Thursday, February 5, 2015

Nuclear Weapons: Understanding Nuclear Binding Energy

Nuclear weapons rely on either a nuclear fusion reaction or a combination of two reactions: nuclear fusion and nuclear fission. Used in bombs, both reactions transform a small amount of mass into a tremendous amount of destructive energy - blast, heat, light and radiation. These are weapons of mass destruction, designed to kill as many people as possible. One nuclear bomb can devastate an entire city. To date only two cities have suffered such a fate.

In 1945, the nuclear bombings of Hiroshima and Nagasaki killed over one hundred thousand people and ended World War ll. If you are interested in an in-depth account based on the recollections of the survivors and the Americans involved, I recommend the HBO documentary called White Light & Black Rain: The Destruction of Hiroshima & Nagasaki, 86 minutes long, released in 2007, available as a DVD at amazon.com. You can also watch it online at cultureunplugged.com. I found it very powerful and offer my thoughts on it: It is one thing to understand intellectually the destruction of a nuclear weapon but it is completely different to really take that information in in a human way. The survivors in this documentary generously offer us a very rare and deeply personal description of what a nuclear bomb does. You might notice as I did that many of the Americans, those involved in caring out the attacks, have an emotional distance. Like most scientists, I was trained to be dispassionate.  I can understand the distance, as I've felt it myself every time I euthanized an animal for research. Now at little older, and hopefully wiser, I look back at that as a risky emotional defence mechanism. As my reader I ask you to form your own opinion. My reason for writing about nuclear weapon technology is pretty simple - an informed opinion is better than an uniformed one.

Rather than deciding, based on the utter devastation, never to use nuclear weaponry again, this marked the beginning of the Cold War between the U.S. and the Soviet Union, both of which amassed huge arsenals of various kinds of nuclear weapons, and have undertaken many test explosions over the decades since.

The Cold War has abated, despite recent Russia - U.S. tensions concerning Russia and Ukraine. Both Russia and the U.S. have cut down on nuclear weapon spending and they have reduced their nuclear arsenals. Russia has begun environmental cleanup of old testing sites. However, neither country has embraced full nuclear disarmament as of yet, and the threat of thousands of aging nuclear devices remains in both countries. For example, chemical high explosives that are used to condense the fissile material in the bombs can chemically degrade over time, becoming less stable. Electronic components suffer from decay. Even the isotopes themselves used in the weapons may be chemically unstable. There are increasing maintenance and security issues over aging nuclear facilities and weapon silos. The youngest nuclear weapon developed by the U.S. is now over 22 years old.

Now we all live in an uneasy world where many countries have active nuclear weapon programs. The world map below represents nuclear weapon development status as of 2007, by colour.


Here is a full list of countries with nuclear weapons, provided by Wikipedia.

In the previous article, Chemical Explosions Versus Nuclear Explosions, we explored how nuclear reactions release energy. In this article, we focus once again on the science of nuclear weapons. Here we will explore why and how certain atoms release energy by diving even deeper into the mystery of nuclear binding energy. Then, in the next article we will explore how nuclear weapon design makes use of the release of nuclear binding energy.

Nuclear Binding Energy = Energy Released During Fission and Fusion

Both fission (breaking apart) and fusion (putting together) reactions release energy from atoms. As we learned in the previous article, a significant contribution to the mass of either a proton or neutron (together they are called nucleons) is in the form of the strong force. A residuum of that strong force holds these nucleons together inside the nucleus. The energy that represents this residual force is called nuclear binding energy. Some of this energy becomes available during a nuclear explosion.

How nuclear binding energy works might seem a bit counter-intuitive at first. We learned in the previous article that mass and energy are equivalent. When an atom undergoes fusion or fission, the total mass of the resulting components is less than the starting mass of the atom. That mass is lost as energy.

We also know that the strong force contributes mass to the proton, so we might expect that its residuum, nuclear binding energy, should contribute mass to the atomic nucleus as well. Therefore, we expect that the mass of the nucleus should be higher than the mass of the individual nucleons that make it up.

However, when we add up the individual masses of the protons and neutrons inside a nucleus, their mass is MORE, not less, than the mass of the nucleus itself. The mass of the nucleus is lower than the mass of its constituent parts. We can take, for example, the helium nucleus, which consists of two neutrons and two protons, shown below right as a very simple atomic diagram.

As we did in the previous article, we will once again use atomic mass units, u.

The mass of one helium nucleus is 4.00153 u.

Two protons = 2 x 1.00728 u.
Two neutrons = 2 x 1.00866 u.

This adds up to 4.03188 u, which is 0.0304 u more than the nucleus mass.

When nucleons combine to form a helium nucleus where does this mass go? This "missing mass" is called the mass defect.

Mass Defect

We can accurately say that the disappearing mass goes straight into energy, but why it does so is not quite as straightforward as it first seems. As we might suspect, mass defect has everything to do with Einstein's mass-energy equivalence. However, unlike the case with proton mass described in the previous article, binding energy does not make a mass contribution to the nucleus. It makes a mass reduction instead.

To understand this, we must switch mental gears a bit and look at it from a different angle. It is most useful to think about this in terms of the potential energy of a system. The nucleus is in a lower potential energy state than the otherwise free nucleons would be in. This is reflected in the lower mass/energy of the bound nucleus state. Mass and energy are not just equivalent; they are the same. Anytime a system settles into a lower energy state, the mass of that system decreases. Even everyday objects follow this rule, though the difference in mass in these case is so small it would be practically impossible to calculate. A compressed spring, for example, has more mass than an uncompressed spring. In the same way, an unbound free nucleon has more mass than it does when it's bound inside a helium nucleus.

Why doesn't a similar mass reduction (mass defect) occur when quarks combine into protons or neutrons? The difference is that the quarks retain their intrinsic kinetic energy inside the proton; they do not find themselves in a lower-energy state.

The Residual Strong Force and the Weak Force Stabilize Nuclei

Considering that in a nucleus we have mutually repulsive protons squeezed in along with neutrons, it seems strange that all this repulsion doesn't translate into a higher (less stable) potential energy state. It does, but the strong force compensates for this electrostatic repulsion. Both neutrons and protons "feel" the powerful residual strong force as an attractive force. It binds them tightly together and stabilizes their arrangement. However, it acts only at very close range and rapidly loses power with distance. Large nuclei must contend with this loss of influence. The extent to which nucleons are stabilized depends entirely on the individual nucleus - how many nucleons are present and whether the number of protons and neutrons is balanced or not. There is great variability in the stability of different atomic nuclei. Some nuclei are incredibly stable. Others last less than microseconds before they fly apart. In these cases, the disruptive energy of the electrostatic forces wins out over the attractive strong force.

Another fundamental force called the weak force contributes to the stabilization of the atomic nucleus but in perhaps a more indirect way. This is the force that mediates beta plus and beta minus decay, described in the previous article. Whereas the strong force is carried out through the exchange of gluons and the electromagnetic force is carried out through the exchange of photons, the weak force is carried out by the exchange of W and Z bosons. It has an even shorter range of action than the strong force - far less than the diameter of a proton but, unlike the strong and electromagnetic forces, it does not have a binding energy - there is no push or pull associated with it. Unstable (radioactive) nuclei, with an excess of protons or neutrons, can stabilize through decay into more stable nuclei containing a better balance of nucleons.

Through beta minus decay, neutrons are converted into protons by emitting an electron and an antineutrino. Neutrons are slightly more massive than protons by an equivalent of approximately 2.5 electrons. The mass in excess of the electron produced is converted into the kinetic energy of the particles. Through beta plus decay, protons are converted into neutrons by emitting a positron and a neutrino. This decay occurs only inside the nucleus and only if there is sufficient nuclear binding energy available to emit a positron. A more common route, though also rare, through which a proton-rich nucleus converts protons into neutrons, is through electron capture. The process is shown in the diagram below left.



This is the decay route undertaken when there is not enough energy to create a positron. Here, a proton captures one of the atom's electrons and becomes a neutron, emitting a neutrino (not shown) in the process. An outer shell electron then transitions downward to fill the now-empty electron shell, shedding its extra energy by emitting an X-ray photon in the process. Less often, an electron (called an Auger electron) is ejected from the atom. Other electrons might also emit photons as well and the nucleus itself, now in an excited state, might emit a gamma ray (not shown), as the atom adjusts back to ground (lowest energy) state.

Though more economical than beta plus decay from an energy standpoint, it is the more difficult maneuver because even the innermost (K-layer) electrons are still very far from the nucleus where the very short-range weak force originates. Therefore it is not easy for the weak force to carry out the electron capture.

This decay mode reveals an important difference between the strong force and the weak force. The strong force acts only between quarks, binding them together. The weak force acts between both quarks and leptons, such as electrons. The exchange of W and Z bosons changes one flavour of quark into another flavour. By doing so, the weak force changes a proton into a neutron and vice versa (both particles are composites of three specifically flavoured quarks). W+, W- and Z bosons, although they are known as virtual particles, have mass. The W+ boson decays into a positron and an electron neutrino during beta plus decay. The W- boson decays into an electron and an electron antineutrino in beta minus decay.

Factors That Affect the Stability of Nuclei

a) Size

Unlike the strong force that acts between the quarks inside protons and neutrons, the residual strong force (nuclear binding energy) decreases very rapidly with distance. It is maximally powerful at a distance of about 1 fentometer (fm; 10-15 m). This is a bit less than the diameter of a proton. By a distance of about 2.5 fm, however, it is approaching insignificance. A nucleus ranges in size from 1.75 fm (hydrogen nucleus) to about 15 fm for the heaviest atomic nuclei. Within large nuclei, the binding energy is weak near the periphery of the nucleus where the disruptive electrostatic repulsive force is still very much a factor. For this reason, there is a size limit to stable nuclei. The largest stable nucleus is lead, Pb. All four of its stable isotopes have 82 protons.

b) Balance of Nucleons

Neutrons stabilize nuclei because they attract both each other as well as protons through the residual strong force. Their presence also pushes the positively charged protons slightly apart, reducing the force of their destabilizing repulsive interactions. Electrostatic force decreases exponentially with distance. For this reason, any nucleus with more than one proton must also have one or more neutrons to stabilize it.

Light stable elements, those below atomic number, Z = 20 (Calcium-40), tend to have an equal number of protons and neutrons in their nuclei (Z = N). There are a few exceptions such as helium-3, hydrogen-1 and carbon-13. As the number of protons in the nucleus increases beyond this, the ratio of neutrons to protons increases in order to stabilize the nucleus. For example, the four stable lead isotopes have 82 protons and a substantial excess of 122 (lead-204), 124 (lead-206), 125 (lead-207) or 126 (lead-208) neutrons. The graph below demonstrates this trend. N (neutron number) is plotted against Z (atomic number; number of protons).
Beginning at N = 20, stable isotopes begin to deviate from Z = N as the ratio of neutrons to protons in the nucleus increases (the belt of stability deviates upward in other words). As we move on to heavier elements, those with a growing excess of neutrons, there is also a trend toward an increasing number of stable isotopes per element. Tin (Sn), has the highest number of stable isotopes of all at 10. The atomic number 50 is one of several magic numbers in nuclear physics, numbers which afford unusual stability to the nucleus.

c) Magic Numbers

There are several numbers that bestow extra stability to the nucleus - 2, 8, 20, 28, 50, 82 and 126. These are magic numbers. In "magic" atomic nuclei, nucleons can be arranged into complete shells within the nucleus. This is reminiscent of the atomic shell model where filled electrons shells afford the atom greater chemical stability. Filled nuclear shells afford higher binding energy and, therefore, greater stability to the nucleus of the atom. Protons or neutrons tend to fill shells going outward from the center of the nucleus. A completely filled outermost shell has more binding energy than a partially filled one because it contributes a number of stabilizing quantum mechanical effects (some of which act a bit like harmonic frequencies), to the arrangement. The shells for protons and neutrons are independent of one another so you can have a magic number for one type of nucleon or the other. You can also have a magic number for both types of nucleon within one nucleus. This is a doubly magic nucleus. Examples of double magic nuclei are helium-4 (2 each protons and neutrons), oxygen-16 (8 of each), calcium-40 (20 of each) nickel-48 (28 protons, 20 neutrons), calcium-48, nickel-78. Lead-208, the heaviest stable nucleus, has a magic number of protons (82) and a magic number of neutrons (126).

Calcium-48 is very neutron-rich for a light element, and that would normally make it radioactive, but its doubly magic neutron (28) and proton (20) shells make it stable. Similarly, nickel-48 is very proton-rich (28 protons, 20 neutrons) and it shouldn't be stable either, but it too has doubly magic numbers (two filled nuclear shells).

Helium-4 has the most unique stability of all the elements. It not only has a doubly magic nucleus (2 protons + 2 neutrons), but it has a filled electron shell (2s) as well. The stability of its nucleus means that helium nuclei, as alpha particles, are easily created during both fusion (of hydrogen nuclei) and fission reactions (as alpha emission). It is one of the most abundant elements in the universe. Its ease of creation left very few free neutrons to create larger elements during the first few minutes after the Big Bang. Its extremely low energy (therefore highly stable) electron cloud also means that it is chemically inert and has the lowest melting and boiling points of all the elements. It is a perfectly symmetrical stable atom, in which its electrons, protons and neutrons all fill orbitals or shells as pairs. Their intrinsic spins cancel each other out perfectly so there is no net orbital angular momentum to the atom, which creates a system with very low potential energy and very high stability. Helium, however, does not have the most tightly bound nucleus, iron-56 does.

It might seem a bit of a puzzle why some atoms fuse (as in hydrogen bombs) and others split apart (as in fission weapons), and yet tremendous energy can be released in both types of reaction.

Trends in Nuclear Binding Energy

Like chemical reactions, nuclear reactions can be exothermic (they release energy) or endothermic (they absorb energy). Whether a particular reaction is endothermic or exothermic depends on the difference in overall binding energy between the reactants and the products. An exothermic reaction releases products that have more tightly bound nucleons. This state of lower potential mass/energy translates into less mass per nucleon in these products. This is the mass that is lost in the reaction, and it is equivalent to the energy output we calculated in the previous article.

Fusion can be exothermic or endothermic depending on which nuclei fuse. The same goes for fission reactions. Even an exothermic nuclear reaction, where energy is released and a more stable system with lower potential energy is reached, is not necessarily a spontaneous one.

Below is a binding energy curve graph. Binding energy per nucleon is plotted against the number of nucleons in the nucleus.


There is an interesting trend in this graph. Up to iron-56, binding energy per nucleon increases as the number of nucleons increases. Helium's unusually high binding energy is an exception to the pattern, evidenced by the upward spike at the left.

After iron-56, the trend reverses; binding energy decreases with number of nucleons. As nucleons are added to a nucleus, each one makes a contribution to the binding energy, so total binding energy increases. However, as nucleons are added, the nucleus itself takes up more and more physical space. Because the residual strong force decreases sharply with distance, a point is reached where nucleon number versus nuclear size achieves a maximum possible total nuclear binding energy, and that point is reached at iron-56. Beyond this, the stabilizing effect of adding extra nucleons becomes more and more outweighed by the electrostatically disruptive effect of protons that are too far away from the influence of the residual strong force. The nucleus grows less and less stable as its physical size increases. Binding energy per nucleon decreases until the nucleus becomes too large to be stable. Uranium isotopes, all large nuclei with 92 protons, are unstable, or radioactive. Even here, however, one can see the stabilizing effect of adding extra neutrons. Uranium-238 with 146 neutrons and 92 protons is quite stable with a half-life as long as the Earth is old, even though it is a huge unwieldy atom.

The stability of nuclei smaller than iron-56 increases with size because, smaller nuclei are well within the range of the residual strong force and there are fewer disruptive protons present, so the progressive filling of nuclear shells is a significant stabilizing influence.

Iron-56 has the highest binding energy per nucleon of all the isotopes. It therefore has the lowest mass per nucleon. It sits at the apex of the nuclear binding mountain with the most tightly bound nucleus of all. Exothermic reactions are possible on either slope of the mountain. Nuclei smaller than iron-56 can undergo fusion in order to release energy and end up with products with more tightly bound nucleons. Nuclei heavier than iron-56 can undergo fission, which also releases energy and also creates more tightly bound nuclei products.

The heaviest element produced in stars (through fusion) is iron-56. Beyond this nuclear size, fusion is an endothermic reaction, which means that energy is absorbed rather than released. Therefore, self-sustaining fusion is no longer possible. The star loses energy. The outer layers no longer supported by radiation pressure, begin to collapse very rapidly onto the core, which is incompressible. This creates a shockwave that expands outward, blowing outer stellar material into space and leaving behind a neutron-dense core (neutron star) or, if the star was extremely massive, a black hole. This explosion is a supernova and it is the only mechanism (with the exception of the s-process in very massive stars) through which nuclei heavier than iron can be produced. Even though these fusion processes (fusion of two large nuclei) are endothermic, there is enough pressure and density in the shock wave to induce them. However, the primary method through which large nuclei are created happens through the r-process, where large nuclei are made by the addition of one neutron at a time (which may later decay into a proton). During rapid compression of the core, many free neutrons are created through the electron capture of protons. These free neutrons are then captured into neutron-rich seed nuclei, such as iron nuclei. All done in a matter of seconds, this neutron capture process leads to new nuclei, which are very large and mostly very unstable. They decay through many different decay processes into heavy stable nuclei. As they cool, they capture electrons and heavy new atoms are born. Through this process, even a collapsing star made purely of hydrogen and helium can create an abundance of assorted heavy atoms.

While endothermic nuclear reactions, which go against the grain of the binding energy curve, are possible, they are very rare and only occur when there is a vast input of energy available. Physicists tend to simplify matters by assuming we are going with the grain of the binding energy curve. In this case, fission and fusion reactions are exothermic - they release energy and create more stable tightly bound nuclei, but this does not mean that they are all spontaneous.

Requirements for Fusion To Take Place

A balloon filled with hydrogen molecules does not spontaneously undergo fusion into more stable helium atoms, even though that reaction is energetically favourable - it would take the system to a lower-energy, more stable, state. The more formal way of saying this is that this exothermic reaction involves a decrease in the enthalpy and increase in entropy of the system. Under normal conditions (of pressure and temperature such as inside a balloon), hydrogen atoms have no interest in fusing. While two hydrogen (H-1) atoms will spontaneously combine chemically into molecules (H2), the two nuclei stay intact and apart during the process because the two positively charged protons repel one another electrostatically. This is called the Coulomb force. In order to fuse, the repulsive Coulomb barrier must be overcome. The trick is to increase the kinetic energy of the atoms. They will move faster and bang into each other with greater force. Two nuclei must crash into each other with enough force to overcome the repulsion and come close enough together so that the strong force can exert its influence. If the walls of the balloon were instead the crushing pressure in the interior of the Sun, two things would happen to the hydrogen atoms. First, they would have too much kinetic energy to hold onto the their electrons so they would exist in an ionized or plasma state. Second, they would spontaneously fuse together into helium nuclei, releasing tremendous energy (increasing the pressure of a system increases the temperature (kinetic energy) of the system).

The rate of fusion depends on the relative velocities of the reacting nuclei in a system. For any two individual nuclei we can get a measure of the probability of them fusing as a function of their relative velocity. This is called the reaction cross section. Individual hydrogen nuclei in any system will have a range of possible velocities. In other words they have a distribution of velocities or a thermal distribution (temperature is the average kinetic energy of a system). If we perform an average over (distribution of cross section x velocity), we get an average called the reactivity. The reaction rate (fusions per volume per time) is a product of the reactivity x reactant density. Below left, fusion reactivity is plotted against temperature for three common fusion reactions: deuterium (H-2) - tritium (H-3), deuterium - deuterium, and deuterium - helium-3.

Dstrozzi;Wikipedia
As we can see, reactivity (and reaction rate as well, assuming constant density) increases rapidly as temperature increases and then plateaus at around 10 billion Kelvin (close to 10 billion°C), which is about 3 times hotter than the core of a very massive star just before it collapses. Reactivity (the chance of fusion happening) is basically zero at room temperature but at temperatures of around 100,000°C (10-2 billion K), it starts to become significant. This tells us that fusion is spontaneous, but it is temperature-dependent.

The Sun's core, where an ongoing proton-proton chain fusion reaction takes place, is about 15 million°C (about 1.5 x 10-2 billion K). This chain reaction is shown below right.

Borb;Wikipedia

Looking at the top and going down, hydrogen-1 fuses into deuterium, where one proton decays into a neutron through beta plus decay. A deuterium nucleus fuses with another hydrogen-1 to create helium-3. Two helium-3 fuse into helium-4, ejecting two hydrogen-1 nuclei in the process. Our Sun is a fairly small star, so the fusion reaction stops at helium-4. In larger stars the pressure and temperature is higher so fusion continues, using the CNO (carbon/nitrogen/oxygen) cycle.

We already know that fusion bombs release far more energy than fission bombs. This is because fusion reactions have a far higher energy density than fission reactions but, interestingly, individual fission reactions are far more energetic than individual fusion reactions. That being said, both types of reactions are vastly more energetic than an individual chemical reaction.

Requirements For Fission To Take Place

Fission is an exothermic reaction when a nucleus larger than iron-56 splits into products that have higher binding energy than the reactant nucleus has. Most fission reactions are binary, producing two nuclei of similar mass. There are two common situations in which fission reactions take place. One situation is the man-made nuclear reaction, either as a nuclear fission weapon or as a nuclear reactor. Both fission reactions must be induced by a collision with the reactant nucleus and a free neutron. The other situation is spontaneous fission, where no free neutron is required to start off the reaction. Spontaneous fission occurs naturally in large unstable isotopes, such as uranium-235. In fact, uranium is the only radioactive element found in significant quantity in nature. Most uranium is found as the isotope U-238 but U-235 and U-234 are also present. U-238 is the most stable and that is why it is most abundant. It has a half-life of about 4.5 billion years, equal to the age of Earth itself. U-235 has a half-life of about 700 million years and U-234, a decay product of U-238 and by far the rarest of the three, has a relatively short half-life of around 250,000 years.

U-235 is of special interest to the nuclear energy and weaponry industries because of the three isotopes, it is the only naturally occurring fissile isotope. This means it is the only isotope that can sustain an induced fission chain reaction. There is an important distinction to be made here. All the uranium isotopes can be made to fission. A high-energy or fast neutron will induce any of them to fission. They are all fissionable. A fissile isotope, however, is a subset of fissionable isotopes. Uranium-235, for example, can be made to fission with a much less energetic slow, or thermal, neutron. It can also be induced into sustained fission, whereas the other two isotopes, once induced to fission, will not sustain fission because the neutrons they emit do not have enough energy to induce further fissions.

The fission of just one atom of U-235 releases an astounding 83.14 TJ/kg! That is about three times the maximum yield (25 TJ/kg) of a fusion bomb, but remember that fission bombs have a much lower energy density so the yield of fission bombs is actually much lower than fusion bombs. When a U-235 nucleus is struck by an energetic neutron, one of many different fission reactions is possible.

An example is shown below. U-235 is struck by a neutron (light blue circle) to create a more neutron-heavy U-236 nucleus.


Uranium-236 has a half-life of 23 million years and it is not fissile. It is considered a nuisance isotope in spent nuclear fuel. However, this atom is now in an excited state, having absorbed the kinetic energy as well as the binding energy of the fast neutron. Its half-life is now "super-boosted" to just 120 nanoseconds. It immediately splits into barium-141 and krypton-92, releasing three new free energetic neutrons ready to continue the chain reaction with three other U-235 nuclei and so on. One or more gamma rays (not shown left) are also released during this reaction.

These fission chain reactions release the energy that drives both nuclear fission bombs and nuclear reactors. The key difference between the two applications is that in a bomb, the chain reaction is uncontrolled. In a tiny fraction of a second all the uranium splits into fission products and an enormous amount of energy is explosively released. In a nuclear reactor, the chain reaction is controlled by inserting control rods into the reactor core. Control rods are made of elements such as boron, cadmium and hafnium, which are especially good at absorbing free neutrons. They have a high capture cross-section in other words. Reducing the density of free neutrons (neutron flux) reduces the number of fission reactions taking place at any given time (the reactivity or reaction rate). There are several types of control and safety measures in a reactor. Inserting and lifting the rods finely controls the overall rate of fission. In order to additionally slow down the rate of reaction, the uranium used in nuclear plants is far less enriched with U-235 than the uranium used in nuclear weapons.

Uranium-235 found naturally in Earths crust is fissile, so why hasn't it all exploded in a nuclear explosion? Every 700 million years, about half of all of Earth's uranium spontaneously decays through a series of beta minus and alpha decays until a stable lead-207 nucleus is created, shown below.

Edgar Boney;Wikipedia

As you can see, no free neutrons are emitted, in contrast to the fission reaction. However, very heavy elements, including uranium-235 and uranium-238, also spontaneously fission, though rarely. This means that a rare free neutron is emitted within natural uranium ore. Couldn't this free neutron start off a chain fission reaction in the fissile uranium-235 component of the ore?

The reason Earth doesn't experience spontaneous fission explosions, is because uranium ore does not have critical mass. At least one neutron from the fission of one U-235 atom must strike another U-235 nucleus in order for a fission reaction to continue in a chain reaction. You need a minimal number of uranium-245 nuclei in close enough proximity to one another in order to accomplish this. To sustain the reaction over more than one generation of fission, there must be no decrease in power or temperature and no decrease in the number of free neutrons available. Several factors come into play when determining what mass is critical (reaction-sustainable) for a particular fissile sample. Nuclear properties such as the fission cross-section, density, shape, enrichment, purity, temperature and surroundings are all crucial factors. A supercritical mass is a mass in which the overall rate of fission increases over time. In a subcritical mass, fission is unsustainable.

All the factors that affect criticality are adjustable, but they interact with each other in often very complex ways, making fission weapon design a delicate matter. Below are some simplified examples of the effects of various factors on critical mass:

Amount of fuel: a mass at exactly critical mass is self-sustaining for exactly one neutron generation. Once achieved, the fuel consumed brings the mass to subcritical and the reaction stops.

Shape: fuel that is shaped into a perfect sphere has the highest reactivity, so a mass that is subcritical in its current shape can be made critical by refining it toward that of a perfect sphere.

Temperature: you might expect an increase in temperature to reduce critical mass. You have faster moving neutrons and should therefore have more powerful decay triggers. Surprisingly, slow (thermal) neutrons make fission much more likely and therefore increase the overall reaction rate. Slow neutrons increase the cross-section of the fissile material (such as U-235). The inverse relationship between cross section and neutron energy is shown below. A higher cross-section (chance of fission) is achieved with slower (lower-energy) neutrons.
There is a naïve way to explain how this works. You can think of the effective size of the neutron as its de Broglie wavelength. The neutron is as wide across as its wavelength. The faster the neutron is (the more kinetic energy it has), the smaller its wavelength becomes, and the less likely it is to hit the target nucleus. It's like trying to hit a snowman with a marble versus a beach ball. This is considered a na?ve explanation because there are many factors involved in the calculation, such as neutron resonances. Nuclear reactor design (thermal neutron reactors in particular) takes advantage of this relationship in order to increase the efficiency of the nuclear reaction. Water molecules are often used as a neutron moderator. Here, the free neutron collides with a proton of near identical mass. A head-on collision will stop the neutron in its tracks but most collisions are glancing blows so they effectively slow the neutrons down. The water reduces the speed of emitted neutrons and increases the probability of fission as a result.

Density of the Mass: this relationship is straightforward. The higher the density, the more neutrons and target nuclei per volume, the lower the critical mass. Therefore, pressurizing a fissile mass reduces its critical mass. Density, however, also introduces a complication to the temperature effect described above. In addition to increasing neutron velocity, heating the mass may also cause it to expand, thereby reducing its density. Thus, reducing the temperature of a fissile material has two independent effects, both of which decrease its critical mass.

Reflector: The use of a neutron reflector also reduces critical mass. A material such as beryllium metal can be used as a reflector because its nuclei tend to scatter neutrons rather than absorb them. Fewer neutrons escape so reactivity increases.

Tamper: In a bomb, for example, a dense shell around the fissile material performs three functions that increase the efficiency of the explosive reaction. First, it momentarily contains or resists the expansion of the fissioning material and therefore increases its density. Second, it acts as neutron reflector especially if it is made of the right material. Third, it slows the neutrons down slightly by absorbing some of their energy when they are reflected (some materials also absorb neutrons better than others). All the neutrons emitted in a fission explosion are fast neutrons, so this slight slowing action is very useful in increasing the cross-section. A possible fourth function: if the tamper is made of depleted (non-fissile) uranium, fast neutron collisions with it will induce some of it to fission by exciting the uranium nuclei, thus increasing the amount of fissioning material.

Conclusion

As you can imagine, nuclear weapon design must take into account these complex interacting factors. In the next article, nuclear weapon design and operation are explored in detail.

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