Building a Reactor
A nuclear reactor is just a system that can get heavy elements to break apart in an efficient manner. The reactor itself, the nuclear core, is not a complicated mechanical contraption. The core has no moving parts or complex mechanisms. It's just an arrangement of special materials, that permits nuclear reactions to occur. There are moving parts to transfer the heat and control the reactions, but it's basically a pile of bricks.
The goal is to get the fuel to keep splitting apart continuously and self sustainably in what is called a chain reaction. This generates heat that is carried away by a coolant, and which we can use for industrial processes like baking cookies or generating electrical power.
Neutrons as multiplying bullets
To split atoms and release nuclear binding energy, we need to launch neutrons at the fuel atoms. Where do we get these neutrons? We could use an accelerator, but that’s not very practical and poses a few safety issues. The spontaneous fission of U-235 or U-238 could be used. But these don’t happen very often. Indeed, 1 kg of natural Uranium generates just 10 such neutrons per second. Which is not very useful.
Fortunately, every time an isotope fissions, or breaks apart, it releases new neutrons that can cause more fissions. A U-235 fission consumes just 1 neutron while releasing an average of 2.43 fast-moving neutrons, with a slight dependence on the original neutron's energy and the target isotope. This gives us 2.43 neutrons to cause an extra fission reaction, which means we have a 1.43 neutron margin for inefficiencies in the system. We’re going to need that margin because we’re going to lose neutrons when they get absorbed by materials or escape from the core altogether.
To sum up, we will use whatever neutrons available to cause fission reactions in U-235 which release 2-3 neutrons each, which in turn cause fission reactions and produce more neutrons, and so on. We will try to get neutrons to multiply like bacteria, and carefully control the growth or decline in the population to sustain a desired number of fission reactions per unit time.
How much power is being produced? The thermal power produced is proportional to the number of fission reactions which is also proportional to the total number of neutrons at any given time.
Criticality
Critical System
What we’ve been doing here is trying to get a critical system. A critical system is just an arrangement of matter that can support a steady state neutron population. Neutron births are balanced by neutron deaths. Neutron births take place when a U-235 fissions or when some fission fragments decay. Neutrons “die” when they are absorbed, escape the reactor core, or are utilized to cause a fission reaction. A reactor core operating at constant power has a constant neutron population, and it is called “critical.”
At a constant 1 MWth, there are approximately 3.1x1016 fission reactions taking place per second (1 J/s divided by 200 MeV/ U fission). The neutrons causing these reactions move at roughly 2200 m/s and have lifetimes of about 20ms. This 1 MWth reactor would have a standing population of 1.5x1015 neutrons (fissions/sec / lifetime), but probably about 2.5x that to account for neutron absorptions.
We can also find the volumetric neutron density which is the neutrons per cm3 or neutrons per mL. For a low power density like 1 W/cm3, like the MMR, we would have 3x109 neutrons per cm3. For higher power density of a typical LWR, on the order 50 W/cm3, it would be 50x higher. This is not much compared to the nubmer of atoms in a cm3, roughly 1022. So neutrons don't collide with each other, but only with atoms.
Sub-critical System
A sub-critical system has a declining neutron population. Deaths exceed births. A sub-critical core’s power is zero or declining. The core is fizzling out.
Super-critical System
A super-critical system has a growing neutron population. Births exceed deaths. As in many biological growth models, the growth is exponential as each new neutron can give birth to generations of daughter neutrons. And just as in all growth models, the growth does not remain exponential indefinitely as negative feedback mechanisms curb the growth and turn exponential functions into logistic functions or even collapse functions.
For bacteria, the growth limiters are the walls of the petri dish and the food supply. For neutrons, growth is limited by many mechanisms, most importantly rising temperatures which either cause more neutrons to die or force the core to fall apart and become sub-critical - more on that later.
Nuclear reactors carefully take advantage of the limiting mechanisms to control the rate of energy release. On the other hand, the design of nuclear weapons aims to overcome the negative feedback mechanisms to achieve a rapid growth of the fissions and fully utilize the nuclear material in a very short period of time.
Core Build-up
Let’s try to make a core. In each simulation, we’ll seed the sample with a starting population of neutrons and watch them propagate through the core, either achieving stability or going extinct. We are aiming to create a self sustaining neutron population in the core.
The first thing we’ll try to do is use the neutrons naturally emitted by the fuel itself. The fuel is just sitting there and spontaneously fissioning and we’re hoping these neutrons are enough to get a chain reaction going.
There’s a little problem. While there are some fissions, illustrated by the blue colored blips, most of the neutrons are escaping! And on top of that, the neutrons are not always doing what we want. We’d like all the neutrons to just cause fissions in the Uranium atoms. But nuclear reactions are probabilistic processes, and fissioning Uranium atoms is just one of the many possible things a neutron can do. The probability of those things occurring depends on the materials and the energy of the neutrons themselves.
The neutrons emitted from the U-235 spontaneous fissions are fast (high energy 2 MeV), moving at about 5% of the speed of light in random directions. At those speeds, the neutrons are not very likely to cause a fission of U-235 or U-238. Instead of causing a fission reaction, they can be scattered or absorbed. In a scattering reaction, the neutron interacts with an atom to slow down and change direction. In an absorption or capture, the target atom absorbs the neutron. So the neutrons travel around the core hitting atoms and slowing down until they are either absorbed, cause a fission, or escape from the core. Absorptions and escapes reduce the neutron population while fissions will grow the population.
There are lots of things to consider. The nuclear reactions are numerous and probabilistic. To control these reactions, we have to know their probabilities and reaction products as well as the energies and directions of the resulting neutrons. We have to know that U-238 and U-235 have very different reaction probabilities for different neutron energies. For example, U-238 is likely to be fissioned only by fast neutrons while U-235 is far more easily fissioned by slow neutrons. On the other hand, U-238 can absorb a neutron to eventually become Pu-239, which is fissile like U-235, easily fissioning with slow neutrons. In current reactors, the fissioning of U-238 contribute 5-10% of thermal power, and the fissioning of the Pu-239 derived from U-238 in the reactor contributes up to 50% of the total thermal power.
Measuring and predicting the probabilities of different reactions for different isotopes and neutron energies is the crowning achievement of nuclear physics. We use cross sections to describe the probabilities for each reaction, for each isotope or material, and across different neutron energies. We show only the fission, scattering, and absorption cross sections. For a given input neutron with a particular energy, we can read out the probability of a particular reaction occurring.
A cross section is basically the practical interaction area of the atoms when they are bombarded with neutrons. While an atom has a physical cross sectional area of a about 1 barn (10-24 cm2), the somewhat complex nuclear physics of nuclear forces and interactions leads to wide ranging cross sections for various neutron energies interacting with different isotopes.
The way to present cross sections for physical materials utilizes their density and is as a probability per unit length. This is called the macroscopic cross section. We can also consider the probability per unit time, which simply multiples the macroscopic cross section by the velocity, but this is less useful when implementing simulations. We can change the materials below.
Let’s use a neutron cannon to seed the core and observe what happens when we change neutron velocity (energy). We can change it from very slow neutrons with velocities of a few hundred m/s to fast neutrons at a few percent of the speed of light.
We can get a lot more fissions at lower neutron energies. But still, lots of the neutrons die by absorption or by escaping from the core. Also, we don't know how to make those slow neutrons yet.
Size
We’ve got to arrange things to make those fission reactions as likely as possible. So let’s keep adding Uranium by adding more blocks. This way, whatever direction the neutron scatters it will come across more Uranium atoms, and any neutrons that pass through one bit may cause a fission in another bit further along. And we can keep doing that. Pressing the reset button will unleash a burst of initial neutrons at the center of the core to test the conditions.
Enlarging the reactor reduces the leakage of neutrons. This has to do with the surface area to volume ratio. As the radius of the sphere increases, the volume increases faster than its surface area, and a smaller fraction of the core’s neutrons are able to escape. One reason reactors have been designed and built to be extremely large is to reduce this leakage of neutrons. Less leakage and the fuel can be used more efficiently, lowering the overall cost of energy.
Unfortunately, this won’t work even for an infinite amount of natural uranium. The U-238 fission cross section is much lower than the absorption cross section. Also the neutrons generated in the fission are too slow to appreciably tap into the fission cross sections on the far right of the spectrum for U-238. We’re losing too many neutrons from absorptions and the neutrons that aren’t absorbed aren't able to slow down to speeds that are likely to cause fission reactions in the tiny amount of U-235. But we have the tools to fix this and create self-sustaining chain reactions. These tools are enrichment, moderation, and size.
Enrichment
Using only the natural uranium available on Earth today, there’s just no way to get a stable, let alone growing, population of neutrons to sustain nuclear reactions. Natural reactors like those found in the Oklo region of Gabon were active roughly 1.8 billion years ago when the enrichment was close to 4% and could not work using today's natural enrichment level. The problem is the U-238 is absorbing a lot of neutrons and it doesn’t easily fission.
We can manually increase the U-235 content in the fuel by various enrichment processes. This is an extra expense but very doable. The U-235 displaces the less fissile U-238, and means fewer neutrons will be lost to the U-238 and more neutrons will be available to interact with the U-235 to cause fissions. U-235 is far easier to fission than U-238, and can utilize both fast and slow neutrons. Just look at the spectrum.
Moderation
We can also slow the neutrons down, or moderate them, so they are more likely to cause fission reactions in the U-235. The U-235 fission releases neutrons moving at about 200 km/s (fast neutrons 2 MeV), and we need to slow them down to 2 km/s (thermal neutrons .1 eV) so they are more likely to cause fissions in U-235. There are several moderating materials that can slow down neutrons without absorbing them that are also resistant to radiation damage. We’ll just use graphite, the same material Fermi used in the Chicago pile and what high temperature gas-cooled reactors use nowadays.
All materials will slow down neutrons to a certain extent, but we want materials that are as efficient as possible per unit of volume. When neutrons hit heavy atoms like Uranium, it's like a ping pong ball hitting a bowling ball and the velocity change in the neutron is very small.Colliding with light atoms like hydrogen and carbon is like a ping pong ball hitting another ping pong ball, and is far more effective at slowing down neutrons. We also want the moderator to be atomically dense so that the chance of interaction is as high as possible. So hydrogen gas is not as effective as liquid water (H20).
All materials will slow down neutrons to a certain extent, but we want materials that are as efficient as possible per unit of volume. When neutrons hit heavy atoms like Uranium, it's like a ping pong ball hitting a bowling ball, and the velocity change in the neutron is very small. Colliding with light atoms like hydrogen and carbon is like a ping pong ball hitting another ping pong ball, and is far more effective at slowing down neutrons. We also want the moderator to be atomically dense so that the chance of interaction is as high as possible. So hydrogen gas is not as effective as liquid water (H20).
There are a few ways we might add graphite. We can add it to the fuel homogeneously so that the uranium and graphite are smeared together.
But mixing the fuel and moderator into a smeared material is not optimal as the U-238 present in the fuel tends to absorb the neutrons as they are slowing down through the resonance absorption region of the spectrum. It is more effective to keep the fuel and moderator separate, clumping the fuel in chunks. This clumping arrangement, also called heterogeneous, allows neutrons to slow down while traveling between fuel chunks and without getting absorbed by the U-238.
Below, you can see the neutron pass from one material to antoher. This is shown in both the 3d model and the 2 spectrum.
Each of the tools discussed can be tuned to achieve different sized cores that achieve criticality. The nuclear core designer will balance these design parameters to achieve a critical core with desired metrics of size, fuel loading, fuel utilization, and ultimately cost.
Other Considerations
Design Options
There are significant limitations on the materials and geometries available for nuclear reactor environments and there’s not much wiggle room on the materials we can use. Most materials are neutron killers and cannot tolerate any significant radiation damage, high temperatures, or corrosive environments. And you have to keep an eye on the materials activation - which refers to the habit of materials to become radioactive when subject to radiation either by absorbing radioactive substances or by reactions with the radiation.
The research and development required to understand materials for use in the complex nuclear environment is often a decades long endeavor with limited success. The knobs we can play with are the ratios and arrangements of materials, the coolant, the enrichment of the Uranium, and the size of the reactor. Later we will be able to choose the power level.
Excess Criticality
To make the reactions self-sustaining, we need to at least achieve a critical system. But in practice we want a system that can be super-critical by simply moving the control rods out of the reactor. This way, we can change the power level and keep burning uranium over time.
The latter point about burning uranium over time concerns the fact that as we burn the fuel, the core conditions are changing. First, the U-235 content is decreasing over time, reducing criticality. In addition, many of the fission products produced by the nuclear reactions will absorb neutrons, so that over time, the reactor by itself becomes less and less conducive to neutron multiplication. On the other hand, some of the fission or decay products are fissionable materials like U-233 which will help extend the life of the core. To account for all this, we have to make the starting configuration sufficiently super-critical that it can tolerate an increasing number of fission products that poison the neutron environment. How super-critical does it have to be? Enough that we can burn an economical fraction of the fuel without compromising safety.
Nuclear Startup
The spontaneous fission of U-235 is not typically relied upon to start-up up a fresh reactor core. The spontaneous fission half-lives for U-235 and U-238 are 3.5*1017 and 8.4*1015 years and produces an average of 1.86 neutrons per fission. 1kg of natural Uranium can be expected to produce about 14 neutrons per second depending on the enrichment, which can be sufficient to start up a nuclear core. However, in practice the startup of a nuclear reactor is aided by a more emissive neutron starter like Cf-252 or Pu-238 and Be so that operators can be really sure there's enough neutrons to get things going.