Critical mass formula: The Los Alamos Primer and Rudolf Peierls paper from 1939

In summary: The formulae are for a sphere of radius 1 with a uniform density of neutrons. After plotting the formulae I found that there is a discrepancy between the two formulae. The Primer gives a smaller value for the critical mass while the Mathematica gives a larger value. I tried to find a reason for this discrepancy but couldn't find anything. I also tried to find a relation between the critical mass and the neutron multiplication number but couldn't find any correlation. I think the discrepancy might be caused by a different definition of the neutrons. I am going to try to find a more recent reference for the calculations in order to compare the results more closely.In summary, Sören Molander has attempted to calculate the
  • #1
sorenmolander
3
0
Dear Forum,

this is my forst time here, so forgive me if I don't conform to any etiquette or rules.

When I did nuclear physics in undergraduate school many years ago, I couldn't help noticing that little or no information was ever given on he issue of neutron multiplication and criticality of Uranium and Plutonium. At the time I thought that such information was still classified but I still was very curious about the basic physics involved. I vaguely suspected that diffusion was involved, but I never attempted to make any calculations.

Some years ago I read Richard Rhodes' book "The Making of the Atom Bomb". This is truly a fantastic source of information for physicists, not only about the Manhattan project but also on the scientific community involved in developing the first quantum theory and what would later become nuclear physics. In the book there is a reference to Robert Serber's "Primer", which was a series of lectures on the basics physics of the nuclear device and also to a paper by Rudolf Peierls. I downloaded a paper version of the "Primer" from a Los Alamos server about 4 years ago and have now also bought the edited version by Serber and Rhodes. I also have a copy of Peierls original paper, where gives a formula for the calculation of the critical mass.

I have tried to compile these two sources into a Mathematica (and html) document and tried to elaborate and compare the derivations of the critical mass in the "Primer" and in Peierls paper. It is curious that there is a typo in Peierl's paper, but with Mathematica it is easy to see the numerical values are the same. Here are the URL:s

http://hem.bredband.net/sormol/critrad/critrad.htm
http://hem.bredband.net/sormol/critrad/criticalMass.nb

The numerical values from the critical mass formula in the Primer and those derived from Peierls formula differ significantly. I would be very interested if anyone could comment on the URL:s above and also give some improvements. Please feel free to use the Mathematica (5.1) notebook as a start.

Best Regards,

Sören Molander
 
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  • #2
sorenmolander said:
Dear Forum, ...
The numerical values from the critical mass formula in the Primer and those derived from Peierls formula differ significantly. I would be very interested if anyone could comment on the URL:s above and also give some improvements. Please feel free to use the Mathematica (5.1) notebook as a start.

Best Regards,

Sören Molander
I might be able to help. But these calculations have been modified substantially over the years in any case, and the latest calcs are certainly classified.
 
  • #3
Nam_Sapper said:
I might be able to help. But these calculations have been modified substantially over the years in any case, and the latest calcs are certainly classified.

I am sure the detailed physics requires much better models than the first estimates that was published 1939 and in 1965, when the Los Alamos Primer was declassified. And I certainly hope that all the important aspects of building a nuclear device will continue to be classified. The reason for my writing a more detailed document up on original - and most likely outdated - calculations was that I had been looking in vain for a "back of the envelope" calculation in textbooks. In all honesty I haven't looked in a textbook in Nuclear physics in a long time, so maybe that has changed. At any rate I hope the Mathematica document might be interesting to someone doing some science history research. I have certainly enjoyed the Primer and Peierls paper from 1939 and going through the calculations in detail was fun.
 
  • #4
I found and fixed some errors in the Mathematica document and also made some changes in the text. In its basic form, Peierls formulae are not useful for numerical calculations for the high-multiplication number regime (see URL ablove). In the derivation a substitution is made in order to be able to plot the formulae together. If the substitution is _not_ made there is a much better numerical corrspondence between this and version given in the Primer.
 

1. What is the critical mass formula?

The critical mass formula is a mathematical equation used to determine the minimum amount of fissile material required to sustain a nuclear chain reaction. It was first described in the Los Alamos Primer, a document written by scientists at Los Alamos National Laboratory in 1943, based on the work of Rudolf Peierls' 1939 paper "On the Theory of Uranium Fission".

2. How does the critical mass formula work?

The critical mass formula takes into account various factors such as the density and shape of the fissile material, the presence of a neutron reflector, and the energy released by each fission event. By calculating these factors, the formula can determine the minimum amount of fissile material needed for a self-sustaining nuclear chain reaction.

3. Why is the critical mass formula important?

The critical mass formula is crucial in the development and maintenance of nuclear weapons and nuclear power plants. It allows scientists to determine the most efficient and safe amount of fissile material to use in these applications. It also helps prevent accidents or unintended chain reactions that could have catastrophic consequences.

4. Are there different versions of the critical mass formula?

Yes, there are different versions of the critical mass formula that take into account different types of fissile materials and varying conditions. For example, the formula used for weapons-grade plutonium is different from the formula used for reactor-grade plutonium. Additionally, different countries may have their own variations of the formula based on their specific needs and research.

5. How has the critical mass formula evolved over time?

The original critical mass formula described in the Los Alamos Primer and Peierls' paper was based on simplified assumptions and did not account for all factors. Over time, as more research and experiments were conducted, the formula has been refined and improved. It has also been adapted to include new types of fissile materials and technology. However, the basic principles and equations of the original formula remain the foundation for all versions used today.

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