Critical mass

A plutonium (simulate) sphere surrounded by blocks of a wolfram carbide neutron reflector. A recreation of a criticality accident occurred in 1945 to measure the radiation produced when an additional reflector block was added, turning to the supercritical mass.

In physics, the critical mass is the minimum amount of material needed to sustain a nuclear chain reaction. The critical mass of a fissile substance depends on its physical (particularly its density) and nuclear (its enrichment and fission cross section) properties, its geometry (its shape) and its purity, as well as whether or not it is surrounded by a reflector. of neutrons. By surrounding a fissile material by a neutron reflector, the critical mass is lower. In the case of a sphere surrounded by a neutron reflector, the critical mass is about fifty-two kilograms for uranium-235 and ten kilograms for plutonium-239.

"Critical" refers to a state of dynamic equilibrium in the chain fission reaction; in it there is no increase in power, temperature and neutron density over time. "Subcritical" refers to the inability to maintain or sustain a nuclear chain reaction over time; by introducing a certain amount of neutrons into a subcritical set, the neutron population will decrease over time (by absorption phenomena in the material or by leakage). "Supercritical" refers to a system in which the number of fission processes per unit time increases to the point where some intrinsic feedback mechanism causes the reactor to break even (go critical) at a higher temperature or power or is destroyed (in which case the critical set is disarmed).

It is possible for a set to reach the critical state at powers very close to zero. If it were possible to do an experiment in which an exact amount of fissile material is added to a slightly subcritical mass, an array with exactly critical mass could be created, and in that case the fission chain reaction would sustain exactly one neutron generation. (since the consumption of the fuel produced by the fission process itself would make the whole subcritical again).

Naked sphere of an isotope

Superior: A fissible material sphere is too small to allow the chain reaction to self-manage, because the neutrons generated by the fission can easily escape from the system. Centre: By increasing the mass of the sphere until reaching the critical mass, the nuclear reaction is self-managed. Inferior: By coating the original dial with a neutron reflector, reaction efficiency is increased and the system is allowed to have a self-sustaining reaction.

The sphere is the shape with the least critical mass. It is possible to reduce the critical mass by surrounding the sphere with a neutron reflecting material, or some other material.

In the case of a bare sphere (without a neutron reflector) the critical mass is more than 50 kg for uranium-235 and 10 kg for plutonium-239.

The following table presents the critical mass of bare spheres of some isotopes with half-lives of more than 100 years.

The critical mass of uranium depends on the degree to which it is present (enriched) in the uranium-235 isotope: for a 20% enrichment of U-235 the critical mass is more than 400 kg; for 15% of U-235, the critical mass exceeds 600 kg.

The critical mass is inversely proportional to the square of the density: if for example the density increases by 1% the critical mass will decrease by 2%, then the volume will be less by 3% and the diameter will be reduced in 1 %. The probability that a neutron per cm traveled collides with a nucleus is proportional to the density, that is, it would increase in our example by 1%, which compensates for the fact that the distance traveled before the neutron leaves the system is 1 % minor. This is something that must be taken into account when making more precise calculations of the critical masses of plutonium isotopes than the indicative values shown in the previous table, because plutonium has a large number of crystalline phases whose densities are highly different from one to another. Yeah.

It is important to note that not all neutrons contribute to the chain reaction. Some escape into the system, and others may suffer radioactive capture.

Let q be the probability that a given neutron causes fission in a nucleus. Considering in a simplified way only the quick or instantaneous neutrons, and if the number of quick neutrons generated in the nuclear fission of an atom is called ν. For example, in the case of uranium-235 ν = 2.7. Therefore the criticality will take place when νq = 1. The dependency with the geometry, the mass and the density is manifested through the factor q.

If the full effective section of interaction is considered σ σ {displaystyle sigma } (usually measured in varn), then the medium free path of a neutron is soon expressed as l l − − 1=nσ σ {displaystyle ell ^{-1}=nsigma } where n{displaystyle n} is the density of atomic number. Most interactions are clashes with change of direction and power of the neutron, so that a neutron will travel a random path until it escapes from the medium in which a fission reaction is found or produced. To the extent that the other mechanisms for the disappearance of neutrons are not significant, then the radius of the critical mass sphere can be calculated in an approximate way as the product of the medium free path l l {displaystyle ell } and the square root of one plus the amount of collision events per fision event (what we call s{displaystyle s}), since the net distance traveled on a random route is proportional to the square root of the amount of steps taken:

Rc l l s snσ σ {displaystyle R_{c}simeq ell {sqrt {s}}}simeq {frac {sqrt {s}{nsigma }}}}}}}}}

Where, you need to remember again, this is just a rough estimate.

Criticality can be expressed as a function of the total mass M, the nuclear mass m, the density ρ, and a factor f that includes geometric and other effects such as:

1=fσ σ msρ ρ 2/3M1/3{displaystyle 1={frac {fsigma }{m{sqrt {s}}}}}{rho ^{2/3}M^{1/3}}

Where the dependence of the critical mass with the inverse of the square of the density is confirmed as previously mentioned.

Alternatively, this can be expressed more succinctly in terms of the areal density of nuclei Σ:

1=f♫σ σ ms・ ・ {displaystyle 1={frac {f'sigma }{m{sqrt {s}}}}{sigma }

Where the factor f has been rewritten as f' to take into account that the two values may differ depending on geometric effects and how Σ is defined. For example, for a solid bare sphere of Pu-239 the criticality is obtained at 320 kg/m², regardless of the density, and for U-235 at 550 kg/m². In other words, the criticality depends on whether a typical neutron "sees" a certain amount of nuclei around it such that the areal density of nuclei is above a certain threshold value.

This applies in nuclear weapons of the type of implosion, where a spherical mass of fossil material that is significantly less than the critical mass, becomes supercritical by rapidly increasing ρ ρ {displaystyle rho } (and consequently also of ・ ・ {displaystyle sigma }See the next section. Indeed, the most sophisticated nuclear weapons programmes allow the manufacture of functional devices with less material than the one required on the most primitive devices.

Beyond mathematics, this result can be explained by a simple physical analogy. If it is considered the smoke that exposes a diesel engine by the exhaust pipe, initially the smoke seems black, but then gradually it is possible to see through it without drawbacks. This is not because the effective section of scattering of all the particles of hollin, but instead the soot has spread in the air. If considered a transparent cube whose side is L{displaystyle L}, filled with hollin, then the optical depth of this medium is inversely proportional to the square of L{displaystyle L}, and therefore proportional to the surface density of the soot particles: it is possible to see through this imaginary cube only with making the cube greater (without varying the total amount of soot contained in it).

Several uncertainties contribute to the determination of a precise value of the critical masses, including (1) precise knowledge of the effective sections, (2) calculation of geometric effects. It is this last problem that gave a strong impetus to the development of the Monte Carlo method in computational physics by Nicholas Metropolis and Stanislaw Ulam. Indeed, even for a solid homogeneous sphere, the exact calculation is not at all trivial. Finally, it is worth noting that the calculation could be carried out if an approximation of a continuous medium for the transport of neutrons were assumed, in such a way that the problem is reduced to a diffusion problem. However, since the typical dimensions of the problem are not significantly longer than the mean free path, such an approximation is of little practical use.

Finally, it should be noted that for certain ideal geometries, the critical mass from a formal point of view could be infinite, and then other parameters are used to describe the criticality. For example, if we consider the case of an infinite plate of fissile material. For every finite thickness, this corresponds to an infinite mass. However, criticality is only achievable when the thickness of the plate exceeds a critical value.

Explosive Design

If two pieces of subcritical material are not assembled fast enough, a nuclear pre-detonation may occur, in which case a very small explosion will separate the parts.

An atomic bomb must be stored in a subcritical configuration until the moment it is to be detonated. In the case of a uranium bomb, it is enough to keep the fuel in the form of separate pieces, the dimension of each of which is less than the critical size either because they are very small or because their shapes prevent reaching criticality. To produce the detonation, the uranium parts are quickly put together. In Little Boy, this was done by shooting a small piece of uranium from a gun-type barrel into a corresponding hole in a larger piece of uranium, a design known as a gun-type fission bomb.

It would also be possible to build an explosive from Pu-239 with a theoretical purity of 100%. But in reality this is not practical because the Pu-239 "armament quality" it is contaminated with small amounts of Pu-240, which has a strong tendency towards spontaneous fission. For this reason, in a revolver-type weapon, a nuclear reaction would occur before the masses of plutonium were in the proper position to produce a magnitude explosion. Even taking into account the impurity in Pu-240, a revolver-type weapon could in principle be built. However, it would not be a very practical weapon, since it would have to be very long to accelerate the mass of plutonium to very high speeds to compensate for the effects mentioned above.

That is why another method is used. The plutonium is placed in the form of a subcritical sphere (or another shape), which may or may not be hollow. Detonation is produced by exploding a charge of a conventional explosive surrounding the sphere, thereby increasing its density (and collapsing the internal cavity if there is one) to produce a configuration that is supercritical. This method is called the implosion weapon.