Re: fission question

From: Steven Sharp (sharp_at_cadence.com)
Date: 07/27/04 

Date: 27 Jul 2004 12:32:41 -0700

"Carey Sublette" <careysub@earthling.net> wrote in message news:<AxlNc.17697$Qu5.9066@newsread2.news.pas.earthlink.net>...
> "John Schilling" <schillin@spock.usc.edu> wrote in message
> news:ce3lcn$53b$1@spock.usc.edu...
> > sharp@cadence.com (Steven Sharp) writes:
> >
> > >With those numbers, in 100 microseconds an initial neutron would
> > >only have increased to 1.00025**10000, or 12.18 neutrons. That
> > >means a growth of only one order of magnitude every 100 usec.
> > >In 1 msec, there would still only be 7.0E9 neutrons, which would
> > >correspond to a fission rate of 7.0E17 fissions/sec.

BTW, I slipped up on the decimal point here. In 1 msec, there would
be 7.1E10 neutrons, and a fission rate of 7.1E18 fissions/sec. It
doesn't affect the timescale much, but I didn't want somebody else to
catch it before me.

> I was going to make an informed estimate about how large a density decrease
> would be required to drive a multiplication factor of 1.00025 below 1, but
> haven't had the time.

Another BTW: I think the multiplication factor was probably a little higher
than this. I found another reference to a reactivity increase in "cents"
with a corresponding multiplication factor that was slightly higher than
my estimate would have given. This one was for Uranium, so I can't apply
it directly. My guess is that the difference is because I just used the
fraction of neutrons that are delayed, but these delayed neutrons probably
have a lower energy, for which there is a higher fission cross-section.
The difference was less than 10%, so I am assuming I was close.

> But since this is only very slightly above critical mass, and the critical
> mass is proportional to the inverse square density, and the density is
> proportional to the cube of linear dimension, this is going to be tiny (it
> makes the critical mass a sixth power of the linear dimension). This
> proprotionate increase in going to be a one, a decimal point, and a bunch of
> zeros before you come to the first significant digit after that.

Agreed. However, the density still only has a linear dependence on the
temperature increase, not the cube of it. The coefficient for volume is
3 times the linear expansion coefficient, but still close to linear, i.e.
(1+delta)**3 is approximately 1+3*delta. That still leaves the critical
mass depending on the inverse square of the temperature change.

To get any further, we need the thermal coefficient value. And as John
Schilling points out, when it is this close to zero, trying to treat it
as constant is probably invalid.

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