The six factor formula

 "The men American people admire most extravagantly are the most daring liars; the men they detest the most violently are those who try to tell them the truth." - H.L. Mencken

Warning: Semi nerd post. This one is about reactor physics.   

Below: The 250 Megawatt (thermal power) Advanced Test Reactor at the Idaho National Laboratory.

Back in my youth, I had to memorize a formula on how to calculate Keff.  "Keff" is the effective neutron multiplication rate in a reactor core.  After learning the formula, and the meaning of each factor, and passing a test, I forgot about it.  However I was reminded of it recently, when I saw one of the Greek symbols used in a different context.

What's the point of figuring out Keff?   If Keff = 1.0, the reactor has a self-sustaining fission reaction going, and the reactor is "critical".  If Keff < 1.0 reactor power is "sub-critical", meaning that power is diminishing.  If Keff > 1.0, the reactor is "supercritical", meaning that reactor power is increasing.  None of these terms indicate a level of danger - they are strictly information on whether reactor power is going up (supercritical), down (sub-critical), or sideways (critical).  

Just FYI, there are much better ways to accurately determine Keff.  These are the calculations that are used for determining the expected control rod position to reach criticality.  The calcs will factor in core age, reactor temperature, time since shutdown, initial neutron levels, and a few other things that I forget about.  

However, this formula is more of a thought-experiment about the different things that might happen to a neutron that just popped out of a split atom in the reactor...  read on if you are interested.

There is an algebraic formula to help understand whether neutron population (and reactor power) are climbing, stable, or declining, and that is the six-factor formula.  It's kind of a neat formula, because it describes probabilities of the "neutron lifecycle" from birth to death.  Here's the top-secret formula, taken from Wikipedia.

k = Є L_f ρ L_th f η

When I was in school, we had a mnemonic device to help recall each variable.  It was a very effective memory trick - I can still remember it today.  "Every Little F*cking Pol*ck Loves The F*cking Navy".  I'm sure there's a more politically correct version available in these less crude and more sensitive days.  Let's look at the terms, and what they mean physically

• "k" or "Keff" or "K∞" = is the neutron effective multiplication factor.  Remember that >1 means power going up, <1 means power going down, and =1 means power is stable.
  
• Є is the fast fission factor (adds neutrons)
  
• L_f is the fast non leakage probablility (neutron loss)
• ρ is the resonance escape probability (neutron loss)
• L_th is the thermal non-leakage probability (neutron loss)
• f is the thermal fuel utilization factor (neutron loss)
• η is neutron reproduction factor (adds neutrons)

So what are all these variables, and how do they physically matter with respect to Keff?  Remember we are just trying to determine if power is going up, down, or sideways.

Let's imagine for a moment that we are a single neutron, freshly born from a fission event inside a reactor core.  We will look at each of the various paths our life might take!  That's what makes this formula interesting. Only two of the six factors have values greater than 1.0 and are therefore the ones that get us up to a self sustaining chain reaction.  Those factors are Є and η.

A prompt neutron from fission will have an incredible amount of energy, which translates to speed.  These are fast neutrons, and the factor Є (fast fission factor) accounts for the likelihood of a fast neutron to fission another atom in the fuel before it does anything else.  That is why this value is greater than one - our fast neutron may fission an adjacent atom and make more neutrons before it's even been slowed down to the correct speed.

Lf is the fast neutron non-leakage probability.  Some fast neutrons will always escape from the core, and so the number remaining will always be less than one  Perhaps the value is 0.95, because 5% of fast neutrons might leak out of the core and be lost from the process

ρ is the resonance escape probability.  Non fissile Uranium 238 has a strong tendency to capture a neutron at several discrete neutron energy levels, and become Uranium 239.  This process of capturing a neutron while the neutron is in the process of being moderated (slowed down by colliding with water or graphite) is called resonance capture.  The less U-238 in the fuel, the less likelihood this event will occur, so greater enrichment of the fuel will reduce this value.

Below:  A graph of the resonance capture energy levels (bottom scale) and cross-sections (left scale) at which resonance capture of a neutron by U-238 is likely.  Some of the peaks are quite high, and these can capture neutrons that are needed to fission U-235, removing them from the process as neutron moderation - slowing down - takes place.

L_th is the thermal neutron non-leakage probability.  Some neutrons that manage not to leak out while they are fast, and manage not to be resonant-captured by U-238.  They get slowed down, but still manage to leak out of the reactor core once they are slowed down.

f  is the thermal fuel utilization factor.  Once the neutron is slowed down, or thermalized (also known as moderated), it may still be lost in the process.  The neutron might absorbed by non-fuel material such as fuel cladding, the moderator, or a fuel rod.  It will then not be available to cause another fission.   

Lastly we have η, the neutron reproduction factor, which is how many neutrons are released from the average thermal (slow neutron) fission, and the value for U-235 is 2.43 neutrons per fission.  It is this large positive factor that allows a reactor to even work.

Nearly all of these parameters can be varied.  They can be varied by fuel enrichment, and by core design - spacing, cladding, moderator, compactness, and addition of burnable poisons.