Criticality widget - Simulation of critical mass, geometry, reflector, multiple masses & delayed neutrons
Published: Aug 10, 2024
Source: Design & implementation by Dr Chris Robbins (Grallator) / Facilitated by Ionactive radiation protection resource
Prelim
We always love presenting new resource at Ionactive, and radiation protection widgets are always a pleasure to release. This new criticality simulation has been created for Ionactive by our long time friend Dr Chris Robbins of Grallator. Criticality of nuclear materials can be explained by fairly simple descriptive principles or by some heavy maths and physics. The goal here was to provide interactive resources that you might experience on a science museum display. So this widget provides visual descriptions of criticality, the effect of increasing or decreasing critical mass, the effective of geometry, the use of a reflector, bringing two sub critical masses close together and the influence of delayed neutrons on duration of power excursions. Have a play with the widget below - supporting notes then follow.
Criticality and delayed neutrons
If you like your maths and physics then please visit the following resource: Criticality and delayed neutrons. If you prefer a simpler description of the outputs supplied by the criticality widget then read on below.
Criticality
Dr Chris Robbins, Grallator
Fissile nuclides have the property that when struck by a neutron they split into two smaller nuclides and, crucially, liberate more neutrons; typically two or three. These liberated neutrons can either be lost from the fissile system by leaking away at the periphery or by being absorbed by nuclides without leading to a fission event, or they can go on to produce another fission event.
When the mass of material is small, the neutrons have a higher probability of escaping than they do of colliding with fissile nuclides and causing a fission event, and the system will be sub-critical.
System is sub-critical
As the mass increases, there is a lower probability that neutrons will travel far enough to escape the system as there's more material to traverse, and a higher probability that a collision that leads to fission will occur. A higher number of fission events will also increase the number of neutrons produced, increasing the scope for fission and further neutron production.
System is still sub-critical but higher probability of a fission
Eventually, a mass is reached where the probabilty for fission equals the probability for loss and the system becomes critical, i.e. it is self-sustaining and maintains a constant fission power output, which may be substantial. Physically, this state is accompanied by a blue flash of light and a sudden increase in heat in the system.
System is now critical (self-sustaining)
If the mass is larger than the critical mass the rate of power output increases exponentially, and the system is said to be in a super-critical state.
System is in a super-critical state
For the cases of being super-critical (and critical), the power output and temperature increase is often enough to disrupt the physical system shutting down the criticality. In the most extreme cases, an explosion occurs leading to rapid disassembly!
Geometry effects
The above examples used a circle (spherical) geometry, which is a minimal surface area object, i.e. it encloses the maximum volume, hence mass, for the minimum surface area. As neutrons escape at the surface, it is reasonable to infer that higher surface area:volume shapes have a higher probability of neutron leakage for a given mass. For example, the rectangle in the image below has 204 mass units, which is just slightly higher than the circular 201 mass units, but is in a sub-critical state because there is a higher probability of neutron leakage due to its increased surface area.
Rectangular geometry - higher probability of neutron leakage due to increased surface area
Adding a reflector
There are some materials that can act as neutron reflectors, i.e. they "bounce" escaping neutrons back into the fissile material. This reduces the neutron loss and lowers the mass at which a system becomes critical. In fact some reflectors also act as moderators, reducing the neutron's energy and making it more likely to cause a fission event. People make quite good moderators and reflectors! In the image below reflected neutrons are shown in red, and the reflector increases the neutron multiplication factor from 0.5 to 0.667.
Using a reflector increases the neutron multiplication factor
With this reflector the critical mass is reduced from 201.062 mass units to 113.097 mass units.
Using the reflector to reduce the critical mass
Geometry effects for multiple items
Sometimes there may be multiple items that individually are sub-critical, but are stored together in a single facility. As each item is sub-critical it by definition does not contain a critical mass of material. However, for multiple items the total mass may be equal to, or even many times more than a critical mass. In such a scenario, it is vitally important that items are stored with sufficient space between them so that neutron leakage from one item does not increase the neutron concentration in another. In the image below, the two sub-critical objects are well spaced and this is the case.
Two sub-critical objects are well spaced
If the two objects are moved closer together, neutrons from one object starts to leak into the neighboring one, shown in red in the image below.
Neutrons from each object leaking into each other
If moved close enough together one or both of the objects may be critical or super-critical.
Critical or super-critical assembly of two objects
How fast does the power increase?
The terms sub-critical, critical and super critical used above refer to how the number of neutrons in a fissile system changes with time. In a sub-critical system they reduce (\(k<1\)), in a critical system they stay constant (\(k=1\)) and in a super-critical system they increase exponentially (\(k>1\)). The super-critical state is the most dangerous as a rapidly increasing power output in a short space of time is often described as an explosion. So how fast does the power increase when a system that is just critical is nudged into being super-critical (which may be caused by adding material, changing geometry, adding a reflector, adding a moderator etc.)?
Case 1 - all neutrons are prompt
When a neutron causes a nuclide to undergo fission the result is two fission product daughters and two or three neutrons. As these neutrons are released immediately the nuclide splits they are called prompt neutrons. These neutrons go on to induce further fission events, and the mean time between neutron production and going on to induce fission is called the mean prompt neutron lifetime. For dense materials such as solid metal the lifetime is very roughly \(10^{-8} s\), while for commercial light water reactors it is roughly \(10^{-4} s\). With a small reactivity increase from \(k=1\) to \(k=1.001\), the system will increase power output by a factor 10 and a factor 100 when the mean neutron lifetime is \(10^{-8} s\) in 23 \(\mu s\) and 46 \(\mu s\) respectively.
These times increase by an order of magnitude as the mean neutron life increases by an order of magnitude, as shown in the table below. However, the timescales are too short for any mitigating action to be taken even for the longest mean neutron lifetime.
| Mean neutron lifetime | x10 power | x100 power |
| \(10^{-8} s\) | 23 \(\mu s\) | 46 \(\mu s\) |
| \(10^{-7} s\) | 230 \(\mu s\) | 460 \(\mu s\) |
| \(10^{-6} s\) | 2.3 \(ms\) | 4.6 \(ms\) |
| \(10^{-5} s\) | 23 \(ms\) | 46 \(ms\) |
| \(10^{-4} s\) | 230 \(ms\) | 460 \(ms\) |
Things get even quicker when a larger reactivity excursion is made, for example with reactivity increase from \(k=1\) to \(k=1.01\), the system will increase power output by a factor 10 and a factor 100 when the mean neutron lifetime is \(10^{-4} s\) in 2.3 \(\mu s\) and 4.6 \(\mu s\) respectively; ten times the reactivity increase reduces the power increase time by a factor of ten.
Case 2 - most neutrons are prompt
In reality, the physics is a little more complicated as not all neutrons are prompt neutrons; a very small fraction of the neutrons are released as a result of decay of the fission products produced when the fissile atom is split. For example, one possible fission product is \(\mathrm{^{87}Br}\), which has a half life of about 55 seconds and which has as one of its decay modes to \(\mathrm{^{87}Kr}\) one which emits a neutron. There are other fission products that also decay by emitting a neutron that have shorter half life values. However, the important point is that these neutrons are produced a long time after the fission event relative to the prompt neutrons, and for this reason they are called delayed neutrons.
The fraction of neutrons that are delayed neutrons is denoted \(\beta\) and is not a lot - values range up to a little over 0.006, i.e. 0.6% of all neutrons are delayed, and its value is generally different for different fissile systems. For small changes in reactivity their delayed nature determines the speed at which power increases. In fact it can dominate so much that the mean prompt neutron lifetime becomes irrelevant. With a small reactivity increase from \(k=1\) to \(k=1.001\), the system will increase power output by a factor 10 and a factor 100 when \(\beta=0.006\) in 140\(s\) and 290\(s\) respectively - many orders of magnitude slower than the case with no delayed neutrons. Such time scales allow ample time for control and mitigation measures to be enacted before a large power excursion occurs.
Power increase duration with 0.6% delayed neutrons
As the reactivity increase becomes larger, the number of prompt neutrons increases which reduces the importance of the delayed neutrons and the time for power increase is reduced. When the new \(k\) value exceeds the value of \(\beta\) there are enough prompt neutrons to sustain criticality without the delayed neutrons. In this case the system is called prompt-critical and the power increases on timescales similar to the case of there being no delayed neutrons.
Prompt-critical system, importance of delayed neutrons significantly reduced (for larger reactivity increases)
The graphs shown so far have all shown exponential behaviour. However, there can be more complex behaviour depending on the system characteristics. For example, when values of \(\beta = 0.06\) and \(\ell = 10^{-4}\), and the instantaneous change in \(k\) is to \(k=1.005\), the power increase shows an initial rapid rise due to the increased prompt neutrons, followed by a slower rise as the effect of delayed neutrons starts to dominate.
Initial rapid power rise (prompt neutrons), then slow rise (delayed neutrons start to dominate)
Zooming in shows this effect in more detail.
Rapid rise following by slower rise
How fast does the power decrease?
The previous cases have all used \(k>1\) to see how power increases. Also of interest is what happens when control is applied to reduce the value of \(k\) in terms of how fast the power output from fission falls. The figure below shows the case when the mean neutron lifetime is \(10^{-8} s\), \(\beta = 0\) and the value of \(k\) is reduced from \(k=1\) to \(k=0.995\). The power drops to 10% of its original value in 4.6 \(\mu s\); systems that have rapid increases in power when super-critical also have rapid reductions in power when they are taken sub-critical.
A power drop to 10% of original value (no delayed neutrons)
Again, delayed neutrons affect the rate of power change. The figure below shows the case when the mean neutron lifetime is \(10^{-4} s\), \(\beta = 0.006\) and the value of \(k\) is reduced from \(k=1\) to \(k=0.995\). The power drops to 10% of its original value in 48 \(s\) in this case. The initial reduction is rapid as prompt neutrons are reduced, however the delayed neutrons then dominate the rate of power reduction.
After initial rapid reduction in power, delayed neutrons dominate further rate of power reduction
Contacts
Dr Chris Robbins can be contacted at this website: Grallator.