Introduction

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Some key safety aspects of Aalo-1, our first product, are:

• Strong negative reactivity feedback
• No means of rapid reactivity insertion
• Natural fission product retention
• Surface area to power ratio sufficient for passive decay heat removal
• Low pressure coolant

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All of the above contribute to implementing three core tenets of reactor safety: redundancy, diversity, and defense-in-depth. Today, we’ll explore just the first point in the list, and will dive into the other topics in later posts. We’ll explain how the thermal vibrations of atoms in a nuclear reactor can naturally control the power level, and how our Aalo-1 reactor design amplifies this effect and achieves a negative feedback loop at all times, similar to the homeostasis mechanism that keeps our bodies at a constant temperature.

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Starting from inherent negative reactivity feedback

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Aalo’s technology promises safety through two key factors. Firstly, the reactor uses uranium zirconium hydride fuel, also known as TRIGA fuel. TRIGAs are research reactors developed by General Atomics in the 1950s. TRIGA fuel has demonstrated its safety for decades in unassuming research settings (nuclear reactors located right on university campuses!), distributed across thirty-four locations in the United States. Due to the use of this special fuel, these reactors can safely take a beating even from inexperienced operators (students). As the fuel gets hotter, the reactor makes less power.

TRIGAs were designed to work with water coolant and fairly low temperature environments, but experiments suggest this fuel works well at higher temperatures, too. Aalo’s approach is to pair this inherently safe fuel with a low pressure, high temperature coolant: sodium. Safety is improved by using a low-pressure coolant and creating a negative feedback loop in the power level with UZrH. No nuclear reactor can safely operate without a negative fuel temperature feedback coefficient, and ours is no exception. It just does so more strongly.

Negative fuel temperature reactivity feedback is a prerequisite for the operation of any reactor. This property applies to any nuclear reactor using enough low-enriched uranium: the Doppler effect in uranium-238 will always decrease reactivity as fuel temperature rises. We call it “Doppler” in analogy to the well-known acoustic effect because the thermally driven vibrations of atoms in the reactor cause them to move with some different relative speed to neutrons in flight. Just like how the pitch of sound shifts when an ambulance drives by, vibrating atoms cause an effective shift in neutron energies. Because uranium-238 atoms are not prone to fission, increased absorption in the 238-isotope compared to 235 slows the chain reaction. This happens because the cross-section spikes in uranium 238 are taller and much narrower than those in uranium-235. Figure 1 compares the two isotopes’ total cross-sections, which is inversely proportional to a neutron's free flight length. High cross sections mean the neutrons don’t travel very far and will collide with a nucleus readily.

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At first glance, the two curves might not look too different. In the spiky region, it’s impossible to tell if the resonances are particularly wider in one nuclide or the other. Now, if we take a closer look, things change a bit.

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In Figure 2, the character of the cross-section becomes clearer. For uranium- 235, the oscillations do not vary the cross-section value as much. On top of that, they are wide enough to overlap with each other partially. The “density of states”, or in other words, the number of excited neutron states for a nucleus, governs the resonant structure. Each spike corresponds to an excited orbital position that a neutron could take. This is exactly the same type of excitation we see in, for example, glow-in-the-dark items. The glow from glow-in-the-dark comes from electrons occupying an excited state, dropping to a lower energy level, and releasing a photon of an energy corresponding to that difference. For neutrons, one might fly through space, occupy an excited state on a nucleus for a bit, then “unstick” from it and leave! The principles are similar and are entirely quantum. This is the quantization effect that gives quantum mechanics its name!

Now, here’s how this comes into play for reactors. If we vibrate an atom, a neutron passing by experiences a different relative speed to the atom than when the atom is stationary. The different energy value corresponding to that different relative speed causes the above cross-section to shift slightly. Pick any point on one of the two curves above to get a feel for this. If you move a little left or a little right from that point, how much does the cross-section change? Figure 3 is a visual guide to how thermal motion effects the cross section. Averaging over thermal motions in all directions, the net effect is to smooth out the cross section curve as temperature rises.

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For uranium-238, you can see it changes very much if near a peak! For uranium-235, while it does change some, it’s not enough to make a discernible difference in reactor operation. The amazing thing about the neutronics simulations we use is that we can analyze how EVERY resonance across nearly four hundred nuclides comes into play in the system's behavior! As you might imagine, it takes a lot of computation to resolve the full behavior of a reactor. Now we can competently ask: what makes Aalo so different? From a Doppler point of view, not so much. It’s just a bit stronger in our reactors than others.

In the present conception of the Aalo-1 reactor, our coolant voiding coefficient is slightly positive if considering a fuel assembly alone. However, this is vastly outweighed by the fuel temperature negative feedback coefficient, which is about 20x stronger. Moreover, the leakage increases accompanied by reduced sodium density partially or completely cancel out the decrease in sodium density.

At this point, to understand how much the coolant voiding effect matters, we should get a sense of scale for reactivity quantities. We’ll provide an intuitive explanation of reactivity, particularly the PCM unit commonly used in reactor physics.

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Building a feel for the scale of reactivity feedback

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If “per cent” refers to one in one hundred i.e. 1e-2, “per cent mille” refers to one in one hundred in one thousand, or 1e-5. This is the unit in reactor physics known as a PCM. Reactivity is the fraction of neutrons produced in the system which are in excess to maintain the chain reaction. For example, if k=2, then reactivity ρ=(k-1)/k=50%. Half of the neutrons produced in a system with k=2 is excess to the steady chain reaction. Naturally, reactivity is a fractional quantity, whether it’s positive or negative. As a consequence, percentages and percent-milles are natural to describe it.

Reactivity is proportional to the time rate of change of the number of free neutrons, but that rate itself is inversely proportional to the prompt neutron lifetime. The neutron lifetime is around a millisecond in the ‘slowest’ of reactors like those with graphite moderator. The situation is a lot jumpier in a fast reactor: on the scale of tens of nanoseconds. This, in effect, can give operators less time to respond. Due to Aalo being a slow neutron reactor, the changes in power are never fast enough to release massive amounts of energy.

For Aalo-1, the fuel’s temperature decreases reactivity at around five PCM per kelvin. On the other hand, increases in coolant temperature decrease its density, and correspondingly, its parasitic neutron absorption increases reactivity by 3/10 of a PCM per kelvin. If the reactor slowly rises in power, the coolant and fuel remain roughly in thermal equilibrium, implying a reactivity removal of around 4.7 PCM per kelvin. Rapid power increases that don’t give the heat time to enter the coolant therefore remove the whole 5 PCM per kelvin.

It's natural to inquire then how much reactivity is too much for the reactor to tolerate. A certain fraction of the neutrons born from fission are released by fission product decay with some delay after fission. This delay can range from microseconds to seconds; reactors would be uncontrollable without delayed neutrons. For systems fueled by uranium-235, that fraction is around 600 PCM. Therefore, any reactivity insertion greater than this threshold quickly shifts the systems response time from the scale of seconds to the scale of microseconds. Fuel might get damaged in a reactor without strong negative feedback loops. Small reactivity insertions are allowable, but any above this threshold, called prompt supercriticality, are intolerable in all but the most robust reactors. Some, such as the TRIGA , can survive reactivity insertions well beyond the prompt criticality threshold because the strongly negative temperature feedback coefficients cancel out the increasing power spike before it deposits too much energy. Our current estimates suggest that Aalo-1 can also safely survive a prompt supercritical reactivity insertion, and we will continue our detailed modeling efforts to confirm this.

The two previously mentioned phenomena, an upper threshold level of reactivity that keeps the reactor responding sluggishly with its power and the Doppler effect, are the two key miracles of nature that allow nuclear power to be controllable. Whoever designed the universe was kind to us: a slightly different set of physical laws could have made uranium-235 have large resonances instead, or no delayed neutrons could exist! In that parallel universe, nuclear power would not be feasible.

There’s one more piece of the puzzle we’re missing. TRIGA reactors add another aspect of reactivity feedback that makes them special, and we’ll explain that now. It’s called the warm neutron effect.

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The warm neutron effect

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The original TRIGA reactors ran on highly enriched uranium. This meant that fuel temperature only had a relatively weaker influence on reactivity if only considering Doppler broadening. Rather than relying on resonant-energy neutrons in the middle of Figure 1 to provide reactivity feedback, TRIGAs rely more on thermal neutrons. These are the ones that would fall into the leftmost part of in Figure 1, i.e. energies around 0.01 eV. Thermal neutrons (and, by extension, thermal reactors) are the namesake of thermal reactors. They’re called thermal because the neutrons maintaining the chain reactors are roughly thermally equilibrated with the materials they’re bouncing around in.

In the thermal range, the effect of temperature on the cross-section is a bit different. The speed of the neutron comes into play in entirely different way from Doppler. At sufficiently low speeds, there are no resonances present. As we mentioned before, thermal neutrons are, roughly speaking, thermally equilibrated in energy with the surrounding materials. Their energies are roughly the same as the energy of thermal vibration of atoms around them. Figure 4 shows what this might look like, with two neutrons bouncing off hydrogen nuclei in a zirconium hydride unit cell.

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Because the uranium-235 cross section in the thermal region increases sharply as the energy becomes low, as shown by Figure 1, increased temperatures tend to cause easier passage of neutrons through the fuel. This is because the higher temperature regions increase the energy of neutrons relative to their energies in cold regions, in a process illustrated by Figure 5. The net effect is that neutrons tend to concentrate in areas of lower temperature, called the “cell effect” in some works studying TRIGA physics.

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In a TRIGA, the warm neutron effect and cell effect decrease reactivity because when the fuel heats up, the neutrons preferentially move to a colder place: the water coolant surrounding the pins. Because so many collisions occur with hydrogen, the hydrogen in the fuel of a TRIGA helps to smack neutrons out of the fuel and have them spend more time in the coolant. This accounts for about 2/3 of the feedback effect in a TRIGA reactor.

Because our coolant is sodium, a material that does not readily collide with thermal neutrons, the neutrons migrate out of the fuel when it heats up. But they cross right through the coolant to the next place they find! As a result, Aalo-1 exhibits only about 60% as much negative feedback effect as a TRIGA, but around twice as much negative feedback as compared to a conventional water-based nuclear reactor. Our reactor exhibits the warm neutron effect, but not the cell effect.

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Conclusion

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We’ve explained how both the Doppler and thermal neutron effects feed into creating negative feedback loops that control reactor power, and how our reactor exhibits a powerful Doppler and warm neutron effect. In contrast to a TRIGA reactor which uses the same type of fuel, our Aalo-1 design does not exhibit the cell effect. However, due to increased resonance absorption, the reactor exhibits approximately doubly strong negative feedback compared to the fuel temperature feedbacks in water-based nuclear reactors in operation across the United States.

In conjunction with our innovatively designed reactor core, these potent negative feedbacks enable straightforward, safe operation of Aalo-1 reactors. Our vision is to usher in the second Atomic Age, and a prerequisite to widespread deployment of nuclear power is an intrinsically safe design. By using a reactor design that naturally controls its power levels, certain classes of accidents are reduced from emergencies to inconveniences.

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