A
critical mass is the smallest amount of
fissile material needed for a sustained
nuclear chain reaction. The critical
mass of a fissionable material depends upon its
nuclear properties (e.g. the
nuclear fission crosssection), its
density, its
shape, its
enrichment, its purity, its
temperature, and its surroundings.
Explanation of criticality
The term
critical refers to an equilibrium fission
reaction (steadystate or continuous chain reaction); this is where
there is no increase or decrease in power, temperature, or neutron
population.
A numerical measure of a critical mass is dependent on the neutron
multiplication factor,
k, where:
 k = f − l
where
f is the average number of neutrons released
per fission event and
l is the average number of
neutrons lost by either leaving the system or being captured in a
nonfission event. When
k = 1 the mass is
critical.
A
subcritical mass is a mass of fissile material
that does not have the ability to sustain a fission reaction. A
population of neutrons introduced to a subcritical assembly will
exponentially decrease, typically rapidly. In this case,
k
1. A steady rate of spontaneous fissions causes a
proportional steady level of neutron activity. The constant of
proportionality increases as k increases.
A
supercritical mass is one where there is an
increasing rate of fission. The material may settle into
equilibrium (i. e. become critical again) at an elevated
temperature/power level or destroy itself (disassembly is an
equilibrium state). In the case of supercriticality,
k >
1.
Changing the point of criticality
The point, and therefore the mass, where criticality occurs may be
changed by modifying certain attributes, such as fuel, shape,
temperature, density, and the installation of a neutronreflective
substance. These attributes have complex interactions and
interdependencies, this section explains only the simplest ideal
cases.
 Varying the amount of fuel
It is possible for a fuel assembly to be critical at near zero
power. If the perfect quantity of fuel were added to a slightly
subcritical mass to create an "exactly critical mass", fission
would be selfsustaining for one neutron generation (fuel
consumption makes the assembly subcritical).
If the perfect quantity of fuel were added to a slightly
subcritical mass, to create a barely supercritical mass, the
temperature of the assembly would increase to an initial maximum
(for example: 1
K above the ambient
temperature) and then decrease back to room temperature after a
period of time, because fuel consumed during fission brings the
assembly back to subcriticality once again.
A mass may be exactly critical, but not a perfect homogeneous
sphere. Changing the shape to be closer to a perfect sphere will
make the mass supercritical. Conversely, changing the shape to be
further from a sphere will decrease its reactivity, making it
subcritical.
A mass may be exactly critical at a particular temperature. Fission
and absorption crosssections decrease with the inverse of relative
neutron velocity. As fuel temperature increases, neutrons of a
given energy appear faster and thus fission/absorption is less
likely. This is not unrelated to doppler broadening of the U238
resonances, but is common to all fuels/absorbers/configurations.
Neglecting the very important resonances, the total neutron cross
section of every material exhibits an inverse relationship with
relative neutron velocity. Hot fuel is always less reactive than
cold fuel (over/under moderation in LWR is a different topic).
Thermal expansion associated with temperature rise also contributes
a negative coefficient of reactivity, since fuel atoms are farther
apart. A mass that is exactly critical at room temperature would be
subcritical in an environment anywhere above room temperature due
to thermal expansion alone.
 Varying the density of the mass
The higher the density, the lower the critical mass. The density of
a material at a constant temperature can be changed by varying the
pressure or tension or by changing crystal structure (see
Allotropes of plutonium). An ideal
mass will become subcritical if allowed to expand or conversely the
same mass will become supercritical if compressed. Changing the
temperature may also change the density, however the effect on
critical mass is then complicated by the temperature effects (See
Changing the temperature), and by whether the material expands or
contracts with increased temperature. Assuming the material expands
with temperature (enriched Uranium 235 at room temperature for
example), at an exactly critical state, it will become subcritical
if warmed, hence lower density, or conversely the same mass will
become supercritical if cooled, hence higher density. Such a
material is said to have a negative temperature coefficient of
reactivity to indicate that its reactivity decreases when its
temperature increases. Using such a material as fuel means fission
reduces as the fuel temperature increases.
 Use of a neutron reflector
Surrounding a spherical critical mass with a
neutron reflector
further reduces the mass needed for criticality. A common material
for a neutron reflector is
beryllium
metal. This reduces the number of neutrons which escape the fissile
material, resulting in increased reactivity.
In a bomb, a dense shell of material surrounding the fissile core
will contain, via inertia, the expanding fissioning material.This
increases the efficiency.Because a bomb relies on fast neutrons
(not ones moderated by reflection with light elements, as in a
reactor) the tamper in a bombis not functioning as a neutron
reflector. Also, if the tamper is (e.g. depleted) uranium, it can
fission due to the high energyneutrons generated by the primary
explosion. This can greatly increase yield, especially if even more
neutrons are generated byfusing hydrogen isotopes, in a socalled
boosted configuration.
Critical mass of a bare sphere
The shape with minimal critical mass and the smallest physical
dimensions is a sphere.Baresphere critical masses at normal
density of some
actinides are listed in
the following table.
The critical mass for lowergrade uranium depends strongly on the
grade: with 20% U235 it is over 400 kg; with 15% U235, it is
well over 600 kg.
The critical mass is inversely proportional to the square of the
density: if the density is 1% more and the mass 2% less, then the
volume is 3% less and the diameter 1% less. The probability for a
neutron per cm travelled to hit a nucleus is proportional to the
density, so 1% more, which compensates that the distance travelled
before leaving the system is 1% less. This is something that must
be taken into consideration when attempting more precise estimates
of critical masses of plutonium isotopes than the rough values
given above, because plutonium metal has a large number of
different crystal phases which can have widely varying
densities.
Note that not all neutrons contribute to the chain reaction. Some
escape. Others undergo
radiative
capture.
Let q denote the probability that a given neutron induces fission
in a nucleus. Let us consider only
prompt
neutrons, and let \nu denote the number of prompt neutrons
generated in a nuclear fission. For example, \nu \simeq 2.5 for
uranium235. Then, criticality occurs when \nu q = 1 . The
dependence of this upon geometry, mass, and density appears through
the factor q .
Given a total interaction
cross
section \sigma (typically measured in
barns), the
mean free
path of a prompt neutron is \ell^{1} = n \sigma where n is the
nuclear number density. Most interactions are scattering events, so
that a given neutron obeys a
random walk
until it either escapes from the medium or causes a fission
reaction. So long as other loss mechanisms are not significant,
then, the radius of a spherical critical mass is rather roughly
given by the product of the mean free path \ell and the square root
of one plus the number of scattering events per fission event (call
this s ), since the net distance travelled in a random walk is
proportional to the square root of the number of steps:
R_c \simeq \ell \sqrt{s} \simeq \frac{\sqrt{s}}{n \sigma}
Note again, however, that this is only a rough estimate.
In terms of the total mass M , the nuclear mass m , the density
\rho , and a fudge factor f which takes into account geometrical
and other effects, criticality corresponds to
1 = \frac{f \sigma}{m \sqrt{s}} \rho^{2/3} M^{1/3}
which clearly recovers the aforementioned result that critical mass
depends inversely on the square of the density.
Alternatively, one may restate this more succinctly in terms of the
areal density of mass, \Sigma :
1 = \frac{f' \sigma}{m \sqrt{s}} \Sigma
where the factor f has been rewritten as f' to account for the fact
that the two values may differ depending upon geometrical effects
and how one defines \Sigma . For example, for a bare solid sphere
of Pu239 criticality is at 320 kg/m², regardless of density,
and for U235 at 550 kg/m².In any case, criticality then
depends upon a typical neutron "seeing" an amount of nuclei around
it such that the areal density of nuclei exceeds a certain
threshold.
This is applied in implosiontype nuclear weapons, where a
spherical mass of fissile material that is substantially less than
a critical mass, is made supercritical by very rapidly increasing
\rho (and thus \Sigma as well), see below. Indeed, sophisticated
nuclear weapons programs can make a functional device from less
material than more primitive weapons programs require.
Aside from the math, there is a simple physical analog that helps
explain this result. Consider diesel fumes belched from an exhaust
pipe. Initially the fumes appear black, then gradually you are able
to see through them without any trouble. This is not because the
total scattering cross section of all the soot particles has
changed, but because the soot has dispersed. If we consider a
transparent cube of length L on a side, filled with soot, then the
optical depth of this medium is
inversely proportional to the square of L , and therefore
proportional to the areal density of soot particles: we can make it
easier to see through the imaginary cube just by making the cube
larger.
Several uncertainties contribute to the determination of a precise
value for critical masses, including (1) detailed knowledge of
cross sections, (2) calculation of geometric effects. This latter
problem provided significant motivation for the development of the
Monte Carlo method in
computational physics by
Nicholas
Metropolis and
Stanislaw Ulam. In
fact, even for a homogeneous solid sphere, the exact calculation is
by no means trivial. Finally note that the calculation can also be
performed by assuming a continuum approximation for the neutron
transport, so that the problem reduces to a diffusion problem.
However, as the typical linear dimensions are not significantly
larger than the mean free path, such an approximation is only
marginally applicable.
Finally, note that for some idealized geometries, the critical mass
might formally be infinite, and other parameters are used to
describe criticality. For example, consider an infinite sheet of
fissionable material. For any finite thickness, this corresponds to
an infinite mass. However, criticality is only achieved once the
thickness of this slab exceeds a critical value.
Criticality in nuclear weapon design
Until detonation is desired, a
nuclear
weapon must be kept
subcritical. In the case
of a uranium bomb, this can be achieved by keeping the fuel in a
number of separate pieces, each below the
critical size either because they are too
small or unfavourably shaped. To produce detonation, the uranium is
brought together rapidly.
In Little Boy, this was achieved by firing a piece of uranium (a
'doughnut'), down a gun barrel onto
another piece, (a 'spike'), a design referred to as a guntype fission
weapon.
A theoretical 100% pure Pu239 weapon could also be constructed as
a guntype weapon. In reality, this is impractical because even
"weapons grade" Pu239 is contaminated with a small amount of
Pu240, which has a strong propensity toward spontaneous fission.
Because of this, a reasonably sized guntype weapon would suffer
nuclear reaction before the masses of plutonium would be in a
position for a fullfledged explosion to occur.
Instead, the plutonium is present as a subcritical sphere (or other
shape), which may or may not be hollow. Detonation is produced by
exploding a
shaped charge surrounding
the sphere, increasing the density (and collapsing the cavity, if
present) to produce a
prompt
critical configuration. This is known as an
implosion type
weapon.
See also
References
 Nuclear Weapons Design & Materials,
The Nuclear Threat
Initiative website.
 Final Report, Evaluation of nuclear criticality
safety data and limits for actinides in transport, Republic of
France, INSTITUT DE RADIOPROTECTION ET DE SÛRETÉ NUCLÉAIRE,
DÉPARTEMENT DE PRÉVENTION ET D'ÉTUDE DES ACCIDENTS.
 Chapter 5, Troubles tomorrow? Separated Neptunium 237 and
Americium, Challenges of Fissile Material Control (1999),
isisonline.org

http://www.lanl.gov/news/index.php?fuseaction=home.story&story_id=1348
 Updated Critical Mass Estimates for
Plutonium238, U.S. Department of Energy: Office of Scientific
& Technical Information
 Amory B. Lovins, Nuclear weapons and powerreactor plutonium,
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 Hirshi Okuno and Hirumitsu Kawasaki, Technical Report, Critical and Subcritical Mass
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