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Page Two - Nuclear
Weapons
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4.1 Elements of Fission Weapon
Design
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Trident Costs Soar |
| Date posted: 2008-06-10 14:30:23 | Article posted by: colin |
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Maintaining and developing the Trident nuclear warheads
stationed on the Clyde is going to cost the British taxpayer a
massive £18.5 billion over 13 years, according to the first
official breakdown of defence nuclear spending.
New figures released by the UK Government after pressure from
MPs reveal that £12.7bn of public money has been spent on
nuclear weapons over the last 10 years. A further £5.8bn is
planned to be spent over the next three years.
The amount of cash being poured into the UK's weapons of mass
destruction is also steadily increasing, from £1.1bn in 2003-04
to a projected £2.1bn in 2010-11.. A raft of new high-tech
facilities is being built at the nuclear bomb factories at
Aldermaston and Burghfield in Berkshire.
Up to 200 nuclear warheads are being stored behind barbed wire
and watchtowers at the Royal Navy Armaments Depot at Coulport,
on Loch Long. As many as 48 at a time are taken to sea from
Faslane eight miles away on Gare Loch by one of four Trident
submarines.
A plan to replace Trident over the next 20 years was agreed by
the former Prime Minister Tony Blair, and backed by MPs in
London last year, despite a major Labour revolt. The plan has
been pursued by Prime Minister Gordon Brown.
The new spending figures emerged in response to questions in the
House of Commons by the Liberal Democrat's defence spokesman,
Nick Harvey MP. "The true cost of the Government's premature
decision on Trident replacement is now coming to light," .
"Britain is spending ever larger amounts of money on the nuclear
deterrent at a time when our troops are struggling with aged
vehicles and a dire shortage of helicopters in Afghanistan. One
has to question the priorities of the Ministry of Defence."
Harvey called on the Prime Minister to "take a lead" by pushing
for nuclear disarmament at key international talks in 2010.
"Instead he is quietly pouring billions into financing a
replacement for Trident," said Harvey.
Scottish anti-nuclear campaigners also attacked the huge amounts
of money devoted to nuclear weapons.
"The Ministry of Defence has cast a small light on the hidden
world of nuclear weapons' expenditure," said John Ainslie, the
co-ordinator of the Scottish Campaign for Nuclear Disarmament
(CND).
"Gordon Brown is doubling the amount spent each year on
Britain's weapons of mass destruction. The Prime Minister should
listen to the people, Parliament and Government of Scotland. He
should scrap the replacement of Trident and put this money to
better use."
The Scottish Parliament has voted against the UK Government's
plans to replace the submarine-launched Trident nuclear weapons
system. The Scottish Government has set up a Working Group aimed
at finding ways of getting rid of nuclear weapons.
On Saturday the Scottish CND is organising a protest outside the
Faslane naval nuclear base near Helensburgh. The event is timed
to mark the 40th anniversary of the first Polaris nuclear
submarine patrol from Faslane, and the 26th birthday of the
Faslane peace camp.
"On 14 June CND supporters will gather outside Faslane to form a
peace chain' of people and banners around the nuclear base,"
said Ainslie.
"It is time we put an end to squandering billions of pounds on
nuclear weapons."
Good reports
have emerged on questions surrounding the W76 warhead
and the Life Extension Programme. It is clear that the
most important impact that this will have is further
supporting the case for the Reliable Replacement
Warhead.
|
 In the line of
fire: conventional weapons combined with radioactive components could
make a lethal combination
Looking down the
barrel: so far, sufficient nuclear technology and know-how have eluded
jihadists
(© Science Photo Library / Van Parys Media)
4.1.1 Dimensional and Temporal
Scale Factors
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:
- the neutron is "absorbed" by the
collision; and
- zero or more neutrons are
emitted.
If the interaction is ordinary
neutron capture, then no neutron is emitted from the collision. If the
interaction is a scattering event, then one neutron is emitted. If the
interaction is a fission event, then the average number of neutrons
produced per fission is emitted (this average number is often designated
by 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. 4.1.1-3
MFP = 1/(cross_total * N)
where N is the number of atoms per
unit volume, determined by the density.
In computing the effective
reactivity of a system we must also take into account the rate at which
neutrons are lost by escape from the system. This rate is measured by the
number of neutrons lost per collision. For a given geometry, the rate is
determined by the size of the system in MFPs. Put another way, for a given
geometry and degree of reactivity, the size of the system as measured in
MFPs, is determined only by the parameter c. The higher the value of c,
the smaller the assembly can be.
An indication of the effect of c on
the size of a critical assembly can be gained by the following table of
critical radii (in MFPs) for bare (unreflected) spheres:
Table 4.1.1-1. Critical Radius (r_c) vs Number of Secondaries (c )
c value r_c
(crit. radius in MFP)
1.0 infinite
1.02 12.027
1.05 7.277
1.10 4.873
1.20 3.172
1.40 1.985
1.60 1.476
If the composition, geometry, and
reactivity of a system are specified then the size of a system in MFPs is
fixed. From Eq. 4.1.1-3 we can see that the physical size or scale of the
system (measured in centimeters, say) is inversely proportional to its
density. Since the mass of the system is equal to volume*density, and
volume varies with the cube of the radius, we can immediately derive the
following scaling law:
Eq. 4.1.1-4
mcrit_c = mcrit_0/(rho/rho_0)^2 = mcrit_0/C^2
That is, the critical mass of a
system is inversely proportional to the square of the density. C is the
degree of compression (density ratio). This scaling law applies to bare
cores, it also applies cores with a surrounding reflector, if the
reflector is density has an identical degree of compression. This is
usually not the case in real weapon designs, a higher degree of
compression generally being achieved in the core than in the reflector.
An approximate relationship for
this is:
Eq. 4.1.1-5
mcrit_c = mcrit_0/(C_c^1.2 * C_r^0.8)
where 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:
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.
In the table below I give some
illustrative values of c, total cross section, total mean free path
lengths for the principal fissionable materials (at 1 MeV), and the alphas
at maximum uncompressed densities. Compression to above normal density
(achievable factors range up to 3 or so in weapons) reduce the MFPs,
alphas (and the physical dimensions of the system) proportionately.
Table 4.1.2-1 Fissile Material Properties
Isotope c cross_total total_MFP density alpha t_double
(barns) (cm) (1/microsec) (nanosec)
U-233 1.43 6.5 3.15 18.9 273 2.54
U-235 1.27 6.8 3.04 18.9 178 3.90
Pu-239 1.40 7.9 2.54 19.8 315 2.20
Values of c and total MFP can be
easily calculated for mixtures of materials as well. In real fission
weapons (unboosted) effective values for alpha are typically in the range
25-250 (doubling times of 2.8 to 28 nanoseconds).
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. Godiva consisted of oralloy (93.71
wt% U-235, 5.24 wt% U-238, 1.05 wt% U-234), Jezebel of weapon-grade
delta-phase plutonium alloy (94.134 wt% Pu-239, 4.848 wt% Pu-240, 1.018
wt% gallium)
Table 4.1.2-2 Properties of Bare Critical Metal Assemblies
Mass, Density, and Measured Alpha are at Delayed Critical (D.C.)
Assembly Material Mass Density Meas. Alpha Del. Neutron Calc. Alpha
Name kg (1/microsec) Fraction (1/microsec)
Godiva Oralloy 52.25 18.71 -1.35 0.0068 199
Jezebel WG-Pu 16.45 15.818 -0.66 0.0023 287
The calculated values of alpha from
the Godiva and Jezebel experiments are reasonably close to those
calculated above from 1 MeV cross section data. Adjusting for density, we
get 270/microsecond for U-235 (1 Mev data) vs 199/microsecond
(experimental), and for plutonium 252/microsecond vs 287/microsecond.
The effective value of alpha (the
actual multiplication rate), taking into account neutron leakage, varies
with the size of the system. If the system radius R = r_c, then it is
exactly one critical mass (m = M_crit), and alpha is zero. The more
critical masses present, the closer alpha comes to the limiting value.
This can be estimated from the relation:
Eq. 4.1.2-1
alpha_eff = alpha_max*[1 - (r_c/R)^2]
= alpha_max*[1 - (M_crit/m)^(2/3)]
Notice that using the two tables
above we can immediately estimate the critical mass of a bare plutonium
sphere:
mass_crit = [(2*1.985*2.54 cm)^3]*(Pi/6)*19.8 g/cm^3 = 10,600 grams The
published figure is usually given as 10.5 kg.
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 * density
The outward force exerted on the
shell is determined by the pressure difference across the shell and the
shell area:
F = dP/dR * thickness * area
From 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:
- uniform expansion of a core; and
- surface escape from a core
initially at constant pressure.
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:
- Reactivity deinsertion is
complete when the rarefaction wave reaches the critical radius of the
core;
- The value of alpha does not
change until the rarefaction wave reaches the critical radius, then it
goes to zero;
- The temperature is uniform
through the core, and no energy is lost.
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).
Figure 4.1.5.3-1. Elapsed
Multiplication Intervals Vs Implosion Time
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.
Figure 4.1.5.3-2. Implosion
Distance and Expansion Distance Plotted Against Implosion Time
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 neutron
source consisted of about 60 g of Pu-240, which produced an average of one
fission every 37 microseconds. 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:
- Whether it is subsonic or
supersonic; and
- Number of geometric axes along
which compression occurs.
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.
Looking at U-235 critical mass
tables for various candidate reflectors we can estimate the achievable
critical mass ratios taking into account differential reflector
efficiency. A steel gun barrel is actually a fairly good neutron
reflector, but it will be thinner and less effective than the target
reflector. M_c for U-235 (93.5% enrichment) reflected by 10.16 cm of
tungsten carbide (the reflector material used in Little Boy) is 16.5 kg,
when reflected by 5.08 cm of iron it is 29.3 kg (the steel gun barrel of
Little Boy was an average of 6 cm thick). This is a ratio of 1.78, and is
probably close to the achievable limit (a beryllium reflector might push
it to 2). 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.
Using the M_c for 93.5% enriched
U-235, the ratio M/M_c for Little Boy was (64 kg)/(16.5 kg) = 3.88, well
within the limit of 4.51 (ignoring the hard-to-estimate effects of the
boron abosrbers). It appears then that the Little Boy design (completed
some six months before the required enriched uranium was available) was
developed with the use of >90% enrichment uranium in mind. The actual
fissile load used in the weapon was only 80% enriched however, with a
corresponding WC reflected critical mass of 26.5 kg, providing an actual
ratio of 64/26.5 = 2.4.
The mass-dependent efficiency
equation shows that it is desirable to assembly as many critical masses as
possible. Applying this equation to Little Boy (and ignoring the
equation's limitations in the very low yield range) we can examine the
effect of varying the amount of fissile material present:
1.05 80 kg
1.1 1.2 tons
1.2 17 tons
1.3 78 tons
1.4 220 tons
1.5 490 tons
1.6 930 tons
1.8 2.5 kt
2.0 5.2 kt
2.25 10.5 kt
2.40 15.0 kt LITTLE BOY
2.5 18.6 kt
2.75 29.6 kt
3.0 44 kt
3.1
If its fissile content had been
increased by a mere 25%, its yield would have tripled.
The explosive efficiency of Little
Boy was 0.23 kt/kg of fissile material (1.3%), compared to 2.8 kt/kg (16%)
for Fat Man (both are adjusted to account for the yield contribution from
tamper fast fission). Use of 93.5% U-235 would have at least doubled
Little Boy yield and efficiency, but it would still have remained
disappointing compared to the yields achievable using implosion and the
same quantity of fissile material.
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 64 kg of uranium in Little Boy
had an isotopic purity of about 80% U-235. The 12.8 kg of U-238 and U-234
produced a neutron background of around 1 fission/14 milliseconds, giving
Little Boy a predetonation probability of 8-9%. In contrast to the Fat Man
bomb, predetonation in a Little Boy type bomb would result in a negligible
yield in nearly every case.
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 shock wave generator: the
high explosive system that generates an initial shock wave of the
appropriate shape;
- The implosion hardware: the
system of inert materials that is driven by the shock wave, which
consists of the nuclear explosive materials, plus any tampers,
reflectors, pushers, etc. that may be included.
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:
- through work done in compressing
the shocked material,
- by adding kinetic energy to the
material (accelerating it), and
- by increasing t
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