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Nuclear reactors depend upon the fact that there are certain chemical elements which, on absorbing one neutron, will release greater than one neutron in response. These elements, known as the fissile elements, include Th-232, U-233, U-235, U-238, and Pu-239. A typical nuclear reactor has fuel made of enriched uranium, which consists of about 2% to 4% U-235 (that is, Uranium with an atomic mass of 235 atomic mass units) for nuclear power reactors and between 20% to 80% U-235 for research reactors, with the remainder being U-238. U-235 will absorb one neutron and undergo nuclear transformation into U-236, which rapidly decays, ejecting fission products and fission neutrons (as well as quite a bit of heat energy). This reaction produces an average of 2.418 neutrons per fission.

So, imagine that we start with one neutron. This one neutron will be defined as being part of the first neutron generation, so the first generation has one neutron in it. This neutron may strike a fissile atom and cause two neutrons to be ejected. This marks the start of the second neutron generation, which contains two neutrons. If these two neutrons each strike a fissile atom, causing both fissile atoms to release two neutrons, then in the third neutron generation, there will be four neutrons.

In the above example, each successive generation increases the number of neutrons by a factor of 2. If generation G contains N neutrons, then generation G+1 will contain N*2 neutrons, while generation G+2 will contain 2*(N*2) neutrons, and so on, such that after X generations have passed, there will be 2x neutrons from one initial neutron. This is known as the neutron multiplication factor, symbolized as k. In this example, k will always be two; however, in reality, there are many factors which change k, such as fuel enrichment and reactor design, which I shall futher explicate below. The more realistic neutron multiplication factor, called the effective multiplication factor, takes these factors into consideration, and is symbolized keff.

keff is defined as:
```	      Number of fission-produced neutrons in generation G
keff	=  ---------------------------------------------------------
Number of fission-produced neutrons in generation G-1```
We can see that whenever keff is greater than 1, more neutrons exist in the current generation than in the preceding one, and that whenever keff is less than 1, fewer neutrons exist in the current generation than in the preceding one, and finally than whenever keff is exactly equal to one, the number of neutrons in the current generation is exactly equal to the number of neutrons in the preceding one. Therefore, when keff is less than one, the reactor is sub-critical; when keff is equal to one, the reactor is critical; and when keff is greater than one, the reactor is super-critical.

Now, to determine how keff might vary, and how it might be controlled, let us trace the life of a single neutron, and define factors to determine the effect that each step will have on keff as a whole.

Fast Fission Factor ε

A neutron is ejected from U-235 following a fission reaction; thus begins our neutron generation. Neutrons released by fission reactions have varying energies. Neutrons with unusually high energy have a probability of being absorbed by U-238 and causing fission, while low energy neutrons can only be absorbed by and cause the fission of U-235. The fast fission factor, symbolized by ε, accounts for those neutrons which are released with great enough energy that they are absorbed by U-238 immediately upon their ejection. Because some small percentage of neutrons have enough energy to fission U-238, this factor is greater than one. The fast fission factor is primarily affected by the concentration of U-238 in the fuel, the geometry of the fuel, and the temperature of the fuel. This factor may be expressed as:

```             Number of neutrons from fast and thermal fission
epsilon = -----------------------------------------------------
Number of neutrons from thermal fission only
```
Fast Non-Leakage Probability Pf

Neutrons with lower energies are absorbed more readily by U-235 and produce more resultant neutrons thereby than higher energy neutrons. Thus, nuclear reactors have components known as moderators, which slow down neutrons in a process known as thermalization. Some fast neutrons, however, will escape from the core before undergoing this process. These neutrons, being no longer in the core, do not participate further in the reaction. Only those neutrons which do not escape continue the process; this is why we are interested in the fast non-leakage probability, symbolized by Pf. This value depends largely upon the surface-to-volume ratio of the reactor. This factor may be expressed as:

```
Number of neutrons that begin thermalization
Pf	= ------------------------------------------------------
Number of neutrons from fast and thermal fission
```

Neutrons which do not escape from the core at this point enter the region of the moderator, which slows down the neutrons through thermalization. The neutrons collide elastically with the atoms of the moderator, losing energy and thus speed in the process. As the neutron slows down, it passes through several energy levels at which absorption by U-238 to produce U-239 is favored. These are known as resonance energy levels, and the process of absorption at these energy levels is called resonance absorption. Only those neutrons which escape resonance absorption may continue in the reaction; hence the resonance escape probability, symbolized p. This factor may be expressed as:

```
Number of neutrons that are successfully thermalized
p	= ------------------------------------------------------
Number of neutrons that begin thermalization
```

As with fast neutrons, a certain percentage of these newly thermalized thermal neutrons will escape from the core without participating in the reaction. The thermal non-leakage probability is symbolized Pth, and is expressed as:

```
Number of neutrons absorbed within the reactor
Pth	= ------------------------------------------------------
Number of neutrons that are successfully thermalized
```

Those neutrons which do not escape from the nuclear reactor are then absorbed within the core. These neutrons can be absorbed by fuel, leading to fission, or by the moderator or poisons, chemical species which absorb neutrons but remain stable; control rods are made of these. Only those neutrons absorbed by fuel can cause the ejection of more neutrons, and thus we have the thermal utilization factor, symbolized f, which is the percentage of neutrons absorbed by fuel. This is expressed as:

```
Number of neutrons absorbed by fuel
f	= ------------------------------------------------------
Number of neutrons absorbed within the reactor
```

Finally, we are left only with those neutrons absorbed by fuel, the only neutrons given the chance to cause a fission reaction. However, not every neutron actually produces fission. Thus, we have the fuel utilization factor, symbolized η, which is the number of fission neutrons produced per thermal neutron capture by fuel. For U-235, this is approximately 2.068. The factor is expressed as:

```
Number of fission neutrons produced
eta	=  ------------------------------------------
Number of neutrons absorbed by fuel
```

We can now redefine keff in terms of these factors as

keff = ε Pf p Pth f η.

As in stoichiometry, all of the units but one cancel out, giving, in effect,

```	      Number of fission produced neutrons in generation G
--------------------------------------------------------
Number of fission produced neutrons in generation G-1
```

as stated above.

Once a reactor has been constructed, the primary methods to control the reaction (by effecting a change upon keff) are to either remove or insert fuel or to remove or insert poisons, such as the control rods. The insertion or removal of fuel strongly influences η and p. The removal or insertion of a control rod most strongly influences f, the thermal utilization factor.

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