Discussion:
Probability Of A Chain Reaction.
(too old to reply)
Patrick D. Rockwell
2010-09-17 00:42:15 UTC
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I've long been interested in probability. Some time ago, I read a
formula for the probability
of a chain reaction is given by

1.5-0.5*((12/k)-3)^0.5.

I got this at the following site, not to mention others.

http://www.newworldencyclopedia.org/entry/Nuclear_reaction


I'd like to know just how this formula is derived. From what I've
read, k
is the average number of neutrons which exit a fission, and in the
above
formula the allowable valuse for k are 1, 2, or 3. Couldn't k have
values outside
that range? Is there a way to make this formula more general? If not,
why? I should
think that it would be possible to make up a hypothetical problem in
probability where
the number of neutrons which exit the fission of the atom in question
is higher than
3.

Any info or insight on this is appreciated. :-)
Tom Roberts
2010-09-18 01:50:14 UTC
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Post by Patrick D. Rockwell
I've long been interested in probability. Some time ago, I read a
formula for the probability
of a chain reaction is given by
1.5-0.5*((12/k)-3)^0.5.
I got this at the following site, not to mention others.
http://www.newworldencyclopedia.org/entry/Nuclear_reaction
Their discussion is oversimplified, and in particular the discussion in which
this formula appears seems to me to be wrong.

K is NOT the number of neutrons exiting a fission, it is the AVERAGE number of
neutrons available to the system for a single initial neutron. It is a
complicated function of the composition and geometry of the system. Nuclear
reactors operate at K=1 to within about 1 part per million, which is why they
are called "critical". Nuclear weapons operate with K values well above 1.

There are a number of miracles that together make nuclear reactors possible.
Principal among them is that about 1% of the neutron flux is delayed by an
appreciable fraction of a second -- this makes it possible to control the
reactor with mechanical control rods (the prompt flux, which is ~99% of the
total, is emitted within a femtosecond or so after a fission, so there is no
hope of controlling it mechanically). The other important miracle is that when
the core of a reactor gets hotter, it expands and reduces the value of K. The
combination of these makes it possible for the operators to set the operating
point of the reactor with mechanical control rods (made of material that absorbs
neutrons, so larger insertion reduces K), and the thermal feedback within the
core makes the power output from the nuclear reaction exactly match the amount
of cooling supplied.

This discussion is still simplified, as it omits several important effects. I am
not expert on nuclear physics or reactor engineering.


Tom Roberts

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