# Talk:Frisch–Peierls memorandum

## Formula for 'energy'

Folks, please sanity-check me. The equation given for energy of the bomb, big E, looks wrong - power of tau/time dimension - the whole thing has the dimensions of power not energy, surely? (Apologies if I'm going mad.) Either it means 'power' which isn't useful, or the tau-to-the-minus-3 should read minus-2 perhaps? Or who knows what.

Could someone take a look for me? CecilWard (talk) 12:23, 1 December 2017 (UTC)

In this case, it doesn't matter at all what is right, but what he wrote years ago. There are many approximations that go into the calculation, and that aren't included. The energy released is growing exponentially, so a tiny increase in the containment time is a huge increase in energy released. The time depends on the inertia of the sphere itself. The term with the square root determines how much the radius can increase before fission stops. I suspect that the 0.2 has units that aren't mentioned. Note that there is no term for the amount of energy per fission, which is important. The energy/fusion divided by m*c**2 for a Uranium atom is about 0.1%, but there are no terms of that order. Gah4 (talk) 16:32, 1 December 2017 (UTC)
OK, from[1] which has a transcription of the memorandum, it should be tau**2. There is a footnote in the transcription about confusion between diameter and radius, though. If you are really interested, read the book, which has annotation (from later) about the original primer. Even so, there is a lot of approximation that isn't described in the memorandum. Gah4 (talk) 17:17, 1 December 2017 (UTC)
Thank you. I have corrected the formula. It was a transcription error on my part. (Note that Serber also transcribed it wrongly.) It appears without derivation in the memorandum. If you have a source that explains how Peierls arrived at it, I can include it in the article. Hawkeye7 (discuss) 19:50, 1 December 2017 (UTC)

I too noticed anomalies, came here to detail them, and discovered the existing thread. I have recapitulated the calculations using the available references. The situation is worse than you might imagine. The original memorandum in the UK National Archives might not contain such anomalies, but comparing the "Atomic Archive" online memorandum with that at Stanford University's version (https://web.stanford.edu/class/history5n/FPmemo.pdf) - they differ in essential detail! In particular, the memorandum text of reference 1 purports to say 4700g yields E=4(10²º) ergs. On the other hand reference 2 purports to say 4700g yields E=4(10²²) ergs, They differ by two orders of magnitude! Furthermore, Reference 1 specifies "For a sphere of radius 4.2 cm" - while reference 2 has "For a sphere of diameter 4.2 cm".

At least one must be wrong, and in fact neither is correct, for both extend the scope of the square root to include the "-1". If necessary I could provide calculations recapitulated to show how both versions are defective. Does arithmetic count as original research?

Carefully working through the arithmetic, using the correct formula, it is possible to work out that Reference 1's final figure of E=4(10²º) ergs is correct, while Reference 2's E=4(10²²) ergs is not. In fact I have now been assured by an academic with access to the original that E=4(10²º) ergs is to be found in the original memorandum.

The situation given of an 8cm radius sphere of U235, which would mass about 32kg, yields about 2.4(10²²) ergs.

```--115.188.58.71 (talk) 15:04, 1 March 2018 (UTC)
```

On a slightly different subject, the article speaks of "time required for a kilogram of uranium to fission." Frisch and Peierls did not address that in the memorandum. The authors do observe that "one might think of about 1 kg as suitable size for a bomb"; and they do calculate the yield of 8kg, 5kg, and 1kg spheres of U-235; but at no time do they calculate the time for a kilogram of U-235 to undergo fission - although Mr Bernstein did. The authors do, however look into the mean time for a fissioned neutron to cause another fission. They give the doubling-life t½ = 2.6×10^−9 seconds, which implies a mean time to increase by e as τ = t½/ln(2) ~ 4(10^-9)s. --115.188.58.71 (talk) 15:52, 1 March 2018 (UTC)

The archives version says "diameter". The reason why they calculated larger bomb sizes was so they could ignore the surface effects. Hawkeye7 (discuss) 21:47, 1 March 2018 (UTC)
I am assured by an academic with access to a photo-facsimile of the UK National Archives typewritten original of the F & P memo, that it says "radius", not "diameter". Even so: it also shows the (handwritten!) square root sign extending only over the ratio of sphere radius to critical mass radius, not including the "-1". Dr Reed of Alma College has also covered this in a private communication, and Bernstein's 2010 paper, "A memorandum that changed the world", discusses this in detail on page 444 of Am. J. Phys., Vol. 79, No. 5, May 2011. He is equally at a loss for some aspects of these calculations. --115.188.58.71 (talk) 01:41, 2 March 2018 (UTC)
Yes that's right; Bernstein says that. I don't know how the bit about diameter crept in. Hawkeye7 (discuss) 02:48, 2 March 2018 (UTC)
If you have any idea where the energy formula come s from, let me know. Hawkeye7 (discuss) 03:02, 2 March 2018 (UTC)
Dr Reed has been in touch with an unpublished paper that I am still working through, that recovers the form of F & P's energy formula, to within a multiplicative constant. When I fully understand it myself, I will let you know. It's not yet in a fit state to publish. --115.188.58.71 (talk) 11:17, 2 March 2018 (UTC)

References

1. ^ Serber, Robert; Rhodes, Richard (1992). The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb. Berkeley: University of California Press. ISBN 0-520-07576-5. OCLC 23693470.

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