann law it would seem that the loss rate from a bare core could eventually match the energy production rate. This doesn't really occur because of the high opacity of ionized high-Z material; thermal energy from inside the core cannot readily reach the surface. But by the same token, the surface can cool dramatically. Since core expansion starts at the surface, and the rate is determined by temperature, this surface cooling can significantly retard disassembly.
When scaling from known designs, most of these issues have little significance since the deviations from the theoretical model used for the derivations affects both system similarly.
The efficiency equations also breaks down at very small yields. To eliminate gamma from the equations I assumed that the core was radiation dominated at the time of disassembly. When yields drop to the low hundreds of tons and below, the value of gamma approximates that of a perfect gas which changes only the constant term in the equations, reducing efficiency by 20%. When yields drop to the ton range then the properties of condensed matter (like physical strength, heat of vaporization, etc.) become apparent. This tends to increase the energy release since these properties resist the expansion effects.
There is another factor that imposes an effective upper limit on efficiency regardless of other attempts to enhance yield. This is the decrease in fissile content of the core. The alert reader may have noticed that it is possible to calculate efficiencies that are greater than 1 using the equations. This is because energy release is represented as an exponentially increasing function of time without regard for the amount of energy actually present in the fissile material. At some point, the fact that the fission process depletes the fissile material present must have an effect on the progress of the chain reaction.
The limiting factor here is due to the dilution of the fissile material by the fission products. Most isotopes have roughly the same absorption cross section for fast neutrons, a few barns. The core initially consists of fissile material, but as the chain reaction proceeds each fission event replaces one fissile nucleus with two fission product nuclei. When 50% of the material has fissioned, for every 100 initial fissile atoms there are now 50 remaining, and 100 non-fissile atoms, i.e. the fissile content has declined to only 33%. This parasitic absorption will eventually extinguish the reaction entirely, regardless of what yield enhancement techniques are used (generally at an efficiency substantially below 50%).
4.1.5.2 Effect of Tampers and Reflectors on Efficiency
So far I have been explicitly assuming a bare fissile mass for efficiency estimation. Of course, most designs surround the core with layers of material intended to scatter escaping neutrons back into the fissile mass, or to retard the hydrodynamic expansion.
I use the term "reflector" to refer to the neutron scattering properties of the surrounding material, and "tamper" to refer to the effect on hydrodynamic expansion. The distinction is logical because the two effects are fundamentally unrelated, and because the term tamper was borrowed from explosive blasting technique where it refers only to the containment of the blast. This distinction is not usually made in US weapons programs, from Manhattan Project on. The custom is to use "tamper" to refer to both effects, although "neutronic tamper" and "reflector" are used if the neutron reflection effect alone is intended.
4.1.5.2.1 Tampers
In the bare core, the fissile material that has been reached by the inward moving rarefaction wave expands outward very rapidly. In radiation dominated matter, expansion into a vacuum reaches a limiting speed of six times the local speed of sound in the material (this is the velocity at the outer surface of the expanding sphere of material). The density of matter behind the rarefaction front (which moves toward the center of the core) thus drops very rapidly and is almost immediately lost to the fission reaction.
If a layer of dense material surrounds the core then something very different occurs. The fissile material is not expanding into a vacuum, instead it has to compress and accelerate matter ahead of it. That is, it creates a shock wave. The expansion velocity of the core is then limited to the velocity of accelerated material behind the expanding shock front, which is close to the shock velocity itself. If the tamper and fissile core have similar densities, then this expansion velocity is similar to the speed of sound in the core and only 1/6 as fast as the unimpeded expansion velocity.
This confining effect means that the drop in alpha as disassembly proceeds is not nearly as abrupt as in a vacuum. It thus reduces the importance of the inaccurate assumption of constant alpha used in deriving the efficiency equation.
Another important effect is caused by the radiation cooling of the core. In a vacuum this energy is lost to free space. An opaque tamper absorbs this energy, and a layer of material one mean free path thick is heated to nearly the temperature and pressure of the core. The expansion shock wave then arises not at the surface of the core, but some distance away in the tamper (on the order of a few millimeters). A rarefaction wave must then propagate back to the surface of the core before its expansion even begins. In effect, this increases size of the expansion distance term ((1/rho_rel)^(1/3)-(1/rho_rel))^2 in the efficiency equation.
4.1.5.2.2 Reflectors
In a bare core, any neutron that reaches the surface of the core is lost forever to the reaction. A reflector scatters the neutrons, a process that causes some fraction of them to eventually reenter the fissile mass (usually after being scattered several times). Its effect on efficiency then can be described simply by reducing the neutron leakage term (rho_rel)^(-4/3) by a constant factor, or by reducing the reference density critical mass terms.
The leakage or critical mass adjustments must take into account time absorption effects. This means that leakage cannot simply be reduced by the probability of a lost neutron eventually returning, and the reflected critical mass cannot be based simply on the steady state criticality value. For example when an efficiently reflected assembly is only slightly supercritical, then multiplication is dependent mostly (or entirely) on the reflected neutrons that reenter the core. On average each of these neutrons spends quite a lot of time outside the core before being scattered back in. The relevant value for alpha_max in this system is not the value for the fissile material, but is instead:
alpha _max = 1/(average neutron life outside of core)
This is likely to be at least an order of magnitude larger than the core material alpha_max value.
4.1.5.3 Predetonation
An optimally efficient fission explosion requires that the explosive disassembly of the core occur when the neutron multiplication rate (designated alpha) is at a maximum. Ideally the bomb will be designed to compress the core to this state (or close to it) before injecting neutrons to initiate the chain reaction. If neutrons enter the mass after criticality, but before this ideal time, the result is predetonation (or preinitiation): disassembly at a sub-optimal multiplication rate, producing a reduced yield.
How significant this problem is depends on the reactivity insertion rate. Something like 45 multiplication intervals must elapse before really significant amounts of energy are released. Prior to this point predetonation is not possible. The number of these intervals that occur during a period of time is obtained by integrating alpha over the period. When alpha is effectively constant it is simply alpha*t.
During insertion, alpha is not constant. When insertion begins its value is zero. If a neutron is injected early in insertion and insertion is slow, we can accumulate 45 multiplication intervals when alpha is still quite low. In this case a dramatic reduction in yield will occur. On the other hand, if it were possible for insertion to be so fast that full insertion is achieved before accumulating enough multiplication intervals to disassemble the bomb then no predetonation problem would exist.
To evaluate this problem let us consider a critical system with initial radius r_0 undergoing uniform spherical compression, with the radius decreasing at a constant rate v, then alpha is:
Eq. 4.1.5.3-1
alpha = alpha_max_0 * ((r_0/(r_0 - v*t))^3 - ((r_0 - v*t)/r_0))
Integrating, we obtain:
Eq. 4.1.5.3-2
Int[alpha] = alpha_max_0*(r_0^3/(2v*(r_0-v*t)^2) - (t-(v*t^2)/(2*rc)))
Which allows to compute the number of elapsed multiplication intervals between times t_1 and t_2.
For example, consider a system with the following parameters with a critical radius r = 4.5 cm, a radial implosion velocity v = 2.5x10^5 cm/sec, and alpha_max_0 = 2.8x10^8/sec. Figure 4.1.5.3-1 shows the accumulation of elapsed neutron multiplication intervals (Y axis) as implosion proceeds (seconds on X axis).
Recall that disassembly occurs when the speed of sound, c_s, integrated over the life of the chain reaction is equal to r - r_c, the difference between the outer radius and the critical radius. Since c_s is proportional to the square root of the energy released, it increases by a factor of e every 2 multiplication intervals. Disassembly thus occurs quite abruptly, effectively occurring over a period of two multiplication intervals. The condition for disassembly is thus:
Eq. 4.1.5.3-3
r(t) - r_c(t) = 2*c_s(t)/alpha(t) for some time t.
Since r - r_c is a polynomial function, and c_s is a transcendental (exponential) function, no closed form means of calculating t is possible. However these functions are monotonically increasing in the range of values of interest so numeric and graphical techniques can easily determine when the disassembly condition occurs. The value of alpha at that point then determines efficiency.
Taking our previous example (r = 4.5 cm, v = 2.5x10^5 cm/sec, alpha_max = 2.8x10^8/sec) we can plot the net implosion distance (r - r_c) and the integrated expansion distance (2*c_s/alpha) against the implosion time. This is shown in the log plot in Figure 4.1.5.3-2 for the period between 1 and 1.3 microseconds. Distance is in centimeters (Y axis) and time is in seconds (X axis). If a neutron is present at the beginning of insertion, we see that the disassembly condition occurs at t = 1.25x10^-6 sec. At this point 52 multiplication intervals have elapsed, and the effective value of alpha is 8.6x10^7/sec. The corresponding yield is about 0.5 kt.
The parameters above approximately describe the Fat Man bomb. This shows that even in the worst case, neutrons being present at the moment of criticality, quite a substantial yield would have been created. Predetonation does not necessarily result in an insignificant fizzle. It is not feasible though to make a high explosive driven implosion system fast enough to completely defeat predetonation through insertion speed alone (radiation driven implosion and fusion boosting offer means of overcoming it however).
The likelihood of predetonation occurring depends on the neutron background, the average rate at which neutron injection events occur. I use the term "neutron injection event" instead of simply talking about neutrons for a specific reason: the major source of neutrons in a fission device is spontaneous fission of the fissile material itself (or of contaminating isotopes). Each spontaneous fission produces an average of 2-3 neutrons (depending on the isotope). However, these neutrons are all released at the same moment, and thus either a fission chain reaction is initiated at the moment, or they all very quickly disappear. Each fission is a single injection event, neutrons from other sources are uncorrelated and are thus individual injection events.
Now neutron injection during insertion is not guaranteed to initiate a divergent chain reaction. At criticality (alpha equals zero), each fission generates on average one fission in the next generation. Since each fission produces nu neutrons (nu is in the range of 2-3 neutrons, 2.9 for Pu-239), this means that each individual neutron has only 1/nu chance of causing a new fission. At positive values of alpha, the odds are better of course, but clearly we must consider then the probability that each injection actually succeeds in creating a divergent chain reaction. This probability is dependent on alpha, but since non-fission capture is a significant possibility in any fissile system, it does not truly converge to 1 regardless of how high alpha is (although with plutonium it comes close).
Near criticality the probability of starting a chain reaction (P_chain) for a single neutron is thus about 34% for plutonium, and 40% for U-235. Since spontaneous fission injects multiple neutrons, the P_chain for this injection event is high, about 70% for both Pu-239 and U-235.
If the average rate of neutron injection is R_inj, then the probability of initiating a chain reaction during an insertion time of length T is the Poisson function: Eqs. 4.1.5.3-4 P_init = 1 - e^((-T/R_inj)*P_chain) If T is much smaller than R_inj then this equation reduces approximately to P_init = (T/R_inj)*P_chain.
When T is much smaller than R_inj predetonation is unlikely, and the yield of the fission bomb (which will be the optimum yield) can be predicted with high confidence. As the ratio of T/R_inj becomes larger yield variability increases. When (T/R_inj)*P_chain is equal to ln 2 (0.693...) then the probability of predetonation and no predetonation is equal, although when predetonation occurs close to full assembly the yield reduction is small. As T/R_inj continues to increase predetonation becomes virtually certain. With a large enough value to T/R_inj the yield becomes predictable again, but this time it is the minimum yield that results when neutrons are present at the beginning of insertion. For an implosion bomb a typical spread between the optimum and minimum yields is something like 40:1.
In the Fat Man bomb the probability of predetonation was 12% (from a declassified Oppenheimer memo), assuming an average P_chain of 0.7 we can estimate the insertion time at 6.7 microseconds, or 4.7 microseconds if P_chain was close to 1. The chance of large yield reduction was much smaller than this however. There was a 6% chance of a yield < 5 kt, and only a 2% chance of a yield < 1 kt. As we have seen, in no case would the yield have been smaller than 0.5 kt or so.
Spontaneous fission is not the only cause for concern, since neutrons can enter the weapon from outside. Natural neutron sources are not cause for concern, but in a combat situation very powerful sources of neutrons may be encountered - other nuclear weapons.
One kiloton of fission yield produces a truly astronomical number of excess neutrons - about 3x10^24, with a fluence of 1.5x10^10 neutrons/cm^2 500 m away. A kiloton of fusion yields 3-4 times as many. The fission reaction itself emits all of its neutrons in less than a microsecond, but due to moderation these neutrons arrive at distant locations over a much longer period of time. Most of them arrive in a pulse lasting a millisecond, but thermal neutrons can continue to arrive for much longer periods of time. This is not the whole problem though. Additional neutrons called "delayed neutrons" continue to be emitted for about a minute from the excited fission products. These amount to only 1% or so of the prompt neutrons, but this is still an average arrival rate of 2.5x10^6 neutrons/cm^2-sec for a kiloton of fission at 500 m. With weapons sensitive to predetonation, careful spacing of explosions in distance and time may be necessary. Neutron hardening - lining the bomb with moderating and neutron absorbing materials - may be necessary to hold predetonation problems to a tolerable level (it is virtually impossible to eliminate it entirely in this way).
4.1.6 Methods of Core Assembly
The principal problem in fission weapon design is how to rapidly assemble or compress the fissile material from a subcritical state to a supercritical one. Methods of doing this can be classified in two ways:
Subsonic assembly means that shock waves are not involved. Assembly is performed by adiabatic compression, or by continuous acceleration. As a practical matter, only one subsonic assembly scheme needs to be considered: gun assembly.
Supersonic assembly means that shock waves are involved. Shock waves cause instantaneous acceleration, and naturally arise whenever the very large forces required for extremely rapid assembly occur. The are thus the natural tools to use for assembly. Shocks are normally created by using high explosives, or by collisions between high velocity bodies (which have in turn been accelerated by high explosive shocks). The term "implosion" is generally synonymous with supersonic assembly. Most fission weapons have been designed with assembly schemes of this type.
Assembly may be performed by compressing the core along one, two, or three axes. One-D compression is used in guns, and plane shock wave compression schemes. Two and three-D compression are known as cylindrical implosion and spherical implosion respectively. Plane shock wave assembly might logically be called "linear Implosion", but this term has been usurped (in the US at any rate) by a variant on cylindrical implosion (see below). The basic principles involved with these approaches are discussed in detail in Section 3.7, Principles of Implosion.
To the approaches just mentioned, we might add more some difficult to classify hybrid schemes such as: "pseudo-spherical implosion", where the mass is compressed into a roughly spherical form by convergent shock waves of more complex form; and "linear implosion" where a compressive shock wave travels along a cylindrical body (or other axially symmetric form - like an ellipsoid), successively squeezing it from one end to the other (or from both ends towards the middle). Schemes of this sort may be used where high efficiency is not called for, and difficult design constraints are involved, such as severe size or mass limitations. Hybrid combinations of gun and implosion are also possible - firing a bullet into an assembly that is also compressed.
The number of axes of assembly naturally affect the overall shape of the bomb. One-D assembly methods naturally tend to produce long, thin weapon designs; 2-D methods lead to disk-shaped or short cylindrical systems; and 3-D methods lead to spherical designs.
The subsections detailing assembly methods are divided in gun assembly (subsonic assembly) and implosion assembly (supersonic assembly). Even though it superficially resembles gun assembly, linear implosion is discussed in the implosion section since it actually has much more in common with other shock compression approaches.
The performance of an assembly method can be evaluated by two key metrics: the total insertion time and the degree of compression. Total insertion time (and the related insertion rate) is principally important for its role in minimizing the probability of predetonation. The degree of compression determines the efficiency of the bomb, the chief criteria of bomb performance. Short insertion times and high compression are usually associated since the large forces needed to produce one also tend to cause the other.
4.1.6.1 Gun Assembly
This was the first technique to be seriously proposed for creating fission explosions, and the first to be successfully developed. The first nuclear weapon to be used in war was the gun-type bomb called Little Boy, dropped on Hiroshima. Basic gun assembly is very simple in both concept and execution. The supercritical assembly is divided into two pieces, each of which is subcritical. One of these, the projectile, is propelled into the other, called the target, by the pressure of propellant combustion gases in a gun barrel. Since artillery technology is very well developed, there are really no significant technical problems involved with designing or manufacturing the assembly system.
The simple single-gun design (one target, one projectile) imposes limits on weapon, mass, efficiency and yield that can be substantially improved by using a "double-gun" design using two projectiles fired at each other. These two approaches are discussed in separate sections below. Even more sophisticated "complex" guns, that combine double guns with implosion are discussed in Hybrid Assembly techniques.
Gun designs may be used for several applications. They are very simple, and may be used when development resources are scarce or extremely reliability is called for. Gun designs are natural where weapons can be relatively long and heavy, but weapon diameter is severely limited - such as nuclear artillery shells (which are "gun type" weapons in two senses!) or earth penetrating "bunker busters" (here the characteristics of a gun tube - long, narrow, heavy, and strong - are ideal).
Single guns are used where designs are highly conservative (early US weapon, the South African fission weapon), or where the inherent penalties of the design are not a problem (bunker busters perhaps). Double guns are probably the most widely used gun approach (in atomic artillery shells for example).
4.1.6.1.1 Single Gun Systems
We might conclude that a practical limit for simple gun assembly (using a single gun) is a bit less than 2 critical masses, reasoning as follows: each piece must be less than 1 critical mass, if we have two pieces then after they are joined the sum must be less than 2 critical masses.
Actually we can do much better than this. If we hollow out a supercritical assembly by removing a chunk from the center like an apple core, we reduce its effective density. Since the critical mass of a system is inversely proportional to the square of the density, we have increased the critical mass remaining material (which we shall call the target) while simultaneously reducing its actual mass. The piece that was removed (which will be called the bullet) must still be a bit less than one critical mass since it is solid. Using this reasoning, letting the bullet have the limiting value of one full critical mass, and assuming the neutron savings from reflection is the same for both pieces (a poor assumption for which correction must be made) we have:
Eq. 4.1.6.1.1-1
M_c/((M - M_c)/M)^2 = M - M_c
where M is the total mass of the assembly, and M_c is the standard critical mass. The solution of this cubic equation is approximately M = 3.15 M_c. In other words, with simple gun assembly we can achieve an assembly of no more than 3.15 critical masses. Of course a practical system must include a safety factor, and reduce the ratio to a smaller value than this.
The weapon designer will undoubtedly surround the target assembly with a very good neutron reflector. The bullet will not be surrounded by this reflector until it is fired into the target, its effective critical mass limit is higher, allowing a larger final assembly than the 3.15 M_c calculated above.
Revising Eq. 4.1.6.1.1-1 we get:
Eq. 4.1.6.1.1-2
M_c/((M - (1.78 M_c))/M)^2 = M - (1.78 M_c)
which has a solution of M = 4.51 M_c. If a critical mass ratio of 2 is used for beryllium, then M = 4.88 M_c. This provides an upper bound on the performance of simple gun-type weapons.
Some additional improvement can be had by adding fast neutron absorbers to the system, either natural boron, or boron enriched in B-10. A boron-containing sabot (collar) around the bullet will suppress the effect of neutron reflection from the barrel, and a boron insert in the target will absorb neutrons internally thereby raising the critical mass. In this approach the system would be designed so that the sabot is stripped of the bullet as it enters the target, and the insert is driven out of the target by the bullet. This system was apparently used in the Little Boy weapon.
4.1.6.1.2 Double Gun Systems
Significant weight savings a possible by using a "double-gun" - firing two projectiles at each other to achieve the same insertion velocity. With all other factors being the same (gun length, projectile mass, materials, etc.) the mass of a gun varies with the fourth power of velocity (doubling velocity requires quadrupling pressure, quadrupling barrel thickness increases mass sixteen-fold). By using two projectiles the required velocity is cut by half, and so is the projectile mass (for each gun). On the other hand, to keep the same total gun length though, the projectile must be accelerated in half the distance, and of course there are now two guns. The net effect is to cut the required mass by a factor of eight. The mass of the breech block (which seals the end of the gun) reduces this weight saving somewhat, and of course there is the offsetting added complexity.
A double gun can improve on the achievable assembled mass size since the projectile mass is divided into two sub-critical pieces, each of which can be up to one critical mass in size. Modifying Eq. 4.1.6.1.1-1 we get:
Eq. 4.1.6.1.1-3
M_c/((M - 2M_c)/M)^2 = M - 2M_c
with a solution of M = 4.88 M_c.
Taking into account the effect of differential reflector efficiency we get mass ratios of ratios of 3.56 (tungsten carbide) and 4 (beryllium) which give assembled mass size limits of M = 7.34 M_c and M = 8 M_c respectively.
Another variant of the double gun concept is to still only have two fissile masses - a hollow mass and a cylindrical core as in the single gun - but to drive them both together with propellant. One possible design would be to use a constant diameter gun bore equal to the target diameter, with the smaller diameter core being mounted in a sabot. In this design the target mass would probably be heavier than the core/sabot system, so one end of the barrel might be reinforced to take higher pressures. Another more unusual approach would be to fire the target assembly down an annular (ring shaped) bore. This design appears to have been used in the U.S. W-33 atomic artillery shell, which is reported to have had an annular bore.
These larger assembled masses give significantly more efficient bombs, but also require large amounts of fissile material to achieve them. And since there is no compression of the fissile material, the large efficiency gains obtainable through implosive compression is lost. These shortcomings can be offset somewhat using fusion boosting, but gun designs are inherently less efficient than implosion designs when comparing equal fissile masses or yields.
4.1.6.1.3 Weapon Design and Insertion Speed
In addition to the efficiency and yield limitations, gun assembly has some other significant shortcomings:
First, guns tend to be long and heavy. There must be sufficient acceleration distance in the gun tube before the projectile begins insertion. Increasing the gas pressure in the gun can shorten this distance, but requires a heavier tube.
Second, gun assembly is slow. Since it desirable to keep the weight and length of the weapon down, practical insertion velocities are limited to velocities below 1000 m/sec (usually far below). The diameter of a core is on the order of 15 cm, so the insertion time must be at least a 150 microseconds or so.
In fact, achievable insertion times are much longer than this. Taking into account only the physical insertion of the projectile into the core underestimates the insertion problem. As previously indicated, to maximize efficiency both pieces of the core must be fairly close to criticality by themselves. This means that a critical configuration will be achieved before the projectile actually reaches the target. The greater the mass of fissile material in the weapon, the worse this problem becomes. With greater insertion distances, higher insertion velocities are required to hold the probability of predetonation to a specified value. This in turn requires greater accelerations or acceleration distances, further increasing the mass and length of the weapon.
In Little Boy a critical configuration was reached when the projectile and target were still 25 cm apart. The insertion velocity was 300 m/sec, giving an overall insertion time of 1.35 milliseconds.
Long insertion times like this place some serious constraints on the materials that can be used in the bomb since it is essential to keep neutron background levels very low. Plutonium is excluded entirely, only U-235 and U-233 may be used. Certain designs may be somewhat sensitive to the isotopic composition of the uranium also. High percentages of even-numbered isotopes may make the probability of predetonation unacceptably high.
The predetonation problem also prevents the use of a U-238 tamper/reflector around the core. A useful amount of U-238 (200 kg or so) would produce a fission background of 1 fission/0.9 milliseconds.
Gun-type weapons are obviously very sensitive to predetonation from other battlefield nuclear explosions. Without hardening, gun weapons cannot be used within a few of kilometers of a previous explosion for at least a minute or two.
Attempting to push close to the mass limit is risky also. The closer the two masses are to criticality, the smaller the margin of safety in the weapon, and the easier it is to cause accidental criticality. This can occur if a violent impact dislodges the projectile, allowing it to travel toward the target. It can also occur if water leaks into the weapon, acting as a moderator and rendering the system critical (in this case though a high yield explosion could not occur).
Due to the complicated geometry, calculating where criticality is achieved in the projectile's travel down the barrel is extremely difficult, as is calculating the effective value of alpha vs time as insertion continues. Elaborate computation intensive Monte Carlo techniques are required. In the development of Little Boy these things had to be extrapolated from measurements made in scale models.
4.1.6.1.4 Initiation
Once insertion is completed, neutrons need to be introduced to begin the chain reaction. One route to doing this is to use a highly reliable "modulated" neutron initiator, an initiator that releases neutrons only when triggered. The sophisticated neutron pulse tubes used in modern weapons are one possibility. The Manhattan Project developed a simple beryllium/polonium 210 initiator named "Abner" that brought the two materials together when struck by the projectile.
If neutron injection is reliable, then the weapon designer does not need to worry about stopping the projectile. The entire nuclear reaction will be completed before the projectile travels a significant distance. On the other hand, if the projectile can be brought to rest in the target without recoiling back then an initiator is not even strictly necessary. Eventually the neutron background will start the reaction unaided.
A target designed to stop the projectile once insertion is complete is called a "blind target". The Little Boy bomb had a blind target design. The deformation expansion of the projectile when it impacted on the stop plate of the massive steel target holder guaranteed that it would lodge firmly in place. Other designs might add locking rings or other retention devices. Because of the use of a blind target design, Little Boy would have exploded successfully without the Abner initiators. Oppenheimer only decided to include the initiators in the bomb fairly late in the preparation process. Even without Abner, the probability that Little Boy would have failed to explode within 200 milliseconds was only 0.15%; a delay as long as one second was vanishingly small - 10^-14.
Atomic artillery shells have tended to be gun-type systems, since it is relatively easy to make a small diameter, small volume package this way (at the expense of large amounts of U-235). Airbursts are the preferred mode of detonation for battlefield atomic weapons which, for an artillery shell travelling downward at several hundred meters per second, means that initiation must occur at a precise time. Gun-type atomic artillery shells always include polonium/beryllium initiators to ensure this.
4.1.6.2 Implosion Assembly
High explosive driven implosion assembly uses the ability of shock waves to instantaneously compress and accelerate material to high velocities. This allows compact designs to rapidly compress fissile material to densities much higher than normal on a time scale of microseconds, leading to efficient and powerful explosions. The speed of implosion is typically several hundred times faster than gun assembly (e.g. 2-3 microseconds vs. 1 millisecond). Densities twice the normal maximum value can be reached, and advanced designs may be able to do substantially better than this (compressions of three and four fold are often claimed in the unclassified literature, but these seem exaggerated). Weapon efficiency is typically an order of magnitude better than gun designs.
The design of an implosion bomb can be divided into two parts:
The high explosive system may be essentially unconfined (like that in the Fat Man bomb), but increased explosive efficiency can be obtained by placing a massive tamper around the explosive. The system then acts like a piston turned inside out, the explosive gases are trapped between the outer tamper and the inner implosion hardware, which is driven inward as the gases expand. The added mass of the tamper is no doubt greater than the explosive savings, but if the tamper is required anyway (for radiation confinement, say) then it adds to the compactness of the design.
If you have not consulted Section 3.7 Principles of Implosion, it may be a good idea to do so.
4.1.6.2.1 Energy Required for Compression As explained in Section 3.4 Hydrodynamics, shock compression dissipates energy in three ways:
Only the first of these is ultimately desirable for implosion, although depending on the system design some or all of the kinetic energy may be reclaimable as compressive work. The energy expended in entropic heating is not only lost, but also makes the material more resistant to further compression.
Shock compression always dissipates some energy as heat, and is less efficient than gentle isentropic (constant entropy) compression. Examining the pressure and total energy required for isentropic compression thus provides a lower bound on the work required to reach a given density.
Below are curves for the energy required for isentropic and shock compression of uranium up to a compression factor of 3. For shock compression only the energy the appears as internal energy (compression and heating) are included, kinetic energy is ignored.
The energy expenditure figures on the X axis are in ergs/cm^3 of uncompressed uranium, the y axis gives the relative volume change (V/V_0). Shock compression, being less efficient, is the upper curve. It can be seen that as compression factors rise above 1.5 (a V/V_0 ratio of 0.67), the amount of work required for shock compression compared to isentropic compression rises rapidly. The kink in the shock compression curve at V/V_0 of 0.5 is not a real phenomenon, it is due to the transition from experimental data to a theoretical Thomas-Fermi EOS.
It is interesting to note that to double the density of one cubic centimeter of uranium (18.9 grams) 1.7 x 10^12 ergs is required for shock compression. This is the amount of energy found in 40 grams of TNT, about twice the weight of the uranium. The efficiency of an implosion system at transferring high explosive energy to the core is generally not better than 30%, and may be worse (possibly much worse if the design is inefficient). This allows us the make a good estimate of the amount of explosive required to compress a given amount of uranium or plutonium to high density (a minimum of 6 times the mass of the fissile material for a compression factor of 2).
These curves also show that very high shock compressions (four and above) are so energetically expensive as to be infeasible. To achieve a factor of only 3, 7.1x10^11 ergs/g of uranium is required. Factoring implosion efficiency (30%), the high explosive (if it is TNT) must have a mass 56 times that of the material being compressed. Reports in the unclassified literature of compressions of four and higher can thus be safely discounted.
Compression figures for plutonium are classified above 30 kilobars, but there is every reason to believe that they are not much different from that of uranium. Although there are large density variations from element to element at low pressure, the low density elements are also the most compressible, so that at high pressures (several megabars) the plot of density vs atomic number becomes a fairly smooth function. This implies that what differences there may be in behavior between U and Pu at low pressure will tend to disappear in the high pressure region.
Actually, even in the low pressure region the available information shows that the difference in behavior isn't all that great, despite the astonishingly large number of phases (six) and bizarre behavior exhibited by plutonium at atmospheric pressure. The highest density phases of both metals have nearly identical atomic volumes at room pressure, and the number of phases of both metals drops rapidly with increasing pressure, with only two phases existing for both metals above 30 kilobars. The lowest density phase of plutonium, the delta phase, in particular disappears very rapidly. The amount of energy expended in compression at these low pressures is trivial. The compression data for uranium is thus a good substitute for plutonium, especially at high pressures and high compressions.
The shock and isentropic pressures required corresponding to the compression energy curves are shown below. The pressures shown on the X axis are in kilobars, the y axis gives the relative volume change (V/V_0).
Since the compression energies of interest vary by many orders of magnitude over compressions ranging up to 3, it is often more convenient to look at logarithmic plots or energy. Figure 4.1.6.2.1-3, below, gives the isentropic curve from 10^7 ergs/cm^3 to 10^12 ergs/cm^3. Since the energy for shock compression is virtually identical to the isentropic value at small compressions, the curve for shock compression is given for compression energies of 10^10 erg/cm^3 (V/V_0 ~ 0.9)
4.1.6.2.2 Shock Wave Generation Systems
The only practical means of generating shock waves in weapons is through the use of high explosives. When suitably initiated, these energetic materials support detonation waves: a self-sustaining shock wave that triggers energy releasing chemical reactions, and is driven by the expanding gases that are produced by these reactions.
Normally a high explosive is initiated at a single point. The detonation propagates as a convex detonation wave, with a more or less spherical surface, from that point.
To drive an implosion, a divergent detonation wave must be converted into a convergent one (or a planar one for linear implosion). Three approaches can be identified for doing this.
4.1.6.2.2.1 Multiple Initiation Points
In this approach, the high explosive is initiated simultaneously by a large number of detonators all over its surface. The idea is that if enough detonation points exist, then it will approximate the simultaneous initiation of the entire surface, producing an appropriately shaped shock from the outset.
The problem with this approach is that colliding shock waves do not tend to "smooth out", rather the reverse happens. A high pressure region forms at the intersection of the waves, leading to high velocity jets that outrun the detonation waves and disrupting the hoped for symmetry.
The multiple detonation point approach was the first one tried at Los Alamos during the Manhattan Project to build a spherical implosion bomb. Attempts were made to suppress the jetting phenomenon by constantly increasing the number of points, or by inserting inert spacers at the collision points to suppress the jets. The problems were not successfully worked out at the time.
Since the war this approach has been used with reasonable success in laboratory megagauss field experiments employing the simpler cylindrical geometry. There is also evidence of continuing US interest in this approach. It is not clear whether this technique has been successfully adapted for use in weapons.
4.1.6.2.2.2 Explosive Lenses
The basic idea here is to use the principle of refraction to shape a detonation wave, just as it is used in optics to shape a light wave.
Optical lenses use combinations of materials in which light travels at different speeds. This difference in speed gives rise to the refractive index, which bends the wave when it crosses the boundary between materials.
Explosive lenses use materials that transmit detonation or shock waves at different speeds. The original scheme used a hollow cone of an explosive with a high detonation velocity, and an inner cone of an explosive with a low velocity. The detonator initiates the high velocity explosive at the apex of the cone. A high velocity detonation wave then travels down the surface of the hollow cone, initiating the inner explosive as it goes by. The low velocity detonation wave lags behind, causing the formation of a concave (or planar) detonation wave.
With any given combination of explosives, the curvature of the wave produced is determined by the apex angle of the lens. The narrower the angle, the greater the curvature. However, for a given lens base area the narrower angle, the taller the lens, and the greater its volume. Both of these are undesirable in weapons, since volume and mass are at a premium.
To create a spherical implosion wave, a number of inward facing lenses need to be arranged on the surface of a sphere so that the convergent spherical segments that each produces merge into one wave. There is substantial advantage in using a large number of lenses. Having many lenses means that each lens has a small base area, and needs to produce a wave with a smaller curvature, both of which reduce the thickness of the lens layer. A more symmetrical implosion can probably be achieved with more lenses also.
It is important to have the lens detonation points (and optical axes) spaced as regularly as possible to minimize irregularities, and to make the height of each lens identical. The largest number of points that can be spaced equidistantly from their neighbors on the surface of a sphere is 20 - corresponding to the 20 triangular facets of an icosahedron (imagine the sphere encased in a circumscribed polyhedron, with each facet touching the sphere at one point). The next largest number is 12 - corresponding to the 12 pentagonal facets of the dodecahedron.
12 lenses, even 20 lenses, is an undesirably small number (although some implosion systems have used the 20 point icosahedral layout). A close approximation to strict regularity can be achieved with more points by interleaving a dodecahedron and icosahedron to produce a polyhedron tiled with hexagonal and pentagonal facets, 20 hexagons and 12 pentagons, for a total of 32 points. This pattern is the same familiar one found on a soccer ball, and was used as the original implosion system lens layout in Gadget, and other early US nuclear weapons.
Designs with 40, 60, 72, and 92 lenses have also been used (although these do not rely on Platonic solids for providing the layout pattern). More lenses lead to a thinner, less massive explosive lens shell, and greater implosion uniformity. The penalty for more lenses is more fabrication effort, and a more powerful and complex initiation system (not a trivial problem originally, but greatly simplified by modern pulse power technology). A simple implosion system could be very massive. The 32 point systems used in early US nuclear weapons had an external diameter of 1.4 m and weighed over 2000 kg. Current systems may be less than 30 cm, and weigh as little as 20 kg, but probably do not follow the same design approach as earlier weapons.
To a degree these multi-lens systems all suffer from the same shortcoming as the basic multi-point detonation approach: strict uniformity of the spherical implosion wave is unachievable. The detonation wave spreads out radially from each detonation point, so each wave produces a circular segment of a spherical wave. If you consider an icosahedron or a "soccer ball", you can see that when circles are inscribed in each of the regular polygons they touch each of their neighbor circles at one point. This marks the moment when the individual wavelets start to merge into a single wave. The gaps left between the inscribed circles however are irregular areas where distortions are bound to arise as the wave edges spread into them, possibly even leading to jetting.
Since the shock wave created by the lens exits from it at the velocity of the slow (and relatively weak) explosive, it desirable to have a layer of powerful explosive inside the lens system (perhaps the same one used as the fast lens component). This layer provides most of the driving force for the implosion, for the most part the lens system (which may well be much more massive) simply provides a mechanism for spherical initiation.
Ideally, the best combination of explosives is the fastest and slowest that are available. This provides the greatest possible refractive index, and thus bending effect, and allows using a wider lens angle. The fastest and slowest explosives generally known are HMX (octogen) and baratol respectively. HMX has a detonation velocity of 9110 m/sec (at a pressed density of 1.89), the dense explosive baratol (76% barium nitrate/24% TNT) has a velocity of 4870 m/sec (cast density 2.55). Explosives with slightly slower detonation velocities include the even denser plumbatol - 4850 m/sec (cast density 2.89) for a composition of 70% lead nitrate/30% TNT; and the relatively light boracitol - 4860 m/sec (cast density 1.55) for a composition of 60% boric acid/40% TNT. Mixtures of TNT with glass or plastic microspheres have proven to be an effective, light weight, and economical slow explosive in recent unclassified explosive lens work (I don't have data on their velocities though).
During WWII Los Alamos developed lenses using combination of Composition B (or Comp B) for the fast explosive (detonation velocity of 7920 m/sec, at a cast density 1.72), and baratol for the slow explosive.
Later systems have used the very fast HMX as a fast explosive, often as a plastic bonded mixture consisting almost entirely of HMX. Plumbatol, a denser and slightly slower explosive, may have been used in some later lens system designs. Boracitol is definitely known to have been used, probably in thermonuclear weapon triggers and perhaps in other types of weapons as well.
The idea of explosives lenses appears to have originated with M. J. Poole of the Explosives Research Committee in England. In 1942 he prepared a report describing a two-dimensional arrangement of explosives (RDX and baratol) to create a plane detonation wave. This idea was brought to Los Alamos in May 1944 by James Tuck, where he expanded it by suggesting a 3-D lens for creating a spherical implosion wave as a solution to making an implosion bomb. A practical lens design was proposed separately by Elizabeth Boggs of the US Explosives Research Laboratory, and by Johann Von Neumann. The Boggs proposal was the earlier of the two, although it was Von Neumann's proposal who directly influenced the Manhattan Project.
The task of developing a successful spherical implosion wave system is extremely difficult. Although the concept involved is simple, actually designing a lens is not trivial. The detonation wave velocity is affected by events occurring some distance behind the front. When the wave crosses from the fast explosive into the slow explosive it does not instantly assume the steady state detonation velocity of the slow explosive. Unlike the analogy with light, the velocity change is gradual and occurs over a significant distance. Since energy can be lost through the surface of the lens, thus reducing the fast wave velocity, the test environment of the lens also affects its performance. The behavior of a lens can only be calculated using sophisticated 2 and 3-D hydrodynamic computer codes that have been validated against experimental data.
Practical lens development generally requires a combination of experimentation, requiring precision explosive manufacture and sophisticated instruments to measure shock wave shape and arrival times, and numerical modelling (computer simulation) to extrapolate from test results. An iterative design, test, and redesign cycle allows the development of efficient, high-performance lenses.
During the Manhattan Project, due to the primitive state of computers and high explosive science and instrumentation, lenses could only be designed by trial and error (guided to some extent by scaling laws deduced from previous experiments). This required the detonation of over 20,000 test lens (and for each one tested, several were fabricated and rejected). When successful sub-scale implosion systems were scaled up to full size, it was discovered that the lenses had to be redesigned.
Assembling the lenses into a complete implosion system aggravates the design and development problems. To avoid shock wave collisions that disrupt symmetry, the surfaces of the lenses need to be aligned very accurately. In a spherical system, the implosion wave that is created is completely hidden by the layer of detonating explosive. The chief region of interest is a small region in the center with perhaps < 0.1% the volume of the whole system. Very expensive diagnostic equipment and difficult experiments are required to study the implosion process, or even to verify that it works at all. Hemispherical tests can be quite useful though to validate lens systems before full spherical testing.
4.1.6.2.2.3 Advanced Wave Shaping Techniques
The conical lens design used by the Manhattan Project and early U.S. nuclear weapons is not the only lens design possible, or even the best. It had the crucial advantage of being simple in form (eliminating the need to design or fabricate complex shapes), and of having a single design variable - the cone apex angle. This made it possible to devise workable lenses with the crude methods then available. Other geometric arrangements of materials that transmit shocks slowly can be used to shape a convex shock into a concave one.
The shock slowing component of a lens, such as the inner cone of a conical explosive lens, does not really need to be another explosive. An inert substance that transmits a shock more slowly than the fast explosive detonation wave will also work. The great range of materials available that are not explosives gives much greater design flexibility. An additional (potential) advantage is that shock waves attenuate as they travel through non-explosive materials, and slow down. This can make lens design more complex, since this attenuation must be taken into account, but the reduced velocity can also lead to a more compact lens. Care must be taken though to insure that the attenuated shock remains strong enough to initiate the inner explosive layer.
By consulting the equation for shock velocity we can see that a high compressibility (low value of gamma) and a high density both lead to low shock wave velocities. An ideal material would be a highly compressible material of relatively high density. This describes an unusual class of filled plastic foams that have been developed at the Allied-Signal Kansas City Plant (the primary supplier of non-nuclear components for US nuclear weapons). It is quite possible that these foams were developed for use as wave shaping materials.
By extending the idea of custom tailoring the density and compressibility of materials, we can imagine that different arrangements of materials of varying properties can be used to reshape shock waves in a variety of ways.
Inserting low density materials, like solid or foam plastics, into explosives can also inhibit detonation propagation and allow the designer to "fold" the path the detonation wave must take. If suitable detonation inhibiting bodies are arranged in a grid inside a cone of high explosive, the same effect as the high explosive lens can be obtained with a lower lens density and with a larger apex angle.
French researchers have described advanced lens systems using alternating layers of explosive and inert material. This creates an anisotropic detonation velocity in the system, very slow across the layers, but fast along the them. A compact lens for producing spherically curved waves has been demonstrated using a cylindrical version of this system, with a slow explosive between the inert layers, and a curved "nose cone"-like surface covered by fast explosive.
It is possible to completely and uniformly cover a sphere with circles if the number of lenses (and circles) is less than or equal to two. A single lens capable of bending a single detonation wave into a complete spherically convergent wave can, in principle, be made so that the resulting wave is entirely uniform. This extends the principle of the explosive lens to its most extreme form. It is also possible to use two lenses, each covering a hemisphere, which meet at the equator of the sphere and can smoothly join two hemispherical implosion waves.
The single point detonation system is illustrated below. This idea makes use of a cardioid-like logarithmic spiral:
fffffff
fssssssssssf
fsssssssssssssf
fCsssssssssssssfD <- Detonator
fsssssssssssssf
fssssssssssf f = fast explosive
fffffff s = slow explosive
C = core
This not a very practical design as given. The thickness of the slow explosive on the detonator side would have to be considerable to achieve the necessary bending. Inserting detonation path folding spacers in the explosive could also dramatically reduce the size (but making manufacturing extremely difficult). A variation on this using the French layered explosive approach has also been proposed.
It is unlikely that a slow explosive would really be used for the inner slow lens component, since the velocity differential is not that great. The high degree of shock bending required strongly encourages using something that transmits shocks as slowly as possible such as an advanced inert material.
Such an implosion system would be extremely difficult to design and possibly to manufacture. The continuously varying 3-D surfaces would require considerable experimentation to perfect, and the surfaces would be a nightmare to machine. Once an acceptable shape were developed, and suitable molds or dies were made, the actual manufacture might be quite easy, requiring only pressing of explosives and plastics into molds, or forming metal sheet in a die. The system would remain quite intolerant in any imperfections in dimensions or material however.
The difficulty in making compact and light implosion systems can be judged by the US progress in developing them. The initial Fat Man implosion system had a diameter of almost 60 inches. A significantly smaller system (30 inches) was not tested until 1951, a 22 inch system in mid-1952, and a 16 inch system in 1955. By 1955 a decade had passed since the invention of nuclear weapons, and hundreds of billions of dollars (in today's money) had been spent on developing and producing bombs and bomb delivery systems. These later systems must have used some advanced wave shaping technologies, which have remained highly classified. Clearly developing them is not an easy task (although the difficulty may be conceptual as much as technological).
4.1.6.2.2.4 Cylindrical and Planar Shock Techniques
Cylindrical and planar shock waves can be generated using the techniques previously described, making allowances for the geometry differences. A cylindrical shock can be created using the 2-D analog of the explosive lens, a wedge shaped lens with the same cross section as the conical version. A planar shock is simply a shaped shock with zero curvature.
A complete cylindrical implosion would require several parallel wedge-shaped explosive lenses arranged around the cylinder axis to form a star shape. To make the implosion truly cylindrical (as opposed to conical) it is necessary to detonate each of these lenses along the entire apex of the wedge simultaneously. This can be done by using a lens made out of sheets of high explosive (supported by a suitable backing) to create a plane shock. The edge of this sheet lens would join the apex of the wedge. This sheet lens need not extend out radially, it can join at an angle so that it folds into the space between the star points.
Some special techniques are also available based on the
peculiar characteristics of the 1-D and 2-D geometries. The basic
principle for these techniques is the "flying plate line charge",
illustrated below.
A metal plate is covered on one side with a sheet of explosive. It is detonated on one edge, and the detonation wave travels across the plate. As it does so the detonation accelerates the plate, driving it to the right. After the explosive has completely detonated the flying plate will be flat again. The angle between the original stationary plate and the flying plate is determined by the ratio between the detonation velocity, and the velocity of the accelerated plate. When this high velocity plate strikes the secondary explosive charge the shock will detonate it, creating a planar detonation.
As described above, the system doesn't quite work. A single detonator will actually create a circular detonation front in the explosive sheet, expanding from the initiation point. This can be overcome by first using a long, narrow flying plate (a flying strip if you will) to detonate the edge of wide plate. This wide plate can then be used to initiate the planar detonation.
The flying strip approach can also be used to detonate the cylindrical lens system described above in place of the sheet lens.
The flying plate scheme can be easily extended to create cylindrical detonations.
This is a cross section view of a hollow truncated cone covered by a layer of explosives. The wide end of the cone is joined to a sheet of explosives with a detonator in the center.
The single detonator located on the axis causes an expanding circular detonation in the explosive sheet. When the shock wave reaches the perimeter, it continues travelling along the surface of the cone. The cone collapses starting at the wide end. The angle of the cone is such that a cylindrical flying plate is created that initiates a cylindrical detonation in the secondary explosive.
Flying plate systems are much easier to develop than explosive lenses. Instrumentation for observing their behavior is relatively simple. Multiple contact pins and an oscilloscope can easily measure plate motion, and well established spark gap photography can image the plate effectively.
4.1.6.2.2.5 Explosives
The choice of explosives in an implosion system is driven by the desire for high performance, safety, ease of fabrication, or sometimes by special properties like the slow detonation velocity needed in explosive lenses.
The desire for high performance leads to the selection of very energetic explosives that have very high detonation velocities and pressures (these three things are closely correlated). The highest performance commonly known explosive is HMX. Using HMX as the main explosive will provide the greatest compression. HMX was widely used in US weapons from the late fifties on into the 1970s, often in a formula called PBX-9404 (although this particular formulation proved to have particularly serious safety problems - causing eight fatalities in a six month period in 1959 among personnel fabricating the explosive). HMX is known to be the principal explosive in many Soviet weapon designs since Russia is selling the explosive extracted from decommissioned warheads for commercial use. The chemically related RDX is a close second in power. It was the principal explosive used in most early US designs, in the form of a castable mixture called Composition B.
In recent years the US has become increasingly concerned with weapon safety, following some prominent accidents in which HE detonation caused widespread plutonium contamination and in the wake of repeated fatal explosions during fabrication. Many of the high energy explosives used, such as RDX and HMX, are rather sensitive to shock and heat. While normally an impact on the order of 100 ft/sec is required to detonate one these explosives, if a sliding or friction-producing impact occurs then these explosives can be set off by an impact as slow as 10 ft/sec (this requires only a drop of 18 inches)! This has led to the use of explosives that are insensitive to shock or fire. Insensitive explosives are all based on TATB, the chemical cousin DATB lacks this marked insensitivity. These explosives have very unusual reaction rate properties that make them extremely insensitive to shock, impact, or heat. TATB is reasonably powerful, being only a little less powerful than Comp B. A composition known as PBX-9504 has been developed that adds 15% HMX to a TATB mixture, creating a compromise between added power and added sensitivity.
Another very strong explosive called PETN has not been used much (or at all) as a main explosive in nuclear weapons due to its sensitivity, although it used in detonators.
Fabricating explosives for implosion systems is a demanding task, requiring rigid quality control. Many explosive components have complex shapes, most require tight dimensional tolerances, and all require a highly uniform product. Velocity variations cannot be greater than a few percent. Achieving such uniformity means carefully controlling such factors as composition, purity, particle size, crystal structure, curing time and curing temperature.
Casting was the first method used for manufacturing implosion components since a very homogenous product can be produced in fairly complex shapes. Unfortunately the most desirable explosives do not melt, which makes casting of the pure explosive impossible. The original solution adopted by the US to this problem was to use castable mixtures of the desired explosive and TNT. TNT is the natural choice for this, being the only reasonably powerful, easily melted explosive available. Composition B, the first explosive used, typically consisted of 63% RDX, 36% TNT, and 1% wax (cyclotol, a mixture with a higher proportion of RDX to TNT, was used later). Great care must be taken to ensure that the slurry of solid explosive and melted TNT is uniform since settling occurs. Considerable attention must be paid to controlling the particle size of the solid explosive, and to monitoring the casting, cooling, and curing processes. Mold making is also a challenging task, requiring considerable experimentation at Los Alamos before an acceptable product could be made.
Pressing is a traditional way of manufacturing explosives products, but its inability to make complex shapes, and problems with density variations and voids prevented its use during WWII. Plastic explosives (that is - soft, pliable explosives) can be pressed into uniform complex shapes quite easily, but their lack of strength make them unattractive in practical weapon designs.
During the forties and fifties advances in polymer technology led to the creation of PBXs (plastic bonded explosives). These explosives use a polymer binder that sets during or after fabrication to make a rigid mass. The first PBX was developed at Los Alamos in 1947, an RDX-polystyrene formulation later designated PBX 9205. Some early work used epoxy binders that harden after fabrication through chemical reactions, but current plastic binders are thermosetting resins (possibly in combination with a plasticizer). Explosive granules are coated with the plastic binder and formed by pressing, usually followed by machining of the billet.
The desire for maximum explosive energy has led to the selection of polymers and plasticizers that actively participate in the explosion, releasing energy through chemical reactions. Emphasis on this has led to undesirable side effects - like sensitization of the main explosive (as occurred with PBX-9404), or poor stability. In the 1970s the W-68 warhead, the comprising large part of the U.S. submarine warhead inventory, developed problems due to decomposition of the LX-09 PBX being used, requiring the rebuilding of 3,200 warheads. LX-09 also exhibited sensitivity problems similar to PBX-9404, in 1977 three men were killed at the Pantex plant in Amarillo from a LX-09 billet explosion.
Normally the explosive and polymer binder are processed together to form a granulated material called a molding powder. This powder is formed using hot pressing - either isostatic (hydrostatic) or hydraulic presses, using evaluated molds (1 mm pressure is typical). The formed material may represent the final component, but normally additional machining to final specifications is required.
PBXs contain a higher proportion of the desired explosive, possess greater structural strength, and also don't melt. These last two properties make them easier to machine to final dimensions. Plastic bonding is very important in insensitive high explosives (IHEs), since mixing the insensitive explosives with the more sensitive TNT would defeat the purpose of using them.
PBX was first used in a full-scale nuclear detonation during the Redwing Blackfoot shot in June 1956. PBXs have replaced melt castable explosives in all US weapons. The PBX compositions that have been used by the U.S. include PBX-9404, PBX-9010, PBX-9011, PBX-9501, LX-04, LX-07, LX-09, LX-10, LX-11. Insensitive PBXs used are PBX-9502 and LX-17.
Table 4.1.6.2.2.5-1. Basic Properties Of Explosives Used In Us Nuclear Weapons
EXPLOSIVE DETONATION DENSITY SENSITIVITY
VELOCITY PRESSURE
m/sec kilobars
HMX 9110 390 1.89/pressed Moderate
LX-10 8820 375 1.86/pressed Moderate
LX-09 8810 377 1.84/pressed Moderate
PBX-9404 8800 375 1.84/pressed Moderate
RDX 8700 338 1.77/pressed Moderate
PETN 8260 335 1.76/pressed High
Cyclotol 8035 - 1.71/cast Low
Comp B 63/36 7920 295 1.72/cast Low
TATB 7760 291 1.88/pressed Very Low
PBX-9502 7720 - 1.90/pressed Very Low
DATB 7520 259 1.79/pressed Low
HNS 7000 200 1.70/pressed Low
TNT 6640 210 1.56/cast Low
Baratol 76/24 4870 140 2.55/cast Moderate
Boracitol 60/40 4860 - 1.55/cast Low
Plumbatol 70/30 4850 - 2.89/cast Moderate
4.1.6.2.2.6 Detonation Systems
Creating a symmetric implosion wave requires close synchronization in firing the detonators. Tolerances on the order of 100 nanoseconds are required.
Conventional detonators rely on electrically heating a wire, which causes a small quantity of a sensitive primary explosive to detonate (lead azide, mercury fulminate, etc.). The primary usually then initiates a secondary explosive, like PETN or tetryl, which fires the main charge.
The process of resistively heating the wire, followed by heat conduction to the primary explosive until it reaches detonation temperature requires a few milliseconds, with correspondingly large timing errors. Conventional detonators thus lack the necessary precision for firing an implosion system.
One approach to reducing the duration of action of the detonator is to send a sudden, powerful surge of current through a very fine wire (made of gold or platinum), heating it to the point of vaporization. This technique, called an exploding wire or exploding bridge wire (EBW) detonator, was invented by Luis Alvarez at Los Alamos during the Manhattan Project. Current surge rise times of a fraction of a microsecond are feasible, with a spread in detonation times of a few nanoseconds.
An exploding wire detonator can be used to initiate a primary explosive (usually lead azide), as in a conventional detonator. But if the current surge is energetic enough, then the exploding wire can directly initiate a less sensitive booster explosive (usually PETN). The advantage of doing this is that the detonation system is extremely safe from accidental activation by heat, stray currents, or static electricity. Only very powerful, very fast current surges can fire the detonators. This type of exploding wire detonator is one of the safest types of detonators known. The disadvantage is the need to supply those very powerful, very fast current surges. A typical EBW requires 5 KV, with a peak current of at least 500-1000 amps. A few kiloamps is more typical of most EBW detonators, but a multi-EBW system would probably try to minimize the required current. With sufficient care in detonator design and construction, inherent detonator accuracies of better than 10 nanoseconds are achievable.
Since WWII, a number of detonator designs based on exploding foils have been developed. Exploding foil detonators could be used to fire the booster explosive directly, as in EBW detonators, but generally this implies the use of different concept called a "slapper" detonator. This idea (developed at Lawrence Livermore) uses the expanding foil plasma to drive another thin foil or plastic film to high velocities, which initiates the explosive by impacting the surface. Normally the driving energy is provided entirely by heating of the foil plasma from the current passing through it, but more sophisticated designs may use a "back strap" to create a magnetic field that drives the plasma forward. Slappers are fairly efficient at converting electrical energy into flyer kinetic energy, it is not hard to achieve 25-30% energy transfer.
A typical slapper detonator consists of an explosive pellet pressed to a high density for maximum strength (plastic bonded explosives can also be used). Next to the exp