Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  As discussed in section 11.6, the theory of a nucleus containing hundreds of strongly interacting component nucleons is extremely difficult, and so physicists struggled to find a theoretical explanation to back up the experimental results. A convincing argument was given by Lise Meitner and Otto Frisch in a January, 1939 Nature paper titled “Disintegration of Uranium by Neutrons: A New Type of Nuclear Reaction.” (Incidentally, it was in this paper that the word fission was first used in the nuclear context having only been used earlier by biologists when discussing cell division.) It was on a Christmas-holiday walk in the snowy woods of Sweden that Meitner and her nephew Frisch realized that they could use the liquid-drop model of the nucleus developed by Bohr. The analogy between a nucleus and a liquid drop is completed by introducing the right sort of volume terms and surface tension to model the nuclear properties. (As a student in 1906, Bohr had done a project—for which he won a gold medal—on a precision measurement of the surface tension of water by the observation of a regularly vibrating jet, so he was equipped with the classical theory to build the liquid drop model.)

  According to Meitner and Frisch: “On account of their close packing and strong energy exchange, the particles in a heavy nucleus would be expected to move in a collective way, which has some resemblance to the movement of a liquid drop. If the movement is made sufficiently violent by adding energy, such a drop may divide itself into two smaller drops.”21 This is illustrated in figure 11.7. Thus the first calculation to explain the fission process began with an analogy that we can picture as in this figure. Meitner and Frisch calculated the energy released in the uranium fission process, and their paper must be ranked as one of the most important in the whole of nuclear physics. (I will say more about Lise Meitner in chapter 13.)

  Figure 11.7. Liquid-drop model of a nucleus undergoing fission. Figure created by Annabelle Boag.

  11.7.1 Details and Implications

  Uranium occurs naturally with a mixture of isotopes, mainly U238 but with about 0.7 percent of U235. The U235 is important because it begins the fission process when neutrons of only comparatively low energies are incident upon it. It becomes an unstable excited U236 nucleus, which then decays to complete the fission process to give

  Note that the number of protons remains fixed at 92 (with 36 in the krypton and 56 in the barium), and the original 144 neutrons finish up in the krypton and barium nuclei leaving three neutrons to be emitted. Many processes of this type may occur (see Dunlap chapter 12) with varying numbers of free neutrons to be emitted, with two being a common number.

  There are two vitally important points to be made about the uranium fission process. First, it releases a large amount of energy, around 200MeV. To give an idea of what is involved, Hans G. Graetzer and David L. Anderson calculate that the complete fission of a pound of uranium would release more than a million watts of power continuously for a year. Physicists soon realized that here was a possible source of power generation, and, of course, we do have nuclear power stations today.

  The second point is the observation that the emitted neutrons were often called prompt neutrons because they may impinge on other uranium nuclei and trigger more fission processes. This we know as a chain reaction, and Enrico Fermi demonstrated experimentally that it can occur (see Graetzer and Anderson for Fermi's paper).

  Physicists soon realized that uranium fission could be exploited in a much more sinister way; if the enormous power involved in a chain reaction were released in a short time, the effect would be explosive—the possibility of an atomic bomb became apparent. Remember that this was 1939, and the thought of nuclear weapons for use in the impending World War II was a frightening possibility. (Readers wishing to explore the whole history of this period and the development of nuclear weapons are recommended to consult the Pulitzer Prize–winning book by Richard Rhodes.)

  The details of a likely nuclear weapon were set out with great clarity and remarkable accuracy in two memoranda written at the University of Birmingham in England and reached the British-government scientific authorities in March 1940. They were written by Rudolf Peierls, a theoretical physicist and German Jew who fled from Germany before the Nazi era, and Otto Frisch (the nuclear theorist we met above), an Austrian Jew who escaped Nazi Germany in 1939. The Frisch-Peierls memoranda were titled “Memorandum on the properties of a Radioactive ‘Super-Bomb’” and “On the Construction of a ‘Super-Bomb’; Based on a Nuclear Chain Reaction in Uranium.” (You can read these documents—and I recommend them as fine examples of scientific writing—in the book by Robert Serber.)

  Albert Einstein wrote a famous letter to President Roosevelt in August 1939 informing him about the possibility of a nuclear bomb and warning him that work on nuclear fission in uranium was being carried out in Germany. (The letter is in the Graetzer-Anderson book.) Einstein suggested that the president should take various actions in response to these developments, and with the receipt of the Peierls-Frisch memoranda in 1941, the way was opened in early 1943 for the Los Alamos laboratories to begin on work to produce a nuclear bomb.

  11.7.2 The Los Alamos Primer

  Scientists and engineers began arriving at Los Alamos in March 1943. To set the scene and explain what the Los Alamos project was all about, a series of introductory lectures was given in April 1943 by Robert Serber, a theoretical physicist who had worked for Robert Oppenheimer, the head of Los Alamos. Lecture notes were taken by Ed Condon, and a typed version was produced. This became the Los Alamos Primer, which Serber published with extended notes and comments in 1992. It is a fascinating and somewhat chilling experience to read the Primer and see the calculations that led to one of the turning points in the history of mankind.

  The Primer opens with a simple statement of intent:

  The object of the project is to produce a practical military weapon in the form of a bomb in which the energy is released by fast neutron chain reaction in one or more of the materials known to show nuclear fission.22

  Serber points out that the energy release in a nuclear fission process of order 170MeV is about 107 times more than the heat energy produced in ordinary chemical or atomic combustion. In order to make the point, he compares the new possibility with an old explosive material and produces the result

  1 kg of U235 ≈ 20,000 tons of TNT.

  This simple but breathtaking comparison would have left no one in doubt about the significance of the project they had all been gathered together to work on.

  Serber then gets down to business by evaluating the chain reaction process with a simple calculation:

  Release of the energy in a large scale way is a possibility because of the fact that in each fission process, which requires a neutron to produce it, two neutrons are released. Consider a very great mass of active material so great that no neutrons are lost through the surface and assume the material so pure that no neutrons are lost in other ways than by fission. One neutron released in the mass would become 2 after the first fission, each of these would produce 2 after each had produced fission so in the nth generation of neutrons there would be 2n neutrons available.

  Since in 1 kg of U235there are 5 × 1025 nuclei [in his 1992 book, Serber notes that there is a mistake here; it should be 2.58 × 1024] it would require about n = 80 generations (280 ≈ 5 × 1025) to fish [note the invented verb!] the whole kilogram.

  Serber has established what sort of total reaction is required, and now he simply makes the vital and somewhat frightening point:

  While this is going on the energy release is making the material very hot, developing great pressure and tending to cause an explosion.

  In fact, for ten percent efficiency, the material would be raised to a temperature of about 10 billion degrees in a millionth of a second. Now, for the first time, comes a hint of the difficulties these bomb designers faced. Some of the neutrons will escape from the material which will now be an expanding hot gas which can ruin the whole process. Serber writes:

  The whole question of whether an effective ex
plosion is made depends on whether the reaction is stopped by this tendency before an appreciable fraction of the active material has fished.

  Here is the great difficulty: Can the total reaction occur in a time before the whole thing has expanded out and effectively stopped the explosion? Serber calculates that the reaction must occur in about 5 × 10–8 seconds or “otherwise the material will have blown out enough to stop it.”

  Serber tests the possibility with a simple order-of-magnitude calculation assuming that neutrons may travel on average (mean free path) about 13 cm between fissions. Here is how he finds the possibility of an explosion occurring:

  Now the speed of a 1 MeV neutron [the type released in the fission process] is about 1.4 × 109 cm/sec and the mean free path between fissions is about 13 cm so the mean time between fissions is about 10–8 sec. [13 divided by 1.4 × 109] Since only the last few generations will release enough energy to produce much expansion, it is just possible for the reaction to occur to an interesting extent before it is stopped by the spreading of the active material.

  The Primer goes on to discuss a whole range of technical matters concerning the fission process, including the possibility of using plutonium as well as uranium, as well as methods of detonation. There is also a section headed “Damage” in which the dreadful effects of a nuclear explosion are described and evaluated. However, perhaps the most crucial question concerns the actual size of the bomb. The enrichment process for extracting U235 from uranium ore was barely developed at that stage so knowing the amount required could decide the viability of the whole project. There was also the question of the delivery of a nuclear weapon and whether it would be feasible to use an aircraft for bombing enemy territory.

  11.7.3 The Size of the Bomb

  So far Serber has evaluated the fission process, checked the number of neutrons involved, and calculated the time scales that must be satisfied. Section 10 of the Primer is headed “Simplest Estimate of Minimum Size of Bomb.” This is a vital calculation to make since the viability of the whole project in terms of a practical weapon rests on the result. Could a nuclear bomb be constructed and delivered to its target?

  Serber uses a diffusion-type equation to calculate the number of neutrons involved; there is a generation term (neutrons produced by nuclear fissions) with a time scale of 10–8 seconds, as discussed above, and a term describing how neutrons disperse in the medium, which depends on the neutron's mean free path. This theory is used to find the critical radius (assuming a spherical bomb). For a block of material with this critical radius, the neutrons generated are balanced by the number escaping at the boundaries; a larger sphere will explode. The calculation needs boundary conditions, and these must be carefully considered. After some analysis, Serber finds approximations for the critical radius Rc and mass Mc:

  Rc ≈ 9cm, Mc ≈ 60kg of U235.

  To this day, it is hard to believe—and terrifying to contemplate—that a sphere of material just a few centimeters in diameter can cause such a devastating explosion.

  Because it is these critical values that are of such vital importance, there is another step that Serber considers. Is there a way to counteract the escape problem for neutrons at the device boundary? Section 11 of the Primer begins:

  If we surround the core of active material by a shell of inactive material the shell will reflect some of the neutrons which would otherwise escape. Therefore a smaller quantity of active material will be enough to give rise to an explosion. The surrounding case is called a tamper.23

  Serber notes that heavy elements (like ordinary uranium U238) should be used for the tamper and gives a mathematical analysis of the role it can play in bomb design. He finally calculates that

  for a normal U tamper the best available calculations give Rc = 6cm and Mc = 15kg of U235 while with Au [gold] tamper Mc = 22kg of U235.

  There were many unknowns in the whole preliminary analysis, and teams of scientists were needed to work on the project. Serber's conclusion in the Primer was that many experiments were needed to measure the neutron and fission properties that went into his calculations. Nevertheless, the calculations that I list as calculation 46, planning for a bomb indicate that the project was indeed viable. The horrific results of the August 1945 bombing of Hiroshima and Nagsaki showed just how effective those calculations were.

  11.7.4 Uncertainty about Heisenberg: Hero or Poor Calculator?

  Albert Einstein, in his letter to President Roosevelt, warned about decisions made in Germany to stop sales of uranium from the Czechoslovakian mines, which invading German armies had taken over. In the Frisch-Peierls memoranda we find another warning:

  If one works on the assumption that Germany is, or will be, in the possession of this weapon, it must be realized that no shelters are available that would be effective and could be used on a large scale. The most effective reply would be counter-threat with a similar bomb. Therefore it seems to us important to start production as soon and as rapidly as possible, even if it is not intended to use the bomb as a means of attack.24

  Obviously, the possibility of a nuclear-fission bomb was understood in Germany just as it was elsewhere. Many scientists fled from Germany and Nazi persecution, but a number of eminent nuclear physicists and chemists remained. Among them was Werner Heisenberg, one of the founders of quantum theory and the recipient of the Nobel Prize for Physics in 1932. Heisenberg was a leading figure in German efforts to develop nuclear weapons. He was not a Nazi, but he was a patriotic German and hoped for a German victory.

  The story of Heisenberg's involvement is complex and murky, and several books have been written on the subject. The central question appears to be: Did Heisenberg deliberately make an error that effectively sabotaged the German nuclear-weapons program? He calculated a critical mass of tons of uranium, rather than the real value of a few kilograms mentioned above. Such a result would immediately signal that the bomb project was not viable. Was that a deliberate error?

  However, the evidence seems to point to the fact that Heisenberg made a genuine and major blunder when he did the calculation; he was just not careful enough with the theory and its evaluation. (Interested readers should consult the article by Jonothan Logan and the book by Paul Rose for detailed and meticulously presented arguments.) There are those who have tried to present Heisenberg as some sort of hero for foiling German nuclear-weapons plans, but the fact casts grave doubts on such a story.

  The most telling evidence comes from the words of Heisenberg himself. A group of German nuclear scientists was interned near Cambridge, England, at a place called Farm Hall from July to December 1945. The group included Werner Heisenberg, Otto Hahn, Max von Laue (famous for his work on x-rays), C. F. von Weizsäcker, Walter Gerlach, and other notable experts. British intelligence officers recorded the group's conversations. On August 6th, the group heard the news that a nuclear bomb had been dropped on Hiroshima. They were clearly stunned and thought a trick was being played. We have Heisenberg's own words:

  I still don't believe a word about the bomb but I may be wrong. I consider it perfectly possible that they have about ten tons of enriched uranium, but not that they have ten tons of pure U235.25

  It is clear that Heisenberg still believes that the critical mass is to be measured in tons not kilograms, and he persists with that line over the next few days. He goes on to suggest that the report on the bomb was a fraud:

  All I can suggest is that some dilettante in America who knows very little about it has bluffed them in saying “If you drop this it has the equivalent of 20,000 tons of high explosive” and in reality it doesn't work at all.

  Heisenberg was always a confident man and sure of his abilities, so he must have been stung by the chemist Hahn's comment:

  If the Americans have a uranium bomb, then you're all second raters. Poor old Heisenberg…You're just second-raters and you may as well pack up.

  (This is a fascinating part of the history of physics, and I recommend the book introduced by Sir Charles Frank to a
nyone wishing to read the Farm Hall conversations in detail and learn more about the participants and what became of them after the war. The transcript of the German scientists’ conversation immediately after learning about Hiroshima bombing and their debates for days afterward makes for truly gripping reading.)

  There are a great many documents and statements related to Heisenberg's war record but amazingly there are still areas of confusion and disagreements about interpretations. Personally, I find the writings of Jonothan Logan and Paul Lawrence Rose most convincing, and I am prepared to believe Rose's final position:

  This German mentality of Heisenberg and his friends, fertilized by astounding powers of self-delusion and rationalization, spun the tissue of deception and self-deception that produced the Heisenberg version and the cocoon of fabrication and denial that has blurred the history of Heisenberg's work on the atomic bomb to the present day.26

  Although Heisenberg did meet his old colleagues, like Niels Bohr, after the war, he never recovered his preeminent position in the world of science.

  11.8 TOWARD A FINAL PICTURE OF MATTER

  The last one hundred years have seen immense progress in our understanding of the basic particles of matter and the way they move and interact. The standard model represents a concise structure for all matter—except, apparently, for the dark matter discussed in section 7.4! There is also the worry about the nature of dark energy. However, the standard model combined with relativistic concepts helps to explain the origin of the universe and its development after the big bang. But not everything is completed. There is still that nagging question of how to combine general relativity and quantum theory. The standard model itself asks some fundamental questions: Why do we find those particular basis particles, and why do the mass and other parameters take on those particular values? No doubt there are many great calculations still to come.