In a recent paper, Michael Pearson offers a thought-provoking speculation on an aspect of the famous story of how Lise Meitner and her nephew Otto Frisch developed the concept of nuclear fission during a walk in a snowy forest in Sweden at Christmastime, 1938.^{1} Pearson's speculation concerns a statement in their *Nature* paper reporting the discovery: “… the surface tension of a charged droplet is diminished by its charge, and a rough estimate shows that the surface tension of nuclei, decreasing with increasing nuclear charge, may become zero for atomic numbers on the order of 100.”^{2} Meitner and Frisch were thinking of George Gamow's liquid-drop model of nuclei, wherein the mutual Coulomb repulsion of protons is counteracted by a “surface tension” effect which originates from the fact that nucleons at the surface of nuclei interact with fewer of their siblings than do those within. The surface tension effect is proportional to the surface area of the nucleus, whereas the Coulomb repulsion is essentially a volume effect. Since the latter grows more rapidly than the former with the increasing nucleon number, the Coulomb force eventually overwhelms the surface tension, leading to instability. Meitner and Frisch's phrasing that the tension is “diminished” and “may become zero” was unfortunate: the tension does not vanish but rather eventually loses out to the Coulomb force; it is the competition between the two that they were trying to express.

Meitner and Frisch did not elaborate on how they arrived at a limiting atomic number of *Z* ∼ 100. Pearson speculates that Frisch's familiarity with the physics of droplet formation in a Wilson cloud chamber would have been sufficient for him to arrive at the *Z* ∼ 100 limit without invoking the mathematical machinery of Legendre polynomials involved in Lord Rayleigh's 1882 theory of vibrating electrified droplets. Later, Niels Bohr and John Wheeler developed their theory of fission, which did involve modeling distorted nuclei with a series of Legendre polynomials.^{3} This theory, which is extremely complicated to say the least, is not something that even an experienced physicist could likely carry out in a few hours or days, especially if no reference works were at hand.

Pearson's paper motivated me to go back to Frisch's own recollections with a view to examining what he had to say on the genesis of the *Z* ∼ 100 limit. (Meitner appears never to have published any memoirs of the event, but a contemporary letter she wrote to Hahn is very telling; this is described below.) My purpose is not to try to analyze Pearson's speculation, but rather to offer readers a fuller picture of and another speculation regarding this pivotal event in physics history.

Frisch published no less than six memoirs in which he mentions the fateful Christmas spent with his aunt; there may be others of which I am unaware. These spanned 25 years from 1954 (a full 15 years after the events involved) to just before his passing in 1979. He also gave an oral interview in 1967, which I believe offers a crucial piece of evidence.

Frisch's first memoir was based on a lecture given in late 1953; this alludes only briefly to his time with Meitner and makes no mention of either the diminishment of the surface tension or the *Z* ∼ 100 limit.^{4} Next was a contribution to a 1967 volume commemorating Niels Bohr. This gives a much more detailed description of the walk and specifically states that their calculations indicated that “… the uranium nucleus might indeed be a very wobbly, unstable drop, ready to divide itself at the slightest provocation …,” but again the *Z* ∼ 100 limit is not explicitly mentioned.^{5} The first mention of *Z* ∼ 100 in a memoir appears to be in an article published later the same year in *Physics Today*: “Then I worked out the way the electric charge of the nucleus would diminish the surface tension and found that it would be down to zero around *Z* = 100 and probably quite small for uranium.”^{6} In a very brief article in the Christmas 1973 edition of *New Scientist*, Frisch characterized this as “a simple calculation,” and in a longer 1978 *New Scientist* article commemorating the 100th anniversary of Meitner's birth, he describes the situation as “… after an hour or so (as I recall it) of arguing and rough calculations we began to understand.”^{7,8} In his 1979 autobiography *What Little I Remember*, Frisch essentially reiterates the story as it appears in the 1967 Bohr volume.^{9}

This all certainly gives the impression of a calculation developed very quickly, but, as pointed out by Ruth Sime in her biography of Meitner, the reality may have been quite different; it is of course not unknown for memory be unreliable and to compress and scramble events after the passage of years.^{10} On December 29, 1938, Meitner, in a letter written to Otto Hahn four or five days after her initial conversation with Frisch, related that “… Otto R. and I have really racked our brains …,” an indication that their understanding did not unfold immediately.

I come now to Frisch's oral interview, which was conducted by Charles Weiner in May, 1967, and can be found on the American Institute of Physics website.^{11} About a third of the way through the interview, Frisch relates how, when he was a research assistant with Otto Stern at the University of Hamburg in the early 1930's, he would go home after spending the day in the laboratory, and after dinner sit down to work on physics problems until late into the night: “But I regularly came home, had dinner at seven, had a quarter of an hour's nap after dinner, and then I sat down happily with a sheet of paper and a reading lamp and worked until about one o'clock at night … It was the ideal life. I'd never had such a pleasant life-ever: this concentrated five hours work every night.” Later, the interview turns to the discovery of fission and the walk with Meitner, and the following very revealing statement appears: “I remember that I immediately at that instant thought of the fact that electric charges diminish surface tension. That's probably one of the problems I had once worked out in those five-hour bouts in Hamburg, when I worked through every conceivable little problem in physics that caught my interest. And so I promptly started to work out by how much the surface tension of a nucleus would be reduced. I don't know where we got all our numbers from, but I think I must have had a certain feeling for binding energies and could make an estimate of the surface tension. Of course we know (*sic*) the charge and the size reasonably well. And so, as an order of magnitude, the result was that at a charge of approximately 100 the surface tension of the nucleus disappears; and therefore uranium 92 (*sic*) must be pretty close to that instability.”

This comment, considered in combination with some possible memory-compression, offers a more nuanced view of the origin of the *Z* ∼ 100 estimate. It may not have been an immediate revelation; during the three weeks that elapsed between his walk with Meitner and the submission of their paper on January 16, Frisch would have had plenty of time to recreate a past Pearson or partial Bohr and Wheeler-like calculation (see below). That his first after-the-fact reference to the *Z* ∼ 100 limit appears in his November 1967 *Physics Today* article suggests that the AIP interview might have triggered recollections of his Hamburg years.

Finally, I develop here in a very condensed form an alternate suggestion for the origin of the *Z* ∼ 100 limit. This is based on a paper I published previously in this journal wherein fission is modeled as an initially spherical nucleus breaking into two spherical product nuclei which are just touching right after the breakup.^{12} The model uses the surface and Coulomb terms of the semi-empirical mass formula to describe the total energy of the nucleus in the form *E _{sphere}* =

*a*

_{S}A^{2/3}+

*a*

_{C}Z^{2}/

*A*

^{1/3}, where

*A*is the mass number and

*a*and

_{S}*a*are calibrating factors whose values, usually quoted in MeVs, can be extracted from fits of the mass formula. Converting to SI units, the 1936-era data Pearson quotes give

_{C}*a*∼ 13.2 MeV and

_{S}*a*∼ 0.58 MeV.

_{C}Now imagine that the initial nucleus fissions into two equal-size product nuclei, both of radius *R _{O}*/2

^{1/3}, where

*R*is the radius of the initial nucleus. Assuming that the fission is into two equal products simplifies the calculation in that the volume and asymmetry terms in the semi-empirical mass formula,

_{O}*a*and

_{v}A*a*(

_{a}*A − 2Z*)

^{2}/

*A*, need not be considered as the energies they contribute cancel out in the before-and-after calculation. Recomputing the total energy of the system as now comprising two spheres and adding in a term to represent the potential energy of two point-charges

*eZ*/2 separated by distance 2(

*R*/2

_{O}^{1/3}) gives

*E*=

_{split}*α*(

*a*

_{S}A^{2/3}) +

*β*(

*a*

_{C}Z^{2}/

*A*

^{1/3}), where {

*α*,

*β*} = {2

^{1/3}, 17/[12(2

^{2/3})]} = {1.2599, 0.8924}. [

*α*and

*β*here are identical to

*α*and (

*β*+

*γ*) of Ref. 12, with the mass ratio

*f*= 1, respectively. Changing the mass ratio of the fission products to any realistic value does not appreciably change the results.] Ignoring the complexities of intermediate distorted shapes and the fission barrier, splitting will be spontaneous if

*E*<

_{split}*E*, which reduces to

_{sphere}Bohr and Wheeler would later prove more rigorously that the factor 0.2599/0.1076 ∼ 2.42 is in fact exactly equal to 2. For a heavy element such as uranium, *A* ∼ 2.6 *Z*, hence giving a limiting *Z* against spontaneous fission of ∼143. This is admittedly somewhat of a stretch beyond *Z* ∼ 100, but the model is obviously an idealization. One immediate consideration is that if the two product nuclei are joined by a thin neck which causes their centers to be more than two radii apart at the moment of scission, this would cause *β* to decrease, which would cause the limiting value of *Z*^{2}/*A* to decrease. The important considerations are that the model relates closely to the semi-empirical mass formula, and that it would not have taken Frisch long to carry out or resurrect such a calculation.

In the end, the question of how Frisch did his calculation remains open, with at least three credible contenders: Rayleigh's vibrating-drop approach, Pearson's cloud-chamber speculation, and the liquid-drop model described here. Beyond this, if Frisch did do some such calculation, might he have come close to *predicting* fission, and why did he not think of applying his analysis to the confusing results of uranium-bombardment experiments? Perhaps he thought only of spontaneous fission, or that the limiting *Z* he arrived at was too far removed from that of uranium to suggest it as a candidate, or he interpreted the limit as a value beyond which nuclei could not possibly form to begin with. If such speculations had occurred to him, he certainly never alludes to them in any of his memoirs.

We will never know for certain how events unfolded in Sweden in 1938; there will always remain an element of mystery to Meitner and Frisch's deduction of the *Z* ∼ 100 limit. Fission is a textbook example of how great scientific discoveries can be very unplanned, discontinuous affairs.

The author was grateful to Michael Pearson for sharing a preprint version of his paper with me, and to him, Ruth Sime, and two anonymous reviewers for comments which led to improvements in this note.