An elementary treatment of the critical mass used in nuclear weapons is presented and applied to an analysis of the wartime activities of the German nuclear program. In particular, the work of Werner Heisenberg based on both wartime and postwar documents is discussed.

## REFERENCES

1.

A text that I found useful is Samuel Glasstone and Milton C. Edlund,

*The Elements of Nuclear Reactor Theory*(Van Nostrand, New York, 1952).2.

Robert Serber,

*The Los Alamos Primer*(University of California Press, Berkeley, 1992). A recent and much more detailed version of this material can be found at Carey Sublette’s web site,*Nuclear Weapons Frequently Asked Questions*, http://www.fas.org/nuke/hew/. Many references to the literature on this subject are listed. It is important to note that neither Sublette nor I, for that matter, have access to classified material.3.

For plutonium the mean free path is about 12.7 cm.

4.

See Ref. 2 for a more realistic discussion.

5.

Purists will note that $\u2207r21/r=\u22124\pi \delta 3(r).$ But we shall apply this result for $r\u22600.$ They will also note that $cos(r)/r$ obeys the second relation. But we will ignore this term because it blows up at the origin.

6.

As I note later, in implosion bombs the efficiency is much higher than in a gun-assembly bomb such as the one that was dropped on Hiroshima.

7.

Jeremy Bernstein,

*Hitler’s Uranium Club*(Copernicus, New York, 2001), pp. 175–176.8.

This discussion is in Ref. 2, but in a form that I, at least, did not find easy to follow.

9.

The fact that several critical masses are involved raises an interesting conundrum; namely, how do you assemble an object that is, say, three times the critical mass, from an object or objects that are less than one critical mass? Reference 2 describes how to do this for the gun assembly by using shapes that are not exactly spherical. For the implosion bomb the key is in the above discussion. You can easily persuade yourself that if you have the same material at two densities, then the critical mass depends inversely on the square of the density. Thus you can begin with a subcritical object, which when compressed has a much lower critical mass, and hence the compressed object will have a mass substantially larger than the critical mass at that density.

10.

The reader who would like to sample this sort of thing can consult Paul Lawrence Rose,

*Heisenberg and the Nazi Atomic Bomb Project*(University of California Press, Berkeley, 1998).11.

All of Heisenberg’s reports can be found in

*Gesammelte Werke/Collected Works*, edited by W. Blum, H. P. Dürr, and H. Rechenberg (Springer, Berlin, 1989), Ser. A, Pt. II. The 1939 paper begins on p. 378.12.

This valuable document,

*Energiegewinnung aus Uran*, February 1942, has not been published. I am grateful to Mark Walker for sending me a copy.13.

Heisenberg actually supposes that there is a nuclear source at the origin which amounts to specifying $N(0).$ This assumption is not relevant to his calculation of the critical radius.

14.

See Ref. 7.

15.

See Ref. 7, p. 128.

16.

The whole notion of treating a chain reaction as successive fissions, one after the other, is a faulty oversimplification, which is what Heisenberg did in his first calculation at Farm Hall. The process actually involves all the neutrons that have been created in the past. This build up is described by an exponential of the form $e(k\u22121)t/\tau ,$ where $k\u22121$ is the effective number of neutrons produced in a fission. The time between fissions, τ, is about $10\u22128\u2009s,$ so by using Heisenberg’s simplified model, it would take about 800 ns to fission 1 kg. But the real exponential buildup is much faster, so it takes about a third less time. Because of the exponential increase in the neutrons, most of the energy is produced in the last few generations.

17.

See Ref. 7, p. 133.

18.

Because they never made any plutonium in a reactor, they did not know that plutonium 240 would be produced, whose spontaneous fission was what led the people at Los Alamos to consider implosion in the first place.

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© 2002 American Association of Physics Teachers.

2002

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