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Criticality. K, $ and C

  1. Jul 11, 2011 #1


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    Hi All.

    I have been reading up about criticality this weekend and I have come across some terms that I could do with some help on.

    I think I understand Keff which describes the 'amplification' of reaction and includes the physical nature of the reactor, i.e. mass, shape, spacing, moderation and reflection. Does it include temperature (I assume that this is a function of spacing)? I understand that Keff < 1 is subcritical and > 1 is critical.

    I also think I understand that there is a relationship between Keff > 1 and the rise time. As Keff increases, rise time decreases. For manually handled criticalities, rise time should be as slow as possible.

    However, I haven't found an explanation of $ and C. Please could someone shed some light for me? (No pun intended).

    Thanks, Jim
  2. jcsd
  3. Jul 11, 2011 #2


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    Actually, k = 1 means critical, k < 1 = subcritical, and k > 1 = supercritical.

    1 $ of reactivity is equivalent to the effective delayed neutron fraction, and it is about ∆k ~ 0.0065. A cent of reactivity would be ∆k ~ 0.000065.

    prompt critical is the condition where the reactor is critical on prompt neutrons, and the neutron population increases as rapidly as the prompt neutron generation lifetime allows.

    The relatively long half lives of the delayed neutron precursors allows for the controlled power ascension in nuclear reactors. A prompt critical, or actually a prompt supercritical system can experience a rapid burst of power that could or would damage a reactor.
  4. Jul 11, 2011 #3

    jim hardy

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    here's my attempt at an explanation. please do not think i am 'talking down', it's just that i idolize Isaac Asimov and have over the years tried to copy his simple step by step thinking.

    You've done the hard work, learning the vocabulary. Now it's just a matter of getting to where you can work it in your imagination. I think you're there just maybe need a little more confidence... i hope this helps.

    K eff - it's a real simple concept and you'll have no trouble with it.
    glance at this page and make sure i dont cross terms below. i am mildly autistic and dyslexic.

    Swap your thinking for a moment away from reactors and to your pocket calculator days.

    Do you recall the old days they talked of "Chain calculation" where if you hit the X key twice it'd keep multiplying your entry(multiplier?) by the same number(multiplicand?)?

    If you set up a chain multiply to multiply (pick any number, maybe say) three, by another number say 1, repeatedly ,,

    every time you hit the = key you get the result of multiplying 3 by 1.

    3 X X 1 = gives 3 as many times as you hit the = key.

    Now if instead you decide to multiply three by two in a chain,

    3 XX 2 = 6 first time you hit =;
    = 12 second time;
    = 24 third time
    and so on.

    Result is 3 X 2^n where n = your particular iteration.

    That is a property of multiplying systems, growth is exponential.

    IF you did that once per second your answer would be 3X2^time.

    If multiplicand is real close to 1, say 1.0001 , it takes a lot of time to get a big change. Try it.
    3 XX 1.0001 = 3.0003 first time
    and 3.003 tenth time

    You could figure the time to change by any given ratio if you knew the multiplicand.

    A common ratio for reactor engineers is two, that interval is known as "doubling time".
    Another popular one is e, 2.718, that math factor you always run across and the basis of natural logarithms.
    The corresponding interval to e is known as "reactor period".
    The old Manhattan Project physicists were mostly young , their wives didn't like the New Mexico wilderness and that term is thought to have been an inside joke.

    OKAY now back to reactor.

    Reactor engineers adopted this lingo because they were dealing with chain reactions which behave like a chain multiplication. A reactor is a real life chain multiplying system.
    Their multiplicand is always real close to 1 and their time interval is way less than one second.

    Change your thinking of time now to be discrete ticks of a clock but maybe 100 ticks per second.

    Remember Arnie's video where he explained that you get two or so neutrons for every fission.

    In a reactor that is exactly critical , exactly one of those neutrons goes on to create another fission. The rest get lost.

    So the number of neutrons present at any time in a perfectly critical reactor remains the same - you're losing exactly enough neutrons to keep it balanced, at every tick of the clock you have the same number of neutrons as previous tick.

    Think of each clock tick as a 'generation of neutrons'. They aren't all born on clock ticks but it's a minor simplification that lets us wrap our mind around the process.

    IF ::
    # neutrons last generation X Multiplicand = # neutrons now = constant , generation after generation

    That means the Multiplicand is exactly 1.

    Keff is your multiplicand.

    If we take 1/100 sec, my clock tick, as time between neutron generations then

    # neutrons in future = # neutrons now X Keff^ (ticks) = # neutrons now X Keff^(time X100)

    Taking the ratio of future to now gives

    (neutrons after a while)/(neutrons now) = Keff^(100t) where t=seconds

    and 1.00000 to any power is still 1.

    What if Keff were 2, ie no neutrons leak away?

    In just a second there'll be 2^100 as many neutrons. That's about 1.2E30
    That's a large number. Remember I said Keff stays close to 1?

    What about Keff = 1.007
    In a second it's at : 1.007^100 = 2.009 as many neutrons.

    Now there's an imaginary reactor with a doubling time of about a second.
    So fast it'd be hard to control, but it gives you the idea about math of multiplying systems and how nicely it describes a reactor..
    In a uranium reactor the neutron generation interval is shorter, less than a milllisecond.
    Of course the process is continuous not discrete but making it discrete in our imagination lets us get our hands around it.

    The effect Arnie talked about, "Prompt Critical", is this.
    Of the neutrons made by fission, EDIT -ALL BUT- about seven per thousand are produced within nanoseconds of fission. The rest are delayed by microseconds to minutes. That's what makes a reactor controllable, the delay. Average of all the delays is in milliseconds.
    Note i used ten millisecond ticks above, one or two might have been closer to truth.
    When Keff gets to 1.007 you are critical on the nanosecond neutrons and the reactor runs away. It no longer has to wait for those last seven neutrons so your generation interval shortens to nanoseconds. Thet's what happened at Chernobyl and probably Borax.

    So Keff is simply the inter-generational multiplicand and it's always real close to one.
    At exactly 1 the reator is critical, below 1 it's subcritical and above it's supercrtical. Like flying an airplane or riding a bicycle you are always slightly off balance but average to a straight and level course.

    Spent fuel pools are by law required to be designed and built to keep Keff< 0.95.

    Hope this helps and you do not feel talked down to. My reactor physics prof spent a whole two hour class on the subject. Maybe you can make it more concise should anybody ask on the board.
    For familiarity for you - my power reactor we started up on a period of a minute or two, thirty seconds is uncomfortable.
    When first starting up our instruments might only detect a few dozen neutrons per second , but the detectors are outside the vessel. Still the neutron population in there is sparse.
    At full power there about 10^14 neutrons per square centimeter/ sec, a strange unit but you see it all the time and symbol is nv. Little research reactors with natural circulation run about 10^12 nv. .
    I measured that number on a university swimming pool reactor. We dropped a piece of copper wire, ordinary #12 house wire, into the core and let it absorb neutrons for thirty seconds . Then pulled it out and measured its radioactivity. By knowing Copper's hunger for neutrons one can back-calculate how many there were in the reactor. We made a map of neutron distribution in the little core - what a fun summer!!!

    old jim

    ps - Pu has prompt critical at Keff about 1.003, and some isotopes give you three neutrons per fission and that's why it makes better bombs.

    hope this helpx. Congratulations on your studies!!!
    Last edited: Jul 11, 2011
  5. Jul 12, 2011 #4


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    Thanks guys for your very accessible and authorititive replies. I will take some time to work through them, there is a lot of 'food for thought' for me there.

    In the meantime, can you clarify one very basic point for me? The $ fraction of neutrons (∆k) are delayed and can have times measured in seconds and minutes. Is this because they are created from the decay of fission fragments? These fragments take some variable time to decay depending on what they are, so delaying the doubling time? (Assuming k <1 but k + ∆k > 1).

    Thanks in advance.

    Jim (maybe 'young Jim' - except I am not...)
  6. Jul 12, 2011 #5


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    You got it - some of the fission fragment are called "delayed neutron precursors".
    They are excited states - and decay by emitting a neutron. That neutron is therefore
    delayed by how ever long the excited state lasts.

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