Simulating a nuclear chain Reaction with Unity3d

A virtual recreation of the "Little Boy" atomic bomb

Introduction

This article was inspired by the article series "Computer Recreations" authored by A.K. Dewdney and published in the Scientific American magazine. In one of the articles Dewdney describes a simple 2d simulation of a nuclear chain reaction. in the original simulation atoms were placed on a small 2d grid. A decay was simulated by approximating neutron paths with lines and using simple geometry to compute collisions with other atoms which then would be destroyed whilst emitting more neutrons. The process was repeated until no further collisions took place.

I will use a similar approach to simulate the uncontrolled chain reaction taking place during the ignition phase of a gun-type atomic bomb with the Unity 3D game engine. The simulation is shown below. In this article I will briefly explain the underlying physics and how i translated some aspects of "Little Boy" into models suitable for a simplified simulation.

What was "Little Boy"?

"Little Boy" was the code name for the type of atomic bomb dropped on the city of Hiroshima on 6 August 1945. You can learn more about the devastating effects of the device in the corresponding Wikipedia page. This article is not about the morality or the human cost of using such a weapon it is an educational article about understanding the basic physical principles behind it.

Fission of Uranium-235

"Little Boy" derived its explosive power from an uncontrolled chain reaction of Uranium 235. A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place. Lets start by looking at the nuclear fission of a single Uranium-235 atom.

\begin{equation} ^{235}_{92}U + ^1_0n \rightarrow ^{236}_{92}U^* \xrightarrow{85\%\ nuclear\ fission} \left\{ \begin{array} \\ ^{144}_{56}Ba + ^{89}_{36}Kr + 3^1_0n \\ ^{144}_{55}Cs + ^{90}_{37}Rb + 2^1_0n \\ ^{140}_{54}Xe + ^{94}_{38}Sr + 2^1_0n \\ ^{146}_{57}La + ^{87}_{35}Br + 3^1_0n \\ ^{160}_{62}Sm + ^{72}_{30}Zn + 4^1_0n \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ ... \end{array} \right. \end{equation}

If an Uranium-235 core is hit by a neutron Uranium-236 is created. In about 85 % of the cases this isotope will fission. One of many different fission reaction will follow with each fission reaction releasing a number of additional neutrons as well as Energy. The average number of released neutrons per fission is 2.45 [1]. The neutrons can then cause new fission reactions in other Uranium-235 atoms thus triggering a nuclear chain reaction. If enough Uranium-235 atoms are present the reaction is self sustaining. The mass required to create a self sustaining reaction is called the critical mass. An example for such an uncontrolled nuclear chain reaction is the explosion of an atomic bomb.

For the simulation we will simplify the model by assuming only the following fission reaction will occur:

\begin{equation} ^{235}_{92}U + ^1_0n \rightarrow ^{144}_{56}Ba + ^{89}_{36}Kr + 3^1_0n \end{equation}

An uranium-235 atom hit by a neutron will fission into Barium, Krypton, 3 Neutrons and Energy. The neutrons released from a fission reaction have a mean energy of 2 MeV which puts them at a velocity of 20000 km/h. Those neutrons may then cause more fission reactions thus eventually triggering a nuclear chain reaction.

Four red spheres arranged on the edges of a cube — Atoms are modeled as spheres.

Neutron beams emitting from the spheres — Emitted neutrons are represented by green and blue cylinders pointing in precomputed, randomized directions.

For the simulation an Uranium-235 atom is modeled as a sphere with a collider element. The simulated bomb will consist of approximately 3000 atoms. Discrete Neutrons will not be modeled because in comparison with the atom they travel with an incredibly high speed. Simulating their movement would cause numerical problems for the detection of Atom/Neutron collisions. Instead the entire neutron path is simulated at once. Therefore neutrons are modeled as small cylinders with capsule collider elements protruding from the atom in random directions. The directions are randomized and precomputed. The cylinders represent the directions in which the neutrons would travel after a fission reaction. The third neutron is not modeled because in the simulation a neutron beam will cause all atoms it touches to fission. It will never be "captured" by the atom. Since the initiating neutron is never captured it kind of acts as the third emitted neutron.

Neutron beams will only be activated once the atoms are hit by a neutron beam themself. The atom is destroyed afterwards.

Triggering the chain reaction

To start a nuclear chain free neutrons are needed. Where are they coming from? In an atomic bomb there are two relevant sources of neutrons.

Neutrons created by spontaneous fission of uranium
Modulated neutron initiators. This are devices designed for creating a burst of neutrons.

Spontaneous Fission of Uranium-235

One source of neutrons is the Uranium-235 itself as it is not a stable element. It has a half-life of 704 million years. Most of the decay will happen in a decay chain called the Actinium series whilst emitting alpha and beta radiation but no neutrons. There is however a small chance that uranium-235 will spontaneously fission. Spontaneous fission of uranium-235 occurs with a tiny probability of 2e-7 percent of all uranium-235 decays and is producing 1.86 neutrons per fission on average.

As soon as the simulation is started each atom may be subject to premature demise by spontaneous fission. The two neutron beams of the atom will be then activated and the atom is removed from the simulation.

Neutron Initiators

Another source of neutrons is a modulated neutron initiator. Initiators in "Little Boy" used a Beryllium/polonium-210 reaction to create a burst of neutrons. A neutron producing reaction is started when the two elements are mixed. In the simulation an initiator is modeled as a cylinder with a number of neutron beams radiating outwards.

Putting it all together

"Little Boy" was a gun-type design atomic bomb. The idea of such a device is that two sub-critical masses of Uranium-235 will be brought together by shooting a projectile element into a target element. The projectile was shaped like a hollow cylinder with the target element fitting right into the projectile cylinder.

The device was constructed in a way that the projectile was stopped as soon as the assembly was complete [3]. The assembly would then contain more than the critical mass and even a single spontaneous fission would trigger the nuclear chain reaction. With such a mass of Uranium a spontaneous fission would be guaranteed to happen within milliseconds. Little boy contained a total mass of 64 kg of Uranium-235 which is more that the critical mass of 52 kg.

To make sure a chain reaction would commence four additional neutron initiators were places on the tamper plug where the projectile cylinder would hit them. Due to the design of "Little Boy" the initiators were not required for the ignition of the device.

Running the Simulation

It turns out you can observe behavior that was of concern when designing the real "Little Boy" bomb in this simplified simulation.

The simulation will result in an uncontrolled chain reaction (if the parameters are right)
Sometimes the bomb will not ignite properly. A stray neutron may trigger a chain reaction before the critical mass is assembled fully. This was the reason why another gun-type design based on Plutonium ("Thin Man") was never built.
The simulated bomb will ignite even if you disable the initiators.
Not all atoms will undergo fission. According to estimates fewer than 2% of the Uranium in "Little Boy" actually underwent fission.

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