Møller scattering: a neglected application of early quantum electrodynamics

Moiler Scattering., a Neglected Application of Early Quantum Electrodynamics

X A V I E R R O Q U I ~

Communicated by K. VON MEs

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 1. M~ller's method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

1.1. The foundations of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 201 1.2. An outline of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 1.3. Previous treatments of electron-electron interaction . . . . . . . . . . . . . . 208 1.4. First results - and errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

2. The Moller formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 2.t. Towards a relativistic scattering formula . . . . . . . . . . . . . . . . . . . . 218 2.2. The deduction of the formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 2.3. The Moller formula and quantum electrodynamics . . . . . . . . . . . . . . 229

3. The experimental testing of the Moller formula . . . . . . . . . . . . . . . . . . 232 3.1. Experimental knowledge of electron scattering around 1930 . . . . . . . . . 233 3.2. Champion's experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 3.3. Subsequent experiments prior to 1947 . . . . . . . . . . . . . . . . . . . . . . 248 3.4. Postwar interest and conclusive test . . . . . . . . . . . . . . . . . . . . . . . 252

Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Introduction*

T h e first a t t e m p t s to bu i ld a q u a n t u m a n d re la t iv i s t ic t h e o r y o f e lec t ro -

m a g n e t i c i n t e r ac t i ons da t e b a c k to the mid- twen t i e s , t h o u g h n o t unt i l the full

* In an article recently published in this Archive, HELGE KRAGH has analyzed the background and development of CHRISTIAN MOLLER's work on electron scattering in the early thirties. The fact that most of my paper was written while I was unaware of KRAGH's work, accounts for the similitudes. This article complements that of KRAGH in that Mt~LLER's actual deduction of his scattering formula, as well as its experimental test, are analyzed in greater detail.

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introduction of the renormalization idea in 1947, would quantum electrodynamics come of age. During this period, the theory faced grave difficulties, which included divergences in fundamental quantities, like the proper energy of the electron or the displacement of spectral lines, as well as the failure - at least until 1937 - in the application of the theory to the analysis and interpretation of cosmic rays. For some of its creators, and most of its critics, these troubles foreboded the imminence and the necessity of a conceptual revolutionJ

Most of the elementary processes that quantum electrodyrramics describes were calculated in first approximation during the early stage of the theory. Besides the phenomena of bremsstrahlung and pair creation and annihilation, this is the case of the relativistic collisions photon-electron (KLEIN-NISHINA formula, 1928), electron-electron (MOLLER formula, 1932), and positron-electron (BHABHA formula, 1936). The formal improvements introduced after the war enable these processes to be easily deduced, and today they represent paradig- matic applications of the theory. Yet_~was it the same in its early period, as has been implicitly assumed? 2

Towards the end of 1930, a young Danish physicist, CHRISTIAN MOLTER, devised a method to treat the collision between two relativistic electrons. Two years later, he presented his doctoral dissertation, where the penetration of rapidly moving electron.s through matter was theoretically analyzed on the basis of the method. As a particular case, he considered the interaction between two free electrons, now known as Moiler scattering, giving the formula that describes it correctly in first approximation. This paper analyzes the relationship between the MOLLER formula and quantum electrodynamics.

In the first place, the provisory and disappointing state of the theory in the beginning of the thirties led us to the questions: how did MOLLER actually work out the scattering formula? Did he intend to probe the then-existing quantum electrodynamics? With which aims did he address the problem of electron scattering in his doctoral dissertation? Were they fulfilled, or did they change as the work progressed? And secondly: did someone try to test the formula? In which context were the experiments carried out? Were they intended to provide early quantum electrodynamics with an experimental basis? In any case, what was their significance for the theory?

1 Quantum electrodynamics did not exist as a single, well-defined theoretical body in the early thirties. Different versions of the theory agreed on the necessity of quantizing the electromagnetic field, but diverged on matter waves' quantization. By "early quantum electrodynamics" I shall refer to quantum electrodynamics before renormalization, or more specifically, to the set of formulations of the theory that included in this period quantization of the electromagnetic field (see also note 1 in KRAGH'S paper). On early quantum electrodynamics see specially DARRIGOL (1984), R~GER (1989), and KRAGH (1990), chapters 6 and 8.

2 Cf for example GIORGI (1990), p. 449: "Perhaps the most impressive successes of early quantum field theory were the calculations of scattering probabilities for a variety of processes [those just mentioned]. . . They were in reasonable agreement with experiment".

Moiler Scattering 199

The method conceived by MOLLER was the subject of the only article he published in 1931. In the first section, I analyze its theoretical foundations and the preliminary results that MOLLER drew from it. An account of previous attempts to deal with electron interaction on the basis of DIRAC'S equation is also included, especially that of BREIT, as it was closely related to the wave-field theory of HEISENBERG and PAULI, the most general formulation of quantum electrodynamics around 1930.

In the second section, MOLLER'S actual deduction of the scattering formula, as well as its relation with early quantum electrodynamics, are considered. The formula appeared in the Annalen der Physik in the summer of 1932, in an article which later that year became MOLLER'S dissertation. A chronological account of MOLLER'S work during this period is also presented, giving particular attention to his changing motivations.

The third section addresses the experimental test of the formula. It focusses on the experiments by the British physicist F. C. CHAMPXON, who tested the MOLLER formula before it appeared in print. The experimental knowledge of electron-electron scattering around 1930 is also briefly reviewed, as it accounts for MOLLER'S surprise in the face of CHAMPION'S attempts. Finally, the surge of experiments around 1950, as well as the conclusive test of the formula in the early fifties, are considered.

1. Moller's method

At the end of September 1930, a 22-year old Russian student, LEV LANDAU, arrived in Copenhagen from Cambridge to spend two months at BOHR'S Insti- tute. 3 MOLLER was then finishing the last of the "not very exciting" papers with which he had started his career in physics. 4 LANDAU had met MOLLER earlier in the year, when he had visited the Institute for the first time; his presence in Copenhagen in October would be, according to MOLLER, "absolutely crucial . . . because h e . . . brought me into the scattering problem". 5

_ J

3 See the Registers Book of the Institute (Archives for the History of Quantum Physics, microfilm no. 35. Henceforward as AHQP-35).

4 Interview with MOLLER by CHARLES E. WEINER, 25 and 26 August 1971, p. 11 (Center for History of Physics, American Institute of Physics. Henceforward as MOLLER interview 1971, AIP). The article referred to contained a general method of evaluating higher-order approximations in BORN'S theory (MOLLER 1930c). In his first paper, MOLLER included relativistic considerations in GAMOW's explanation of radioactive disintegration (MOLLER 1929); his next two articles dealt with the application of BORN's perturbation theory to the anomalous scattering of a particles by light atoms (MOLLER 1930a, b). Except for MOLLER (1930a), these articles appeared in such a prestigious journal as Zeitschrift ffir Physik, not least because of MOLLER's command of German.

5 MOLLER interview 1971 (AIP), p. 13. See KRAGH's paper for the early career and background of MOLLER, and AASERUD (1990) on the working atmosphere at BOHR's Institute.

200 X. ROQUI~

The problem of the collision between two relativistic electrons became the subject of the only article MoI.I~EI~ published in 1931: "The Collision Between Two Particles, Taking Account of the Retardation of the Forces". 6 It described a method of treating the interaction between two relativistic particles in first order of perturbation theory. MoI~LE~ was indebted to LANDAU for some "absolutely essential remarks" that he did not detail in his article, though they were probably two. The first referred to the possibility of relativistically generalizing a method developed by HANS BEa'rt~ in a recent article, "Towards a Theory of the Passage of Fast Corpuscular Radiation Through Matter", which had appeared in the middle of 1930 and was to influence later developments greatly. 7 LANDAU'S second remark referred to the symmetry of the final result, in spite of the asymmetrical appearance of the method. 8

The relativistic analysis of the passage of charged particles through matter was a question of more than purely theoretical interest. The debate over the constitution of cosmic radiation that extended between 1928 (first hints that the radiation might be corpuscular in nature) and 1937 (discovery of the "meson", today's muon), reached a high point during the early 1930s. The possibility that high-energy electrons formed the incident primary radiation, rendered a theoretical analysis of the interactions they experienced in their passage through matter more necessary. Nevertheless, the application of his calculations to cosmic rays interested MOLLBR only after having developed his method. Initially, his interest in scattering was of a more fundamental character, and it was limited to the possibility of generalizing its theoretical treatment through the inclusion of relativity, in much the same way as he had analyzed radioactive decay in his first paper.

MoLI.Ert entitled his dissertation, which depended essentially on his article of 1931, the same as BETHE'S article, changing only "corpuscular radiation" for "electrons". What did he have to add to BEa'nE's thorough treatment?

Bethe had treated the collisions and stopping phenomena in the nonrelativistic case. He had written the matrix element for the transition in such a way that it looked as if one particle in its transition creates a charge distribution which then acts on the other through a Coulomb potential. And then the rather obvious idea came to do it relativistically; instead of using the Coulomb static potential, to introduce a retarded potential corresponding to the charge and current which corresponds to such a transition. 9

MOLLER, in few words, had relativistically generalized BETrtE'S analysis. He himself would later describe his treatment as a generalization of BORN'S method, thus denoting one of its bases. Yet this is too simple a characterization, and we now consider it more carefully.

6 MOLLER (1931). 7 BETHE (1930). 8 MOLLER interview 1971 (AIP), p. 13. 9 Interview of MOLLER by THOMAS S. KUHN, 29 July 1963 (AHQP/Oral History

Interviews-4), p. 15. Henceforward as MOLLER interview 1963, AHQP/OHI-4.

Moller Scattering 201

1.1. The foundations of the method

MOLLER'S method was essentially based on three distinct theoretical developments: BORN'S perturbation method, DIRAC'S theory of the electron, and the correspondence principle. Around 1930, they were more firmly grounded than any of the existing versions of quantum electrodynamics, and they gave MOLLER'S analysis theoretical soundness and certainty. I

BORN'S perturbation method is an approximation method that makes it possible to consider aperiodic processes, like collisions, on the basis of quantum mechanics. MAX BORN introduced it in 1926, along with the probabilistic interpretation of the wave function, x~ It will suffice here to say that BORN'S collisions theory was rapidly embodied in quantum mechanics, and it soon proved indis- pensable in the applications of the theory. In 1930 MaLLER could safely use it - even to the support of his argument.

The necessity of including relativity in the frame of the new quantum mechanics had clearly shown itself to its creators from the outset. It is well known that SCHRODINGER tried initially to fincl a relativistic equation, Similar attempts to bring relativity, even general relativity, into quantum mechanics, were numerous during 1926 and 192Z The so-called KLEIN-GORDON equation, the most successful of these, failed however to account for the hydrogen atom. DIRAC proposed in January 1928 a new equation that, in spite of its uncertain physical interpretation, was impressive for its formal simplicity and the natural inclusion of spin. a x DIRAC'S was a first order equation - something essential for transformation theory, and one of the determining factors in DIRAC'S proced- u r e - and described the electron by four-component wave functions. The main difficulty of the equation, its negative energy solutions, was immediately recognized though it seems not to have initially represented a grave problem.

While a student in BOHR'S Institute, MOLLER soon assimilated DIRAC'S theory and made thorough use of it in his first article} 2 Although by the end of 1930 the difficulties of interpretation of the negative energy states were clearly perceived, the successful application of the theory to scattering processes and its formal and aesthetic appeal had secured it a place in physics.

It was only natural for MOLLER, who had had BOUR and KLEIN as his mentors, to adopt the correspondence principle as a basis for his work. As has been argued by KRAOH, the correspondence approach was highly esteemed in Copenhagen, probably more so than anywhere else. 13 This is the most characteristic element, yet also the most subtle, of MOLLER'S treatment of electron- electron interaction.

io BORN (1926). li DIRAC (1928a, b). On the genesis of DIRAC's theory see KRAGH (1981), MOVER

(1981a); on its reception see MOVER (t981b, c). The most complete account to date of DIRAC's life and work is to be found in his scientific biography by KRAGH (KRAGH 1990).

12 MOLLER (1929). Soon after DIRAC'S theory appeared, MOLLER reported it in one of the frequent seminars at the Institute (MOLLER interview 1971, AIP, p. 2).

13 See section 2.3 in KRAGH's paper.

202 X. ROQUI~

MOLLER'S understanding of correspondence stemmed from KLEIN'S adapta- tion of the principle to its use within quantum theory. While the correspondence principle had played a principal role in the development of quantum theory and in the creation of the new quantum mechanics, the introduction of HEISENBERG'S quantization rules in 1925 significantly diminished its significance. 14

At the end of 1926, KLEIN showed anew the significance of the principle in an article that would prove essential to MOLLER'S method: "Electrodynamics and Wave Mechanics from the Standpoint of the Correspondence Principle". In KLEIN'S formulation, the principle becomes a method to transpose the electromagnetic field equations into quantum mechanics; the key elements of this reinterpretation are the charge and current densities: is

As we try here to connect wave mechanics directly with the electromagnetic field equations, we will assume, that the electromagnetic phenomena corresponding to the magnitudes Pn and ], give, in the sense of Bohr's correspondence principle, a quantitative expression of the observable actions related to the presence of an atom in a certain stationary state [-represented by hi.

KLEIN showed in his article some examples of this procedure: the case of an electron in a central field, the perturbation of an atom by an external field, and the Compton effect, among others.

The wave-mechanical expressions for the charge and current densities become the privileged junction between classical expressions and its quantum equivalents. This transposition to a different conceptual frame is characterized in KLEIN'S article by the term "korrespondenzmfigig", which might be translated as "according to a principle of correspondence". What is denoted as "korres- pondenzmfiBig" is the way of considering Maxwell equations in wave mechanics, as well as the way of evaluating in the new theory well-known expressions of classical electromagnetism, like the continuity equation or the relation between fields and potentials.

MOLLER followed KLEIN in his use of the term "korrespondenzmfigig" and in the generalization of quantum nonrelativistic expressions through the charge and current densities. But the critical dependence of the method on the principle is never very explicit in MOLLER'S papers. In his article of 1931, MOLLER talks only of "analogical generalization of Born's theory" ("sinngem/ige Verall- gemeinerung der Bornschen StoBtheorie"), or of "making plausible" the relativistic generalization. 16 MOLLER cites KLEIN'S article only to observe that his calculation of the transition probability has "a certain resemblance" ("gewisse ~,hnlichkeit") with KLEIN'S method of evaluating the radiation emited by an atom.17 In the article that contains the most detailed exposition of the method and its application to the stopping of relativistic electrons, MOLLER characterizes

1,* Cf. JAMMER (1966), pp. 109-118; MEYER-ABICH (1965). is KLEIN (1927), p. 419 (translations by the author). 16 MOLLER (1931), p. 787. 17 Ibid. , p. 788.


it more explicitly as "korrespondenzm/igig", but only on one occasion. 18 In the remainder of the article he continues referring to plausible or analogical generalizations. The application of the principle is masterfully exemplified, but the significance of this procedure is not evaluated in depth.

Those who wrote about MOLLER'S method, however, did notice its critical dependence on a correspondence argument and pointed to the correspondence principle as one of the most characteristic, elements of MOLLER'S work. Soon after the appearance of the article of 1931, LEON ROSENFELD clearly stated this conceptual filiation. By September 1931 ROSENFELO, since February in Copen- hagen, had completed a treatment of the relativistic many-body problem on the basis of "Heisenberg's refinement of the correspondence principle". 19 He had obtained "a generalization of the method used by Moiler in the relativistic treatment of the scattering problem, in agreement with the results deduced in another way by Breit". 2~ On September 30, MOLLER wrote to BETHE that "Mr. Rosenfeld has carried out a rather interesting generalization of the method used in my paper, by means of which he deduces quite general relativistic expressions for the exchange between any number of particles, naturally only in the region where the interaction can be considered small, thus, where the relativistic many-body problem according to the correspondence principle makes sense".21

When, in the beginnings of 1932, DxRac presented his alternative quantum electrodynamics, the method was also more explicitly related than in MOLLER'S article with the correspondence principle. 2z DIRAC stressed the role played by the correspondence principle in the development of quantum theory, and he compared MOLLER'S method with the methods based on the correspondence principle in use before the advent of the new quantum mechanics. 23 Though disliking its lack of generality, DIRAC praised MOLLER'S method as "a definite advance in the relativistic theory of the interaction of two electrons". 24

OPPENHEIMER also reacted to the appearance of MOLLER'S article, and together with J. F. CARLSON, he applied the method to the calculation of the

18 MOLLER (1932), p. 533: "Das hier benutzte korrespondenzm/iBige Verfahren zur Behandlung der Relativit/itseffekte ist prinzipiell ein N/iherungsverfahren, wo die Wech- selwirkung zwischen dem Atom und dem einfallenden Elektron als St6rung behandelt wird".

19 ROSENFELD (1931), p. 253, referring to HEISENBERG (1931). 20 Ibid., p. 253. See 1.3. on BREIT'S treatment. 21 MOLLER to BETHE, September 30, 1931 (AHQP-59; my emphasis). 22 DIRAC (1932). See KRAGH (1990), pp. 132 ft. 23 Ibid., p. 455: "The method by which Moiler obtained his result may be compared

with the methods of the Correspondence Principle in use before the introduction of Heisenberg's matrix theory, for calculating Einstein's A and B coefficients from classical models. In certain cases the result obtained was unambiguous (usually those cases for which the result was zero) and was then presumed to be correct. In general, however, there was ambiguity, so that one could get no reliable accurate result".

24 Ibid., p. 455.

204 X. RoQut~

collision between a high-energy electron and the "hypothetical elementary neu- tral particle whose existence was tentatively suggested by Pauli ' , referring to PAULI'S "neutron". 25 OPPENI-IEIMER & CARLSON had realized the potentiality of the method and, along with a brief but precise description, they valued it favorably in the introductory remarks of their article? 6

Quite recently Moller has given a beautiful method of treating the relativistic impact of two electrons. This method is based upon a refinement of the correspondence principle; it neglects higher powers of the interaction energy between the electrons, and the effect of radiative forces; but within these limits it is strict and unambiguous, and enables one to take account, not only of the relativistic variation of mass with the velocity o f the electrons, but of the retardation of the forces between them, of the spin forces, of interchange and the exclusion principle.

Apart from the sympathetic reaction that these comments denote, they are atso clearly related to the difficulties that the attempts to quantize the electromagnetic field had encountered. These had led some physicists to resort to the old idea of correspondence, that had so significantly guided the quantum theory in its beginnings. 27 It is difficult to assess to what extent this revival of the principle represented a real alternative to the attempts of field quantization, like those of HEISENBERG and PgtJLI, or merely a provisional solution. Ques- tioned on this by KtJHN, MOLLER answered: "I think one looked upon [the correspondence principle approach] as a preliminary thing. I mean something like the Heisenberg-Pauli theory would always appear as something more fundamental". His evaluation seems influenced by later developments. 2s

In any case, these were the fundamentals of MOLLER'S analysis, the basis of the method he developed between October 1930 and May i931, when he submitted his article. Little can be said of its elaboration. MOLLER was still relatively unknown outside Copenhagen, and no correspondence remains from this period - indeed, this article would increase his contacts with other physicists. Our analysis is therefore limited to the published article.

25 OPPENHEIMER ~r CARLSON (1932b). See also the previous communications OPPENHEIMER & CARLSON (1931), (1932a). Today's neutron and neutrino had not yet been "discovered", and PAULI'S hypothetical particle accounted for the failure of statis- tics of the nuclei as well as the continuous fl spectrum. OPPENHEIMER & CARLSON did not specify either the mass or the spin of the "neutron", and considered "the most general particle which satisfies the wave equation proposed by Pauli". OPPENHEIMER & CARLSON (1931) contained the first mention of PAULI'S neutron, about which they had heard PAULI lecture in Pasadena (cf. BROWN 1978, V. ME~ENN 1987).

26 Ibid., p. 765. The characterization of the method as "based upon a refinement of the correspondence principle", stemmed probably from ROSENFELD'S paper mentioned above.

27 On the reappraisal of the correspondence principle circa 1931 see DARRIGOL (1986), ME~ENN (1989), ROGER (1989).

28 MOLLER interview 1963 (AHQP), p. 12. See the full quotation in KRAGH's paper, section 2.3.


1.2. An outline of the method

In his article of 1931, MOLLER first describes the quantum procedure that he was later to generalize, not only to make its introduction easier, but also to "make plausible" the generalization. In Bogy's theory, a collision is described as the transition between two stationary states of the total system. In the collision between two particles they are considered, as MOLLER notices, to occupy initially certain stationary states I~1) and t~b2), represented by plane waves,

11~1 ) = ale i (k i " r l -~ l t )= e-i~'~l~ol(rl) ) ,

11//2) = e-iC~ ) ,

where k = p/h and co = E/h. The interaction between particles 1 and 2 moves them to certain final stationary states, I~q,) and [~b2,). According to BORN'S theory, the probability that this transition takes place is given in first approximation by the square modulus of the matrix element

= (~0~,, ~02,1 v ( r ) l ~ o l , ~ 0 2 ) ,

where V(r) is the potential that describes the interaction. BEXHE had evaluated this matrix element through what apparently was no

more than a clever mathematical observation. It led however to an interesting physical interpretation. 29 The matrix element

( ~ 0 1 ' 1 V l q ) l ) ,

which in the plane wave representation and for Coulombian interaction is

f e ei(k 1 _kl,).rl d3rl Ir2 - rll

can be interpreted as the scalar potential V'(r2) created by the charge distribution

p = e . e i ( k l - k r ) ' r l .

The potential V' can now be directly calculated by means of the potential equation, A V '= -4rip, of a well-known general solution in terms of p, and it is possible to write

= (~o2 ,1V' l~02) ,

which might quite evidently be interpreted as the action on particle 2 of a perturbation V'.

The interaction is therefore considered to take place in two successive steps in which only one particle is implied. As MOLLER explains, "to a certain transition of particle 1 corresponds a certain charge density p, which induces through the potential equation a certain scalar potential. This potential acts as a perturbation on particle 2, and causes the transition of this particle to another state". 3~

29 BETHE (1930). 30 MOLLER (1931), p. 787.

206 X. ROQUI~

MOLLER'S method consists in the relativistic generalization of this argument, on the basis of the correspondence principle. The stationary states that the particles occupy before and after the collision are again described by plane waves, which, however, are now solutions of the D~RAC equation,

~1 = U(S1) ei(kl"rl --c~ = e - i c ~ ,

~12 = e - i~2 t ~o2(r2) .

The amplitudes u(s) are "four-component magnitudes ( 'Gr6Ben') which determine the direction of spin". 31 MOLLER remarks that for each value of the momentum p of one of the particles there are two proper solutions, corresponding to the possible values of the Spin variable s, thus neglecting, as he simply acknowledges in a footnote, the negative energy solutions. 32 This position on the negative energy states, interpretation of which was still uncertain, was not unusual even by 1930, despite the growing evidence that these states were an essential part of the theory. MOLLER'S attitude, in any case, was justified because he limited his treatment to the interaction between electrons, which are described by the positive energy solutions.

KLEIN'S understanding of correspondence must now be applied to obtain an expression for the transition probability. MOLLER starts from the charge and current associated to the transition of particle 1 according to DIRAC'S theory, 33

P = - e l 0 1 ' 0 1 ,

j = e iOI,~( i )0i .

The potentials associated to these densities can be obtained directly from the general expression for the potentials created by a charge and current distribution (LI~NARD-WIECnERT potentials). MOLLER simply says that the potential equations

1 024 Aqb c2 Or2 ----4rip , AA

"can be solved directly" to obtain

h2ei ~b~,l~l 4 =

n ( P l - P l ' ) 2 '

h2ei 01,~(1)Ol A = m

1 t~2A 4 n . C 2 ~t 2 -- C J '

7Z ( P l - - P l ' ) 2 '

where p = (E/c,p) represents the four-momentum. 34

31 MOLLER (1931), p. 788. Following a usual convention, the amplitudes of the positive energy solutions of the DIRAC equation are here represented by u(s). MOLLER represents them by a(s).

32 MOLLER (1931), p. 788n. 33 The superscript in DIRAC matrices ~ indicates that they only act on the wave

functions associated with the corresponding particle. 34 MOLLER does not use four-vectors, but rather explicitly states their scalar and


This potential is treated like a perturbation which acts on particle 2. According to DIRAC'S theory this particle satisfies the equation

( ih ~ - ich~(2) " ~7 + flm2 c2 ) ~t2 = -e2(q~+~(2) 'A)O2 .

The probability of a transition (1 ,2)~(1 ' ,2 ' ) taking place in the unit time is then, "according to well-known formulas of perturbation theory", as given by

4re 2 P(1,2 --* 1',2') = -~-1~[26(E, + E2 - E r - E2,) , (1)

where r = (~02,1U(r2)l~02) = S q~*2, U(r2)q)2d3r2 ,

and U(r2) is the spatial part of the perturbation function, --e2( ~ + ~(2).~). Taking account of the plane wave representation, the matrix element writes

- ele2h2 fq~l ' u[,ua - ~:(2),(UI,~(I)Ul) (p l ~ l , ) 2 ei(k~-kr)'r2tpzd3r2 .

If the delta function in expression (1) secures energy conservation, on integrating the last expression momentum conservation is obtained. One should only observe that by definition of the delta function

I el(k1 + k2 - k r - k2,)" r2 d3r2 = h3(~ (Pl + P2 - P l" - P2') , (2)

and therefore we have finally for the matrix element a6

_ ele2 h2 u*2,u2u~,ux - (u*2,~(2)u2)"(u~,fi(1)Ul) ha6(pt +P2 - P r - P 2 ' ) (Pl -- Pl') 2

(3)

This equation essentially describes the interaction. MOLTER is pleased to recognize directly its relativistic invariance and its symmetry, particularly satisfying given the asymmetry of the method. He then discusses the meaning of the different terms and the relation with previous treatments. There had been other attempts, before MOLLER'S, to deal with electron-electron interaction on

vectorial components. Yet he often refers to them in the text and I have used them to transcribe some of his expressions.

35 MOLLER (1931), p. 789. 36 Introducing the notation 7 u, with 7~ fl and ~= 7~ and noticing that

u~, ua = ~r 7~ and u~, ~a)u 1 = ul' ~ul, we write the second fraction as

g 2 , ~'u U2 /g l , ~/~Ul ~

( P l ' - - P l ) 2 '

the same as we should write directly for the interaction between two distinguishable DIRAC particles.

208 X. ROQUI~

the basis of the Dirac equation, which we now examine. We shall thus not only appreciate the peculiarities of MOLLER'S treatment, but also show how the problem was approached from quantum electrodynamics and how little hope the theory gave that it could be solved.

1.3. Previous treatments o f electron-electron interaction

Given the success of DIRAC'S electron theory in its immediate application to the hydrogen atom, and its enthusiastic reception, it is understandable that it would shortly be applied to the helium atom. In June 1926 HEISENBER~ had solved the helium atom according to quantum mechanics through the application of the Schr6dinger equation to the many-body problem. 37 The British physicist J. A. GAUNT, then professor of physics at Trinity College, Cambridge, applied the DIRAC equation to the helium atom in the middle of 1928. The fundamental character of the new equation justified the reconsideration of the problem, as "Schr6dinger's equation is likely to be supplanted as the foundation of wave-mechanics by the equation recently put forward by Dirac". 3s

GAUNT obtained the magnetic interaction energy between two electrons, from the energy associated in DIRAC'S theory to the movement of an electron in a magnetic field of potential vector A, -e~:" A. This potential vector was created by the other electron, and GAUNT resorted to the expressions for the charge and current densities associated in DIRAC'S theory to a wave function 0, - -e0tO and ecOt~b. The similitude with MOLLER'S method goes no further, because GAUNT grossly simplified the resolution of the problem, merely stating that "if we are right in attributing a scalar potential - e / r to the charge on the electron, we should also expect a vector potential due to the moving charge A = e?/r." He thus obtained for the interaction Hamiltonian 39

e 2 Hint = - - (1 - ~1. ~2) . (4)

r

MOLLER'S treatment contains GAUNT'S. Having explicitly written the denominator of expression (3) for the matrix element, MOLLER has only to suppress the term (El. - El) 2 to show it. In this way, the formulas he has obtained, among them the expression for the matrix element, "reduce to those obtained when the

37 HEISENBERG (1926a), (1926b). 38 GAUNT (1929a), p. 513. See also GAUNT (1929b), submitted like the previous

article on November 6, 1928, which contained more detailed calculations. 39 GAUNT'S equation could be directly derived from the magnetic interaction energy

between two electrons,

Ein t : 1 - - C---- T - j ,

associating DIRAC's ~ matrices with v/c - as indeed HEISENBERG did at about the same time; see footnote 41.


collision is treated according to the equation of the two-body problem put forward by Gaunt". Thus, however, not only relativistic invariance is lost, but also "the retardation is obviously disregarded", the influence of which "is in most cases of the same order of magnitude as spin interaction". 4~ This omission he judges as an important flaw of GAUNT'S expression.

In the context of their work on the quantization of the electromagnetic field, HEISENBERG and PAULI had independently arrived at GAtmT'S interaction potential. On November 30, 1928, GREGORY BREIT, then working with PAULI in Zurich, wrote to EHRENFESTff 1 "Pauli and Heisenberg have set up an equation for two electrons similar to DigAc'S equation:

{ : ( ) "( eAt"~ po+~Al+-A1o'+Eo~[ p[+~A~ +E~Zk P~'+c kJ

+~14mc +~t4tmc--~(1--2.a~t')}qb=O (5)

where the last term represents the magnetic interaction. It can be deduced by the method of amplitude quantization, or by interpreting the e's as velocities". In the same letter, however, BREIT did more than simply convey HEISENBERG and PAULfS results: he himself had significantly contributed to the two-electron problem, having just deduced an equation which considered retardation. The Breit equation is the major attempt to describe electron-electron interaction on the basis of the DIRAC equation, earlier than that of MOLLER.

GRZGORY BREIT, a Russian-born physicist educated in the United States, had been interested in DIRAC'S equation since its appearance. 4z In 1928, having spent two years in high voltage experiments, BREIT thought the moment had arrived to come back to theory, with which he had only "spasmodically" been concerned during this time. In May 1928, BRErT left the experiments in the hands of his collaborator, MERLE TUVE, to seek "a good foundation in the principles of the new quantum mechanics" in Europe. 43

BREIT wanted to begin his stay in Europe at Leiden, but EnRENFEST recommended that he work in Zurich with PAULL Following his advice, BREIT arrived in Zurich at the beginning of September, and he remained there for the rest of

40 MOLLER (1931), p. 791. To suppress this term implies disregard of retardation, because it comes directly from the solutions of the potential equations in terms of the charge and current densities (LII~NARD-WIECHERT potentials). Only by taking the retarded value of these densities can the energy term, which is embodied in an exponential function of time, appear when the integration over space that the general solution prescribes is effected.

41 BREIT to EHRENFEST, November 30, 1928 (AHQP/EHRENFEST Archive-18). BREIT later explained that the interaction energy had been "suggested independently by Pauli and Heisenberg. Heisenberg arrived at the result by interpreting Dirac's ct's as expressions for the velocity - (v/c) while Pauli got it by considerations forming an extension to those of Jordan and Klein with quantizing wave-amplitudes" (BREIT 1929, p. 553n).

42 See BREIT (1928). 43 BREIT to EHRENFEST, May 21, 1928 (AHQP/EHR-18).

210 X. ROQUI~

the year. 44 In his correspondence before his travel with EHRENFEST, BREIT expressed his interest in quantum mechanics, but he did not mention any theoretical question that particularly interested him. PAULI introduced him to the problem of electron-electron interaction, a problem he and HEISENBERG had lately considered while attempting to quantize the electromagnetic field.

In the first place, BREIT obtained from equation (5) an expression for the energy of the helium atom which closely resembled that of HEISENBERG. He tried then to consider retardation, in which he succeeded, except for a term of dubious interpretation: 45 "I have also tried then to consider retardation, always in the same approximation, (v/c) 2. With the interpretation of ~ , ~ i as the components of the velocity, this works with the help of Darwin's old (1920) article. The result contains only one incomprehensible term, which could perhaps be checked on the ortho-helium structure".

Pushed by PAULI, BREIT at tempted then to consider retardation exactly. 46 Although this at tempt failed, he favourably impressed PAULI, who on December 24 wrote to EHRENFEST: 47 "I was very glad that you sent Mr. Breit to me in Zurich, as I have been extraordinarily well-satisfied with him. However, I would be even more glad if you could soon send me an equally able man, who will not run off to America after a cout)le of weeks in pursuit of 5 million (not dollars, but) volts!".

Back in the United States, BREIT completed his analysis and published the results. As he had explained to EHRENFEST, BREIT set up "an approximate wave-equation which takes into account terms of the order (v/c) 2 in the interaction of two electrons". 4s The equation was deduced from DARWIN'S classical Hamiltonian for two electrons, which included terms that described the free electrons, their interaction with an external field, and also the interaction between them

Hint_ele2 ele2 Ip~p2+pl"rp2"r 1 r 2c2ml m2 r3 , (6)

where the indexes 1 and 2 distinguish the electrons, and r = r E -- r 1.49 The first term obviously corresponds to electrostatic interaction, the second is the term associated with retardation.

44 EHRENFEST to BREIT, June 1, 1928, and BREIT to EHRENFEST, November 30, 1928 (AHQP/EHR-18). In this last letter BREIT said that he had already spent some three months in Zurich.

45 BRErr to EHRENFEST, November 30, i928 (AHQP/EHR-18). 4.6 Same letter: "Pauli has pushed me then to the exact consideration of retardation.

A couple of days ago we still had hopes, but now we have left it. He and Heisenberg had already tried it in the spring". In the same letter BREIT said he intended to leave Zurich in the middle of January.

47 PAULI to EHRENFEST, December 24, 1928 (HERMANN, V. ME~/'ENN and WEISSKOPF (1979), letter [212]. Henceforth as PAULI Briefe I, [212]). It seems that BREIT had once again become interested in high voltage.

4s BREIT (1929), p. 553 (submitted on May 31, 1929). 49 DARWIN (1920). See formula (15) on page 546.


DARWIN'S Hamiltonian suggested to BREIT the idea of trying for electron- electron interaction in DIRAC'S theory

Hint = ~ [ 1 ~(i). ~(2)2 ~(1)"/" ~(2) " rl2-F2- A '

on the basis of the relation

dr

c d t

(7)

which had been "communicated to the writer by letter by Dr. P. A. M. Dirac in the summer of 1928". s~ BREIT noticed that if the effect of retardation was not taken into account, and only the coulombian and magnetic interactions were considered, instead of (7)

e 2 Hin t = - - [1 -- ~(1). ~(2)] (8)

r

was obtained, "the equation used by Gaunt and Eddington, and claimed by them to be correct", sx

BRE1T'S analysis, based like those of HEISENBERG and GAUNT on classical models of electron-electron interaction, had the advantage of including retardation. BREIT was, however, perfectly aware of a rather significant aspect of his interaction formula: it could also be deduced from "the new Heisenberg-Pauli theory of wave-fields", to within terms of the first order in the Coulomb interaction. HEISENBERG and PAULI'S was indeed a new theory. The attempts of both physicists to build a quantum theory of the electromagnetic field had faced the difficulty of applying to the field the canonical quantization procedure. In January 1929 HEISENBERG found a way to overcome the problem by means of a formal trick. That finally enabled him and PAULI to write their "quantum dynamics of wave-fields". As HEISENBERG was leaving for the United States on March 1, the article was rapidly written, PAULI doing most of the work. It was completed by mid-March. 52

HEISENBERG'S travel gave BREIT the opportuni ty to meet the other creator of the theory, this time without having to cross the Atlantic. BREIT met HEISEN- BERG in Boston and Chicago, and he discussed with him the new theory and its

50 BREIT (1929), p. 555. DIRAC's letter is not preserved among the BREIT Papers (Yale University, Beinecke Rare Book and Manuscript Library; microfilm copy at the Center for History of Physics, AIP New York). Two letters from BREIT to DIRAC (June 20, July 26, 1928), relating to BREIT'S stay in Europe, are to be found among the DIRAC Papers (The Florida State University, Tallahassee; copy in Churchill College Archives Centre, Churchill College, Cambridge).

51 Ibid., p. 557. ARTHUR EDDINGTON followed GAUNT'S treatment in the article BREIT cited, EDDINGTON (1929).

52 HEISENBERG & PAULI (1929). See PAULI Briefe I, pp. 482 ft., DARRIGOL (1984), pp. 479-487.

212 X. ROQUI~

application to the electron-electron problem. 53 He could thus include in his article a section in which he deduced the interaction Hamiltonian (7) from the Heisenberg-Pauli theory. Once finished the paper, BREIT wrote to EHRENFEST: 54

Heisenberg-Pauli's quantized waves give in a certain sense the same result that is obtained according to the correspondence principle ("nach dem Korrespondenzprinzip") in the configuration space, in the approximation (v/c) 2. Heisenberg, Pauli and Oppenheimer believe that this equation suffices for the helium-problem, but leads nowhere ("gibt es doch nichts Gutes") in the exact solution of the two electron-problem. The approximation (v/c) a yields quite unreasonable results, which might be connected with Dirac's +m difficulty.

The equation was still worth applying to the helium atom. Yet by the time BREIX submitted the article in which he applied his equation to helium, his initial confidence in the equation, which "appeared at the time as a likely one", had much diminished. 55 OPPENHEIMER had recently subjected to close examina- tion the question of the proper energy of the electron, which ranked among the less satisfactory aspects of HEISENBERa & PAULfS article. He showed clearly that "it is impossible on the present theory to eliminate the interaction of a charge with its own field, and that the theory leads to false predictions when it is applied to compute the energy levels and the frequency of the absorption and emission lines of an atom", s6 According to the theory, the energy levels were not only infinitely displaced from their positions according to BOHR'S theory, but the energy differences between adjacent levels were also infinite. OPPENHEIMER concluded that "the present theory will not be applicable to any problem where relativistic effects are important, where, that is, we cannot be guided throughout by the limiting case c ~ oo". 57

When BREIT published the article where he applied his equation to the calculation of the fine structure of helium, the effects of OPPENHEIMER'S clear statements were evident. OPPENHEIMER had clearly shown that BREIT'S equation could be derived only "if certain infinite terms of the interaction energy are systematically neglected". 5s BREIT related the failure of quantum mechanics to account for the electromagnetic interaction between two electrons with two

53 BREIT to EHRENFEST, July 16, 1929 (AHQP/EHR-18): "Heisenberg habe ich in Boston und auch in Chicago aufgesucht und bei ihm gearbeitet". In his article, BREIT expressed his thanks to HEISENBERG "for discussions about the new Heisenberg-Pauli theory of wave-fields and his encouraging interest in this work" (BREIT 1929, p. 553n).

54 Same letter, my emphasis. 55 BREIT (1930), p. 383. 56 OPPENHEIMER (1930), p. 461. OPPENHEIMER worked on this problem in Zurich

with PAULI in the first half of 1929. They did not manage to publish an article together, and finally a second article by HEISENBERG & PAULI appeared (HEISENBERG & PnULI 1930). See KIMBALL SMITH & WEINER (1980), pp. 121 ft.; PnULI Briefe I, [231] and [2321 .

57 Ibid., p. 477 (my emphasis). 58 BREIT (1930), p. 384.


"difficult questions: (1) the size of the electron i.e. whether the electron can be located at a point and (2) the Dirac jumps to states of negative energy". BREIT'S conclusion clearly conveyed his pessimism: 59 "Neither of these questions can be answered at present and it seems that no satisfactory purely theoretical solution of the two electron problem can be obtained before this is done." The Heisen- berg-Pauli theory, the most general formulation of quantum electrodynamics, gave little hope therefore that the problem MOLLER was to solve some months later could even be tackled.

Two years later, BREn" was to make more definite statements in a review article on "Quantum Theory of Dispersion" for the Review of Modern Physics. 6~ While preferring to justify KLE~N'S correspondence method by means of "the theory of the light quanta", BREn observed that this theory "has not gone so far much beyond the justification of the correspondence method a n d . . , it cannot claim to be logically consistent on account of the well-known divergence difficulties". BRZlT praised KLEIN'S correspondence method "as a particularly clear way of stating our knowledge about the probabilities of spontaneous emission and of scattering", and "highly recommended" it "by its simplicity". "In practi- cal applications", he added, "it has the additional advantages of avoiding too complicated calculations and of enabling one to visualize the phenomena in terms of charge and current distributions". 61 When BRHT commented on his previous work on scattering, OPPZNHEIMER'S negative appraisal was still apparent: 62 "The interaction between two particles can also be treated according to [Heisenberg-Pauli's] quantum electrodynamics, and it is possible to explain by this means the interactions of the electron spins of two particles as well as the orbital and orbit spin interactions. It should be remembered, however, that the divergence difficulties of the theory make it impossible, according to Oppenheimer, to arrive at a unique interpretation of the results". After being deeply involved for more than two years with the two-electron problem and with the application to it of Heisenberg-Pauli's theory, BREn" finally concluded: 63 "It will be seen from the above review of the work on the two electron problem that the theory of light quanta is not a very satisfactory tool for its discussion. Results can be obtained, but without additional physical considerations they are not unique".

In 1931, MOLLER only mentioned BREn"s treatment in a footnote, comment- ing on its application by HUGH C. WOLFE to the scattering of high velocity electrons in hydrogen. WOLFE, a National Research Fellow who worked at Caltech with OPPENHEIMER between 1929 and 1931, had considered this problem towards the end of 1930 as a test of the interaction energy between two electrons. He compared the simple electrostatic interaction, and GHENT'S and BRHT'S formulas. WOLFE said of BREIT'S formula that it had "the most theoretical justification", stressing the fact that it could be obtained from the

59 Ibid., p. 384. 6o BREIT (1932b), (1933). See also BREIT (1932a). 61 BREIT (1933), p. 129. 62 Ibid., p. 131. 63 Ibid., p. 136.

214 X. ROQUE

Heisenberg-Pauli theory. Yet he added that BRHT'S results were "no better than those of Gaunt" when the formula was applied to helium, hence the interest "to find another place to use these formulas, where it may be possible to determine which gives results in better agreement with experiment". 64

WOLFE computed the cross section according to the different interaction potentials, but he could not test them for lack of experimental data. In the last section we shall see why so few experiments had been done by 1930 on the scattering between free electrons.

1.4. First results - and errors

After the brief reference to GAUNT'S and BREIT'S treatments, M~LLER derives the differential cross section from expression (3) for the transition probability. He considers P2 = 0, takes the Z axis along the momentum of the incident particle, and introduces polar coordinates in the pr-space. Then

d3pl ' = [Pr[ dp~, sin OdO&p = [PI'I E r dE1, sin OdO&o . c

MOLLER calculates the cross section from the transition probability dividing by the number of scatterer particles, ut2u2 V (V, volume of the r2-space) and by the flux of incident particles, - c u ~ ( ' ) u l . He finally obtains, after integrating over P2' and Et,, and adding over the final spins 65

h ~1 ~2 da(O) 4[prlEz, 6^2^2 = c2 ~ (S l szp l lA ls l , s z ,p l , ) s inOdOd~o, (9)

SI 'S2,

where

(S1 SZpl [A lSl,S2,pl,)

1 U2,U2Ul ,U 1 (10) = -

This expression containedan~error "which nobody discovered, which I corrected in the second paper": 66 MOLLER had separately integrated over Pz' and E r not taking into account that those magnitudes were related. A factor 2 was missing, as H. J. BHABHA discovered some years later while applying MOLLER'S method to electron-positron collision. 67

6,* WOLFE (1931), p. 592. 65 The integration over P2' is readily done taking into account expression (2) for the

delta function: 2~i

I a 3 p 2 , 1 I + , 1 - - , 2 + , 2 3r212 = h3 S a 3 r 2 = V .

66 MOLLER interview 1971 (AIP), p. 13. See MOLLER (1932), p. 565n. 67 BHABHA to MOLLER, October 13 and 29, 1935 (MOLLER Papers, Niels Bohr

Archive, Copenhagen).


MOLLER observes that the calculations can be carried out exactly, though "they yield in general rather complicated expressions". 6s He therefore restricts his attention to the nonrelativistic and ultrarelativistic limits.

In the nonrelativistic limit, the velocity of the incident electrons allows a development in fl = v/c. MOLLER indicates that up to the order f12 the cross section is then given by

da(O) e*sin202dOdq~ { 1 1 1 = m2v 4 ~ + COS 4 0 sin 2 0 COS 2 0

4 ~ + cos4~ sinE0cos20 cosa0 ' (11)

and that the same expression is obtained in this approximation from BREIT'S equation, according to WOLFE'S calculations. To show the equivalence with WOLFE'S result 69

e4 sin 20 2dO dq~ f 1 1 1 dtr(O)

m2v 4 ~ + Cos 4 0 sin 2 0 cos 2 0

4 ~ +cos4~ sin 20cos 20 '

since WOLFE integrates over W' = El, + E2, and p' =Pl" +P2' , this expression is to be multiplied by the transformation factor 1 - fie sin 20/4 (in order f12), as MOLLER notices.

As a matter of fact, MOLLER'S result is obtained in this way except for the 5 3

term cos2--- ~ , which is replaced by co-~0" As MOLLER stresses the equivalence of

both results, this was prol~ably a misprint - a suspicion we will later confirm. MOLLER characterizes his formula (11) as "the generalization of a formula by Mott, which considers the exchange but not the relativity". 7~ MoTa-'s formula results in this limit from suppressing the f12 term

da(O) = e4sin202dOdq~ { 1 1 1 } (13) m2/) 4 ~ + COS 4 0 sin 2 0 cos 2 0 '

Suppressing still the last term in this expression, we obtain the classical scattering formula: 71

dtr(o)=e4sin202dOdq~ { 1 + 1 } m294 ~ ~ . (14)

68 MOLLER (1931), p. 793: "[Die Rechnungen] verlaufen ganz wie bei Wolfe [(1931)], der Unterschied liegt nur in dem Ausdruck ffir die Matrixelemente der Weehselwirkung'.

69 WOLFE (1931), equation (18) III. A trivial error in this expression has been corrected.

7o MOLLER (1931), p. 794. MOLLER cites MOTT (1930); see p. 47. 71 DARWIN (1914). DARWIN generalized in this paper RUTHERFORD'S classical ana-

lysis of the collision of an ~ particle with a nucleus, introducing the reduced mass of the system. In this way he could consider the passage of ct particles through helium.

216 X. ROQUE

MOLI~ER then calculates the quotient F between his expression for the cross section and the classical one, obtaining 72

F = A(O) + B(O) ,

sin2 0_cos2 0 ,] 3/~ 2 sinE 0 cos2 0 A(O) = (1 - / / z ) 1 - sing 0 + cosg0 j 2 sing0 + cos40 '

//2 sin z 0(1 + 3 sin 2 0 cos 2 0)

B(O) = ~- sing 0 + cos4 0

It can easily be seen that boldface 3 in B(O) corresponds to the numerator of the last term in (11). This means that MOLLER had rightly computed the nonrelativistic limit. It is therefore surprising that MOLLER did not mention this error in his correspondence until December 1931, in letters to HEISENBZRG (December 4) and CHAMPION (December 7). 73 MOLLER had previously had the opportuni ty to point out the error. On September 30 he had sent an offprint of his article to BEXHE, asking in turn for an offprint of BETHE'S important article on stopping. There is no mention of it, nor is there in the first letter that MOLLER sent to CHAMPION, on November 4. Most probably, MOLLER did not notice the error until he had the general formula available - a first, mistaken version - and he compared it with the limits he had obtained earlier, as suggested by the fact that he first commented on the error in the letters that contained this formula.

In the ultrarelativistic limit (~ >> 1), for dispersion angles 0 such as sin20 >> 1/~, MOLLER obtains

e g sin 20 dO do (tan 4 0 + tan z 0) (15) do(O) = 4m2v4

an expression he compares with the corresponding one according to BREIT'S equation 74

da(O) e 4 sin 20 dO dcp (tang 0 + tan 2 0 Sill g 0) 4mZv 4

MOLLER only adds that, according to this expression, the angular dependence of the cross section essentially differs from that for slow electrons.

Formula (15) for the ultrarelativistic cross section was not correct. Again, MOLLER realized this only when he had at his disposal the right general

72 An error in this expression has been corrected: the factor of the last term in A(O) reads in MOLLER'S paper 3/32/4 (p. 794).

73 AHQP-59. The carbon paper copies of these typewritten letters which were microfilmed and incorporated in the archive do not contain the formulas. The draft of CHAMPION'S letter, which also is extant, does contain the correction.

74 WOLFE did not compute the ultrarelativistic limit. MOLLER might have calculated it from the complicated general formula that WOLFE obtained from BREIT's Hamiltonian (WOLFE 1931, equation (21) III).


formula. He referred to this error only in his correspondence with HEISENBERG, w h 0 a t the end of 1931 was preparing an article in which the most significant experiments on cosmic radiation would be confronted with the theoretical predictions. HEISENBERG dealt with it in the stopping of high-energy electrons and 7 rays, and he thus became interested in MOLLER'S results for the ultrarelativistic limit. The resulting correspondence contains valuable information on the evolution of MOLLER'S work, which we shall use in the next section.

Only the last of these letters need concern us here. At the beginning of February 1932, MOLLER reported to HEISENBERG the correct version of his scattering formula, which he had obtained a few days before. This letter is no longer extant, though HHSENBERG had just received it upon answering MOLTER on February 15:75 "Thank you very much for your letter! Your present result is very satisfying to me. Your formula

da(O) - e4dO dq) (tan3 0 + tan 0) (16) m2c 4

is exactly the classical formula". After showing in a very simple way how this result stemmed from a classical argument, HHSENBERG added: "I like very much the fact that your formula is the same as the classical one, as the classical formula accounts well for the transition effects, but your former one not at all. Unfortunately I have already sent my paper - I will still try to modify it". HEISENBERG referred to his article on cosmic radiation, which had been submitted to the Annalen der Physik on February 13. He could finally indicate there that MOLLER'S result was equivalent to the classical one. 76

Before achieving this satisfying outcome, the disagreement of the calculations had led HEISENBERG to question the applicability of MOLLER'S method to high-energy electrons. HEISENBERG'S objections, as we shall see, would prove decisive in making MtOLLER desist from applying his results to cosmic radiation, as he intended initially to do. MOLLER clearly expressed this intention at the end of his article of 1931, the last paragraph of which considered the possibility of experimentally testing the ultrarelativistic cross section. MULLER observed that this extreme case could not be attained by means of artificially accelerated electrons, but "it could be realized in the electrons produced by the Aurora borealis, and also in the corpuscular rays that accompany the height,radiation. Formula [(15)] would thus be of use in the calculation of the stopping and scattering [of cosmic radiation], to which I hope to return later". 77 I simply

"75 HEISENBERG to MOLTER, February 15, 1932 (AHQP-59). HEISENBERG showed that to MOLTER with a simple calculation, the one he had included in his article.

76 HEISENBERG (1932), p. 433: "MOLTER I-(1931)] hat den StoB zweier freier Elek- tronen nach der Bornschen Methode unter Ber/icksichtigung der Retardierung behandelt und (nach Verbesserung eines Rechenfehlers in der zitierten Arbeit, f/Jr dessen briefliche Mitteilung ich Hrn. MOLTER zU groBem Dank verpflichtet bin) genau das klassische Resultat [(16)] erhalten".

77 MOLTER (1931), p. 795. MOLTER talks in rather general terms of a "corpuscular radiation", reflecting the uncertainty about its nature and about the constitution of

218 X. ROQUr

note here that MOLLER did not mention at this juncture the fi radiation, which was also known to consist of relativistic electrons.

In mid-1931, therefore, MOLLER had at his disposal a method which, at least in his limiting cases, led to promising results. If applied to the calculation of the stopping of relativistic electrons, it could even prove significant in the analysis of the cosmic rays, one of the most active research areas at the time. Equipped with these perspectives, MIOLLER was to undertake the work that led him to complete in a year his hitherto most extensive and important article.

2. The Moiler formula

Having developed the method, MOLLER could work out his doctoral dissertation. He may have asked BOHR: 78 "Wouldn't it be interesting to try to continue this and calculate the stopping phenomenon of relativistic particles?". The calculation of stopping was to be the most important aim of MOLLER'S work during the following months, as the numerous references to this problem in his correspondence show. Interest in stopping needed no justification. From the beginning of the century, a proper understanding of stopping processes had shown itself to be essential to the analysis of radioactive radiations and the constitution of atoms. This question figured prominently in the Copenhagen agenda, as BOHR kept a constant interest in the analysis of the passage of rapidly moving particles through matter, to which he devoted two of his first and one of his last papers. 79 The measure of stopping was a common means of identifying particles, and around 1930 the necessity for a relativistic treatment, which could help to determine the nature of cosmic rays, was clearly felt.

When MOLLER began working on histhesis, the application on his results to cosmic radiation was a major stimulus. By 1931, research on this phenomenon had achieved great importance, which would only increase in the next decade. Its relevance to MOLLER'S work, however, gradually diminished as his calculations proceeded, and in the end MOLLER exclusively considered fi radiation. Before we examine in detail the deduction of his scattering formula, we will first analyze its elaboration process and the simultaneous change in MOLLER'S aims, which are uniquely shown in his correspondence.

2.1. Towards a relativistic scattering formula

Late in the summer of 1931, MOLLER began working on the application of his method to the calculation of the stopping of relativistic electrons. He had

cosmic radiation in general. The term he uses, "height-radiation" ("H6henstrahlung") had been the most common denomination in German since the first investigations of the phenomenon in the beginning of the century, and remained so until the late thirties. It was finally displaced by MILLIKAN and CAMERON's more suggestive "cosmic radiation".

78 MOLLER interview 1971 (AIP), p. 18. 79 BOHR (1913), (1915), and (1948). See p. 235, and section 5 of KRAGH'S paper.


expressed this intention at the end of his article of 1931, submitted in May, and it had not changed by the end of September, when MOLLER sent to BETHE an offprint of his article, asking for an offprint of BETHE'S paper on the stopping of rapidly moving particles, "as I would now like to try to calculate the stopping of extreme fast particles (fl rays, height-radiation) according to the same method", so

Two weeks later, on October 14, MOLLER received an unsigned letter from Cambridge. The correspondent was F. CLIVE CHAMPION, a research graduate student then working on /3 radiation, who had read MOLLER'S article and was interested in the general scattering formula. MOLLER had by then advanced "a good piece" ("ein gutes Sttick") in his calculations, as he explained to MAX DELBRt?CK while inquiring about the unknown correspondent. The remarks that followed referred exclusively to stopping, which most worried MOLLER. What he had done suggested that, owing to retardation, fast electrons would be more stopped than was expected according to SCHR(SDINGER or classical theories. MOLLER had not yet at his disposal an expression for stopping, but he discussed in his letter to DELBROCK an expression for the number of ions formed in hydrogen per cm path - a formula he added to the typewritten letter, which is missing from the extant carbon copy. As may be deduced from DELBROCK'S answer, the primary ionization was proportional to 72 in the high-energy limit, which led DELBR/3CK to express the first doubts about the correction of the calculation, since this proportionality "would signify an enormous ionization for the height-radiation electrons, while SKOBELZYN, MILLIKAN and BLACKETT only find about twice the ionization". 81

On November 2, on failing to receive an answer, CHAMPION wrote again to MOLLER. This time he signed the letter and MOLLER replied immediately upon receiving it on November 4. MOLLER, who took a great interest in CHAMPION'S work, was optimistic with regard to his calculations, and hoped to be able to send him the general formula "in a short time" - a month as it turned out. s2 CHAMPION, however, would not be the only one to take interest in MOLLER'S work. At the end of November, MOLLER received a letter from HEISENBERG, whom he had frequently met in Copenhagen. HEISENBERG'S insightful criticisms and comments would prove decisive in MOLLER'S final success.

HEISENBERG was at this time preparing the review article on cosmic radiation we have referred to above. His letter was motivated by OPPENHEIMER & CARLSON'S application of MOLLER'S method to the calculation of the stopping of fast electrons: 83

s0 MOLLER to BETHE, September 30, 1931 (AHQP-59). He refers to BETHE (1930). 8t MOLLER to DELBRfSCK, October 14, 1931 (AHQP-59). DELBRI]CK's suggestion,

"Sollte das 7 2 - 1 nicht noch mit unter den Log not kommen?", agrees with the expression MOLLER finally obtained. See MOLLER (1932), formula (93).

82 CHAMPION to MOLLER, November 2, 1931; MOLLER to CHAMPION, November 4, 1931 (AHQP-59).

83 HEISENBERG to MOLLER, November 28, 1931 (AHQP-59). HEISENBERG refers to

220 X. ROQUI~

A couple of days ago an article appeared by Oppenheimer . . . who main- 47regN

tains that according to your calculation "m----~ log E would result for the

stopping power of extremely hard incident electrons. This seems to me very implausible, but I do not think like looking through all the calculations. Have you ever calculated this problem? If so, what did you find out?

HEISENBERG was also interested in the ultrarelativistic limit for small scattering angles, a case MOLLER had not considered in his article.

Significantly enough, MOLLE~'S answer first addressed HEISENBERG'S second question, to which he replied with a first, mistaken version of the general scattering formula. Though the extant copy does not contain it, it could only have been the same expression as MOLLER sent to CHAMPION three days later, on December 7. An extant draft of this letter does contain MOLLER'S earliest formula for the differential scattering cross section:

e4sin202dOdcp4[7 + 3 + (7 -- 1)cos 0cml(7 + 1) dtz(O) - m 2 7 4 v 4 [7 + 1 - - (7 - - 1)cos 0cm] [(7 + 3) 2 - - (7 - - 1) 2 cos2 0cm] f f ' (17)

(4 3) = 272(7 + 1) sinTOr m sin~O~ m

(7 -- 1)2 (411 + (7 + 1)2] + 3 -- 272) + (7 -- 1)4 sin2 0~m (18) + ~ - sin 2 0era 1 ~

(0cm the scattering angle in a center of mass system, 0 the scattering angle in a lab system; 7 and v refer to the incident electron). This expression, consider- ably more complicated than the correct one, still led to the limits MHLLER had previously calculated. This he pointed out to HEISENBERG and CHAMPION, as well as the error in the ultrarelativistic limit mentioned above. 84

With regard to stopping, MOLLE~ was at that moment doing the calculations for hydrogen, and he explained to HEmENBERG that the result might easily be generalized to light atoms. MHLLER had obtained an expression for the number of ions formed per cm path which reduced exactly to BETHE'S one in the nonrelativistic limit. But he did not fail to express his doubts about the significance of some of its terms. One of them in particular puzzled him, as even when it did not contradict "the existing experiments on the ionization by fast Jill-rays . . . it would signify an incredible increase of height-radiation ionization". 85

OPPENHEIMER & CARLSON (1931); see their formula (1). The formula gives the energy loss per cm path through a gas in which there are N electrons per cc.

8,* MHLLER to HEISENBERG, December 4, 1931; MHLLER to CHAMPION, December 7, 1931 (AHQP-59).

85 MOLLER to HEISENBERG, December 4, 1931 (AHQP-59).

Mailer Scattering 221

The letters to HEISENBERG and CHAMPION also contained the relations between the scattering angles in the center of mass and lab systems, 0em and 0, which MOLLER had used: a6

2 - (7 + 3) sin2 0 (7 + 1)sin 20 cos 0cm = 2 + (7 -- 1) sin2 0 ' sin 0cm -- 2 + (Y -- 1) sin2 0

These relations, as HEISENBERG noticed at once, were not correct. If the one for cos 0or, were right, HEISENBERG obtained for the sine

sin 0r = x/2(~ + 1)sin 20 2 + ( 7 - 1) sin20 '

which agreed with the expression he himself usually applied,

t a n ~ = ~ 1 tan0 .

This was the first remark he made to MOLLER on December 10. The second one concerned the stopping formula deduced by MOLLER on the basis of the scattering formula, and its significance for cosmic radiation: s7

Your final [stopping] formula . . . is indeed extremely interesting, but I would like to point out that it would no doubt signify a sharp contradiction between theory and experiment, and show that the whole method of calculation with retarding fields is no longer admissible. It must be considered that 7 could be at least 1000 for height-radiation electrons. However, I do not understand at all how such divergences with classical theory would be possible.

HEISENBERG'S negative estimates, together with those expressed by DELBR~CK and MOLLER himself before, finally led MOLLER to leave aside cosmic radiation. At the same time, news from the Cavendish laboratory of a possible test of his formula by means of fl rays made him reconsider the importance of this radiation. Both aspects are clearly displayed in MOLLER'S answer to HEISEN- BERG, which allows us to locate in mid-December 1931 the final redirection of his calculations: as

I also think that the expression for q~ is no longer valid for such great values of 7 as height-radiation requires, as the approximation procedure used probably becomes senseless in this region. For fast fl rays, on the contrary,

86 MOLLER does not use the term "center of mass system" but rather refers to "the Lorentz system in which the momenta of both electrons before the collision are opposite,

i.e . . . . a system which moves with velocity with reference to the rest system"

(MOLLER to HEISENBERG, December 4, 1931; AHQP-59). He distinguishes the magnitudes in the center of mass system by an asterisk.

s7 HEI-SENBERG to MOLLER, December 10, 1931 (AHQP-59). s8 MOLLER to HEISENBERG, December 15, 1931 (AHQP-59).

222 X. ROQU~

the method used might be reasonable. There are experiments in progress at Cambridge which should provide a test of the scattering formula for free electrons with the help of Wilson's photographs.

In the same letter MOLLER stated that the error in the 0cm -- 0 relations did not affect the scattering formula. However, HEISENBER~ found yet another error, which induced MOLLER to look through all the calculations and finally led him to the correct formula. HEISENBERG had found the error through conditions of symmetry, and he communicated it to MOLLER on December 17: "You wrote me the formula

e 4 sin 20 2dO &o 4(7 + 1)[~ + 3 + (7 - 1)cos 0~m]

dtT(O) = m2v4 741_7 ~,, 1 _ (7 - 1)cos 0era ] [-(7 + 3) 2 - (7 - 1) 2cOs2 0crn'] ~ '

/

but I am sure that at the indicated point it should read '3' not '1', as otherwise there is no invariance under 0~m ~ 0~m + ~ in the moving system". 89

Little more is known until January 25, when MOLLER sent to Moa-r and CHAMPION the correct scattering formula. MOLLER and MOTT were well acquainted with each other since M o ~ ' s stay in Copenhagen in 1928, and they had probably met again at the Cavendish in May 1929, when MOLLER visited the laboratory with Bong. MOTT had sent to MOLLER an offprint at the end of November 1931; upon answering him MOLLER explained, among other things, that he had already finished the calculation of the scattering formula, simply adding: "If it has any interest for you, I shall report upon this another time". MOTT'S reaction denotes the significance of the problem: "I should be M O S T interested to see your results on the collision between two particles, as soon as you can conveniently send them". 9~

The sending of the results was delayed by more than a month, not only because MOLLER had to correct the error pointed out by HEISENBER6, but also because Christmas long interrupted his work, as his letter to MoTr of January 25 implies: 91 "I have just now come back from Holiday and have not thought about physics since before Christmas". When did MOLLER correct what he qualified in the letter he wrote to CHAMPION on the same day as a "little fault"? 92 No doubt, after HEISENBER~'S observation, but HEISENBERG'S letter could not have reached Copenhagen before December 19, too short a time before Christmas, even more when it is considered that MOLLER most probably spent his holiday at his home town, Hundslev, as he usually did. It is also hardly plausible that, if MOLLZR had obtained the correct formula before Christ- mas, he would have delayed its communication so long.

MOLLER, therefore, would have checked up on his calculations when return- ing from the Christmas holidays, in mid-January, and obtained the right scat-

89 HEISENBERG to MOLLER, December 17, 1931 (AHQP-59). 9o MOLLER tO MOTT, December 9, 1931; MOTT to MOLLER, December 15, 1931

(AHQP-59). 91 MOLLER tO MOTT, January 25, 1932 (AHQP-59). 92 M~LLER to CHAMPION, January 25, 1932 (AHQP-59).


tering formula shortly before January 25. The new formula

e4 sin 0cm doom dcp 8(]; + 1) dff(0) = m274/)4[.(7 -t- 3) 2 - - (7 - - 1) 2 cOs2 0cm'] f f (19)

- the factor ~- did not change, see (18) - was simple only if expressed in terms of 0~m, as MOLLER remarked to MOTT. In the letters to CHAMPION and MOTT, MOLLER emphasized the necessity of the symmetry in 0~m = 7C/2, and also expressed his confidence in having finally removed all the errors. Naturally, MOLLER also sent his new result to HEISENBERa, though not immediately. As we have seen above, MOLLER'S result very much satisfied HEISENBERG, who nevertheless kept questioning the expression for stopping. 93

We know nothing of the subsequent development of MOLLER'S work. The writing of the extensive article which contained his results must have occupied him intensively till the end of April. It was submitted to the Annalen der Physik on May 3. By mid-July, MOLLER had already revised the second proofs, which could not prevent the article from having a number of misprints, including one in the scattering formula.

MOLLER had done the paper with the idea that it could become his doctoral dissertation. BOHR managed to get the article recognized as such, and MOLLER had only to add a Danish introduction. After a vacation in Nor th Zealand and a stay of two or three weeks at Hundslev, MOLLER returned to Copenhagen in mid-August in order to prepare the introduction, which he did not finish until October. 94 It is highly significant that he entitled this twenty-five page r6sum6 "An overview over theory and comparison with experiments", when references to experiments were so scarce in the article. No doubt this MOLLER owed to CHAMPION, who had completed his experiments in June. 9s

MOLLER defended his dissertation on November 28. It was highly praised by his opponents, BOHR and KLEIN. 96 The doctoral degree did not immediately affect MOLLER'S professional status, who for some time remained a lecturer in BOHR'S Institute.

2.2. The deduction of the formula

The article that was the core of MOLLER'S dissertation contained, according to its title, "a theory of the passage of fast electrons through matter". In its first

93 HEISENBERG to MOLLER, February 15, 1932 (AHQP-59): "Ihre 72-Glieder ffir die Bremsung dagegen glaube ich Ihnen wohl nicht. Sic sind mit den Experimenten auch bei Sekund/irelektronen der Energie e ~ 30me 2 in krassestem Widerspruch".

94 See MOLLER to BOHR, July 25, 1932 (AHQP/BSC-23). 95 "Oversight over Teorien og Sammenligning med Eksperimenterne", preface to

MOLLER's dissertation. MOLLER commented on CHAMPION's work in the last two pages of his introduction. He reproduced CHAMPION's results, along with his own theoretical predictions, and those of MOTT and RUTHERFORD, but omitted the predictions of the classical corrected theory, which also agreed well with the observed values. He concluded that the experiments "definitely support" ("afgjort til Gunst") his scattering formula.

96 See the first section of KRAGH's paper.

224 X. ROQUE

paragraph, MOLLER confessed his true subject and clearly characterized his treatment. It deserves to be quoted in full: 97

The object of the present work is to treat the passage of hard fl rays through matter in agreement with quantum theory and relativity. All the physical phenomena tied with the passage of rapidly moving electrons through matter, like stopping, scattering, ionization and excitation of the atoms, can be reduced to the interaction of the electrons with the atoms; radiative forces, on the contrary, do not practically play any role. A theory of these phenomena therefore requires a quantum-theoretical treatment of the relativistic many-body problem. At present there is still no consequent general theory of this problem, and one has provisionally to content himself with obtaining an approximate treatment suitable to the present problem, by means of the convenient generalizations of the nonrelativistic theory.

The ambiguity of the title is readily solved in the first sentence, where "fast electrons" are identified with fl electrons. The continual reference to cosmic radiation during MOLLER'S work has vanished from its final outcome. This allows MOLLER to neglect radiative forces and concentrate his analysis on the two-body interactions, electron-electron and electron-nucleus. For him, the lack of a "consequent general theory of the relativistic many-body problem" - i.e.,

a quantum electrodynamics free from divergences - makes it necessary to treat provisionally the problem in an approximate form. While MOLLER does not present his method as an alternative to the attempts to quantize the electromagnetic field, he does not mention them either.

MOLLER'S presentation of his method in the first section of his article, "w 1. Theoretical foundations", is more complete and extensive than the original, schematic one, but it still does not include any reflexion on the conceptual basis of the method. MOLLER deduces from nonrelativistic quantum mechanics the probability for the transition (mlm2)~(n ln2)

47z 2 P = - ~ - 6(Em, + E,,2 - E,1 - E , 2 ) l ( n l n z [ U t m l m 2 ) [ 2 , (20)

an expression he "adopts" in the relativistic case, where "naturally, the 'matrix elements' in the right hand side look different from those in the Coulombian case". 9s He then notices, in much the same terms as in his previous article, that in order to obtain a "plausible relativistic hypothesis" ("plausiblen relativisti- schen Ansatz") for the matrix element, it is necessary to start from the charge and current densities associated with the transition of one of the particles. Nowhere in the article was he to describe the transition to a relativistic treatment with greater precision.

The use of retarded potentials is now more explicit. M~LLER builds the charge and current densities corresponding to the transition of particle 1 from

9'7 MOLLER (1932), p. 531. 98 Ibid., p. 537.


its initial to its final state, 1 ~ 1'

p(1) = _ e O l , 01 = - e o I , 91 ei~~ , (21)

j(1)= ecO*l:~(1)O1 = ec01,~(1)01ei,O,,., (22)

2= ) = - i f ( E l - El , ) , and remarks that the corresponding scalar and vec- (CO 11'

torial potentials are given in Maxwell's theory by

r ,

A(1)(r2) = f[j(1)]r d3rl

(23)

(24)

(r = It1 - 1 " 2 1 ) , where "the square bracket is to indicate that for p(1) and jo) the retarded values, i.e. the values at time t - r/c, are to be introduced". 99 Taking these expressions into account, he writes for the potentials

�9 '*'(r2) f< O1 r = " e~~ , (25)

A(l'(r2) eei~,,~ f ~o*~,~l,01 r = - e'~,,'cdSrl . (26)

The operator related to these potentials is, according to DIRAC's theory,

_ e [ # O ) ( r z ) + ~(2). A ( 1 ) ( r 2 ) ] , (27)

and the corresponding matrix element for the transition of particle 2

- e ~I 0'2' [~m(r2) + ~(z). AO)(r2)] 02 , (28)

that is to say, according to (25) and (26),

ei(O3u, + to=,)t f f Otl,fflYz, eZ( 1 -- ~m.r - ~r ffjlffJ2ei~O~,~d3rl d3r2 . (29)

This expression is not in general symmetric in the two particles, but it is for the physically-allowed transitions, in which Co22' = - m ~ l ' according to the delta function in expression (20).

This formulation of the method displays more clearly than the previous one the origin of the different terms. The exponential factor inside the integral stems from retardation, and cancels out when c ~ ~ . GAUNT'S interaction term,

~(1~. fi(2~ - - e 2 , , is also immediately recognized. As MOLLER says, "it is satisfying

r that this term appears here of its own, as soon as relativistic invariance is demanded". 1~176

99 Ibid., p. 536. lOO Ibid., p. 538.

226 X. ROQUE

Having considered in the second and third sections of his article the excitation of the lower atomic levels, MOLLER treats in the fourth one the upper levels and the limiting case in which the energy imparted by the incident electron to the atomic one is much higher than the ionization energy. This case may well be considered as an interaction between free electrons. The stationary states that both electrons occupy before and after the collision are represented by solutions to the DIRAC equation

01 = U(Sx)e ilk''r' -o,,t) , (30)

where k = p / h , co = E / h .

The differential scattering cross section for those transitions in which the momentum of the incident electron after the collision lies between P r and Pl ' + d p r , and that of the atomic electron between P2, and P2, q- dp2,, is 1~

4g 2 dtT(p l , ,p2 , ) = h~f~zl ) ~ (E I ' -1- E 2, - - g 1 - - E2)

1 x ~ ~" ~, I (P l ,S l , , p z ,Sz , l V l P l S l , p 2 s 2 ) l Z d 3 p l , d a p 2 , . (31)

81S2 51,S2,

MOLLER obtains this expression from (20) in the usual way: dividing by the incident electron flux, "~) a z ; summing over final spins, because the electrons' polarization is not measured; and averaging over initial spins, as the incident beam and the atomic electrons are not polarized.

The matrix element follows directly from the general expression (29), introducing the solutions to the Dirac equation:

<1', 2'1Vr 1, 2) = e2{utz,UzU~,Ul - ( u ~ , f : t Z ) u 2 ) ' ( u ~ , $ r (32)

- COS r d 3 r l d 3 r 2 . (33) r c

The total system has been till now implicitly represented by means of the asymmetric proper functions

1 1 ' + , 2 ' + > , 11 ' + ,2 ' >, I i ' 2' 2' , - - , + > , 1 1 ' - , - > (34)

where I I ' + , 2' - > denotes the state in which electron 1 has a momentum P r and up polarization, an electron 2 a momentum P2, and down polarization. In order to consider exchange, MOLLER needs to introduce functions of the total system antisymmetrical under exchange of both electrons. Following

lOl MOLLER considers that the atomic electron is initially at rest (P2 = 0), and he describes it by means of the solutions to the DIRAC equation for the hydrogen atom. He verifies at the end that this is equivalent to considering this electron as free (p. 562). In order to facilitate the comparison with the present treatment, we consider more generally a free electron of any initial momentum.


OPPENrIEIME~, he adopts as representation of the final states of the total system the functions 1~

1 2' , 1 ' , ]a') = ~ [11' + , + ) - - 1 2 ' + + ) ]

[b') = ~ [ 1 1 ' - , 2 ' - ) - 1 2 ' - , 1 ' - ) ]

= - - 1' , 2 ' 1' , Ic') �89 ) 12'-, + ) + 1 1 ' - + ) - 1 2 : + , - ) 3

= , - - , - 2' 1' . Id') �89 2' ) 12' 1 ' + ) I 1 ' - , + ) + 1 2 ' + , - ) 3

Initial states are analogously represented by the functions I a), I b ), I c ) and I d) . Exchange may be taken into account only replacing in (31) the sum over sl, s2, s,, and s2, by the quadratic sum S over the matrix elements corresponding to the 16 possible transitions between the initial states, la), Ib), I t) , Id) and the final ones, la'), [b'), Ic') and Id'). The matrix S has the form

S = ~ I(b'l g l a ) l 2, (35)

and the matrix elements that it contains are all similar. MOLLER calculates one of them as an example:

= 2' IV[1 + , 2 + ) + ( 2 ' - , 1 ' - IVI2 +, 1 + ) (b'l V ia} �89 [ (1 ' - , -

- ( i ' - , 2 ' - I g]2 +, 1 + ) - (2' - , 1' - I Nil +, 2 + )-] .

The two former and last terms are equal, due to the symmetry of the matrix element under exchange of both electrons. Therefore,

( b ' l g l a ) = ( 1 ' - , 2 ' - I g [ 1 + , 2 + ) - ( 2 ' - , 1 ' - I g l l + , 2 + ) ,

where both terms are distinguishable only by the exchange of indexes 1' and 2'. Noticing expression (32) for the matrix element, we may more explicitly write

(b'[ Via ) = eZ[ {u~,u~ u*2,u2 -(U*l,~mu~)"(ut2,~(2)u2)}f

- {u~2,ul ul,u2 - (u*2,~t)ul)'(u~,~2)u2)}O] , (36)

where f is given by (33) and

g= ffei(kz-kv)'rxei(kl-k2)'r~COS(O)l;fD2'r)d3rld3r2 . (37,

As MOLLE~ indicates, all the matrix elements in S are of the same form. We may notice that the terms in the matrix element are distinguished by the permutation of final momenta: they thus correspond to the FEYNMAN diagrams

102 OPPENHEIMER (1928); see p. 363. The first three states are symmetrical on spin (triplet); the last one is antisymmetrical (singlet).

228 X. ROQUE

by means of which we now calculate this process. 1~ This analogy shows itself more clearly when integrals f and g are evaluated, 1~

h 5 f - / z ( p 1 _ pl,) 2 c~Q~I q-P2 - -P l ' - -P2 ' ) ,

h s g -- /r(p I _ p2,) 2 ~(Pl q- P2 - - P l ' - -P2 ' ) ,

as they only differ in the exchange of 1' and 2', and we easily recognize in them the transferred fou r -momen tum corresponding to each diagram.

Gather ing these results together, MOLLER obtains for the scattering cross section (31) l~

4h9e 4 d a ( p l , , p 2 , ) = ~ 6~(pa + Pz - P r - P 2 ' ) 1 S ' d a p r d 3 p 2 ' , (38)

where S' denotes the sum over the different matrix elements of the form (36). The scattering cross section is now more precisely interpreted as the number of collisions in which, after colliding, o n e of the electrons has a m o m e n t u m P t ' and t h e o t h e r a m o m e n t u m P2,. MOLLER notices that "owing to exchange we can no

103 With the notation 7 u, with 7 ~ fl and ~ = 7~ the matrix element becomes (disregarding factors)

/,11,~/~/1 ~2,~#U2 U2'])gUl lil,7#/,/2 <b'[ V i a ) = (Pl - P r ) z (Pl - Pa') a (50)

This is the matrix element we now directly write from the FEYNMAN diagrams corresponding to this process.

1o4 MOLLER bases this calculation on a result of MOLLER (1930C), according to which for any r might be written

COS 0) 1 -- 091, = -- 0 k 2 - - ( c ~ 1 7 6 22kdk ' c

where in the integral in the right member the Cauchy principal value is to be taken. MOLLER shows that this integral is equivalent to

1 e ik ' r

so that 1 ~ f f e i(k2 k~,-k).r2 e,(k~-kl, k).,l

@

This integral may easily be evaluated as a delta function. los The difference in a factor t / h 3 with formula (62') of MOLLER (1932) stems from

the normalization factor of ~2, which we consider included in u2.


longer say which of the two expelled electrons is the one originally bound to the atom, and which the incident one". 1~

MOLLER gives next the scattering cross section as a function of the scattering angle 0 of one of the electrons, leaving till later the calculation of the sum S'. The delta function in (38) expresses conservation of energy and momentum. MOLLER uses this principle to write the magnitudes implied in the cross section in terms of the scattering angle 0 and the velocity v of the incident electron. Momentum and energy are conserved in any Lorentz system, and MOLLER calculates the kinematics of the collision in the simpler one, the centre of mass system. In this way he obtains for the scattering cross section

do-(0) = -~e4m 2 sin 0cm dOem [-(g -t- 3) 2 - (7 - 1) 2 cos2 0era] �88 . (39)

It only remains to calculate S'. MOLLER remarks what our notation immediately displays, that the matrix elements (36) are relativistic invariants, and he calculates them one by one in the centre of mass system. This last, formidable calculation leads him to the scattering cross section 1~

27ze 4 sin Ocm dOem 2(7 + 1) &r(O) = m2] )zv 4

�9 sing-0~m sin / 0 ~ -~ 47- 2- 1 + sin g 0~m , (40)

now known as the Moiler scattering formula. As MOLLER indicates, the relations 0 - 0~m may be used to express the result as a function of 0, but the resulting expression would be much more complicated.

The scattering formula does not occupy a prominent position in MOLLER'S article. Once deduced, MOLLER applies it, together with the results of preceding sections, to calculation of stopping. He limits any appraisal of the method used, as well as of the scattering formula obtained, to the brief introduction. Value judgements are scarce there and do not include any mention to quantum electrodynamics. Did this mean that the MOLLER formula was not related to this theory?

2.3. The Moiler formula and quantum electrodynamics

MOLLER did not refer to quantum electrodynamics in any of his articles on scattering. His treatment of electron-electron interaction was grounded on firm theoretical developments, avoiding the grave problems that affected the quantum theory of the electromagnetic field. Yet the MOLLER formula and quantum electrodynamics were much more directly related than MOLLER'S silence might suggest - indeed, their relation was soon to be clearly displayed. MOLLER'S

106 MOLLER (1932), p. 560. lo7 In MOLLER'S article the denominator of the first factor is written as m274v 4

(MOLLER 1932, formula 74). This is clearly a misprint.

230 X. ROQUI~

second article on electron scattering had been submitted to the Annalen der Physik on May 3, 1932. A month later, on June 9, an article by BETHE & FERMI was submitted to the Zeitschrift fiir Physik, where "the relations between the interaction formulas of Breit and Moller and quantum electrodynamics" were discussed. 1 os

BEa'HE had been interested in the stopping of fast particles from the beginning of his career. In 1932, having spent the previous year at Cambridge, a Rockefeller Fellowship enabled him to work at Rome with FERMI'S group. There he addressed the problem of generalizing the calculation of stopping including relativity. 1~ MOLLER'S first article not surprisingly aroused his interest; on March 25, BETHE wrote to MOLLER: 110 "These days I have been intensely occupied with your important work on the scattering of relativistic electrons. I find it wonderful that you can treat the problem in such a simple way!". BEa'I~E had taken MOLLER'S article as a starting point to calculate stopping, having known through HEISENBER~ about MOLLER'S initial troubles. In fact, the reason for BETHE'S letter was simply to know if MOLLER had corrected his errors and intended to publish his results.

MOLLER'.S answer is lost. He apparently suggested BETHE publish a brief note in Die Naturwissenschaften, and he also advanced the contents of his next article. On April 30, BEa'HE informed MOLLER that the note, "though I said only that which was most necessary", had exceeded the size required by Die Natur- wissenschaften, and that he had therefore sent the article to Zeitschrift far Physik, hoping not to have forestalled MOLLER'S paper. 111

MOLLER'S second scattering article and BETRE'S extended note were submitted within a day of each other. Soon after, BETHE considered together with FERMI a more fundamental question: how were MOLLER'S treatment, BREIT'S formula and quantum electrodynamics related? Their work seems to have progressed very rapidly, to judge by BETHE'S recollections. 112 Quantum electrodynamics meant naturally FERMI'S who had given the theory a more

lo8 BETHE d(z FERMI (1932), p. 296. lO9 Cfi BERNSTEIN (1979), GALISON (1987). MOLLER's work, however, is not men-

tioned in either account. 11o BETHE to MOLLER, March 25, 1932 (AHQP-59). 111 BETHE to MOLLER, April 30, 1932 (AHQP-59). See BETHE (1932). 122 Cf BERNSTEIN (1979), p. 31 (private communication): "FERMI and I wrote a

paper comparing three methods of treating the relativistic elecron-electron interaction - unifying electro-magnetic quantum theory with relativity. The research took two days. Then he said, 'Well, now we have solved it, now we will write a paper.' So on the third day he himself sat down at the typewriter - there was no secretary in the institute. His procedure was to state a sentence in German - he spoke excellent German, while I spoke hardly any Italian - and I would either approve it or modify it. When he came to an equation, we would agree on it, and I would write it down in longhand. That was the paper. It was a nice paper, and even though Fermi did by far the larger part of it, it had my name on it along with his. I felt very happy about that, and I learned a lot from it. It taught me how to write a scientific paper simply and clearly. My stay in Rome came to an end much too soon."


accessible formulation than DIRAC, o r HEISENBERG d~ PAULI, and had recently presented it in Review of Modern Physics. 113 BETHE & FERMI explored the relationship between both "hypotheses" for the interaction between two electrons - "that apparently set out from completely different points" - and stated explicitly the approximations needed in deducing them from quantum electrodynamics. 114 Let us simply note that BETHE & FERMI used the Coulomb gauge and obtained MOLLER'S equation (3) adding to the matrix element for static Coulombian interaction, the contribution arising from the exchange of a transverse photon between the two electrons. What I wish to stress here is that they clearly and concisely showed that MOLLER'S interaction formula followed on from quantum electrodynamics. 115

Despite this, the M~LLER formula was paid little attention in the scarce texts devoted to the new theory during the 1930s and the 1940s. Thus, in PAULfS famous Handbueh article (1933), MOLLER'S work was cited only on the last page, together with BREIT'S, in a footnote to the statement that "the fact that the proper energy [of the electron] comes out infinite according to the theory, prevents also a consistent relativistic treatment of the many-body problem". 116 To PAULI the troubles with proper energy clearly diminished the value of any approximation to the electron-electron problem. Being devoted to the radiation field, HEITLER'S The Quantum Theory of Radiation omitted the MOLLER formula from its first and second editions (1936, 1944); significantly, it was included in the third (1954). 117 MtOLLER'S work was also absent from WENTZEL'S Einfiihrun 9 in die Quantentheorie der Wellenfelder (1943). 118 Another influential, albeit less- known text, KRAMER'S Quantentheorie des Elektrons und der Strahlung (1938), cited MOLLER'S 1931 article in a footnote as an example of "how in the case of feeble interaction between fast electrons can one calculate in a rigorous relativistic way". 119

Accounts of the formula after renormalization are rather different. In the first postwar systematic treatise about the theory, JAUCH & ROHRLICH'S The Theory of Photons and Electrons (1955), a chapter is devoted to the electron- electron system, in which "MflLLER scattering" prominently figures. 12~ Here, the collision between two electrons is considered as a fundamental process interesting in itself, the problem being introduced in a general, logical manner a long

113 FERMI (1932). See also FERMI (1930). 114 BETHE & FERMI (1932), p. 296.

115 Ibid., w "Ableitung der Mollerschen Formel aus der Quantenelektrodynamik". For an illuminating discussion, see HEITLER (1936), 3 rd ed., pp. 231-236.

116 PAULI (1933), p. 272. In 1941, however, PAULI used "the well-known method of Moller" to calculate the cross-sections for the scattering of mesotrons by a Coulomb field, and by electrons (PAULI 1941, p. 229).

117 HEITLER (1936). See in the third edition "w The scattering of two electrons", p. 231.

118 WENTZEL (1943). 119 KRAMERS (1938), p. 301. 12o JAUCH & ROHRLICH (1955), chapter 12. The authors used the general term

"electron" for both electrons and positrons.

232 X. ROQUE

way from preceding phenomenological introductions. The presentation set a standard for most future accounts of "MMler scattering", from texts not far from JAUCH & ROHRLICH'S, such as KXLL~N'S "Quantenelektrodynamik" (1958), up to present ones) 21 They invariably include a deduction of the formula and references to its successful comparison with experiment.

This sample, however incomplete, may suffice to assess that renormalized quantum electrodynamics reappraised the MOLLErt formula. In the concluding remarks, I shall suggest some causes for this change of status. Next, however, we shall turn to the experimental probing of the formula and find that this change affected not only the theoretical judgement on the formula, but also the significance attached to a thorough test of it.

3. The experimental testing of the Moiler formula

The experiments of ASHKIN, PAGE & WOODWARD in 1954 are often cited as the most decisive among postwar attempt s to test the MOLLZR formula. 122 By then the status of quantum electrodynamics had very much improved with respect to that of its early formulations, and FEYNMAN'S approach to the theory had given new relevance to MOLLZR'S interaction: This may help explain the renewed interest of experimental physicists in the formula around 1950, when a number of research teams were applying the newly-built accelerators to distinguish it from MOTT'S or RUTHZRFORD'S formulas.

This interest is a striking contrast to the indifference the formula encountered in the early stage of the theory: Together with those of ASHKIN, PAGe, & WOODWARD, the experiments of 1932 by the British physicist F. C. CHAMPION are usually mentioned: A typical account might read: "After the theory of Moiler indicated that deviations from the Mort formula might be expected for relativistic collisions, efforts were made to measure the scattering of fast beta- particles. Champion found good agreement with the Moller theory for 250 collisions of radium E beta-particles in nitrogen found in cloud chamber pic- tures". 123 Nevertheless, this causal account is untenable, because CHAMPION'S

experiments were conceived before the MOLLER formula existed. Furthermore, these experiments remained almost unique during the thirties - indeed not until 1941 was another article reporting a test of the formula to appear.

121 To quote but one example, ITZYKSON & ZUBER (1985). i22 ASHKIN, PAGE & WOODWARD (1954). For example: JAUCH • ROHRLICH (t955)

state that MOLLER'S results "are found to be in excellent agreement with observations", referring to the "beautiful experiments by Ashkin, Page and Woodward" (p. 261); MOTT & MASSEY (1933) say that "confirmation of [the MOLLER formula] for incident electrons in the energy range 0.6 to 1.2 MeV has been provided b y the coincidence counter experiments of Ashkin, Page and Woodward" (3 ~d ed., 1965, p. 818). These experiments remain the classical test of Mf~LLER & BHABHA formulas at low energies (Cf ITZYKSON d~ ZUBER 1985, p. 281).

123 SCOTT, HANSON, & LYMAN (1951), p: 638.


While working on his thesis, MOLLER did not show much concern for a possible test of his results. Perhaps, as he would later recall, MOLLER doubted the significance of his calculations: 124

Of course, my confidence was not so big, that I was really very surprised when Champion experimentally could show that my formula was obviously in better agreement with the experiments he had done than the non-relativistic [ o r m u l a . . . I was rather surprised that one could by such a formal generalization get to something which was really there in nature.

Yet his attitude may have had another justification: around 1930, experiments on the scattering of relativistic electrons were scarce and not decisive. We begin by considering the experimental situation right at the beginning of the thirties, when both CHAMPION and MOLLER were about to begin their work.

3.1. Experimental knowledge of electron scattering around 1930

Since its discovery and later identification with the: electron, at the beginning of the century, the fi radiation constituted a unique source of high-energy electrons. With energies of the order of 1 MeV, fi rays amply surpassed t h e energy of photo electrons or incandescence electrons (102 eV), while the presence of electrons in the natural source of most energetic particles, cosmic radiation, was hotly discussed at the beginning of the thirties: The scattering of relativistic electrons was therefore first investigated by means of fl particles, which moving with a velocity near that of light (0.9c), displayed relativistic effects sufficiently.

The fl radiation was much less easily handled than another, better known product of disintegration, the e radiation. The fl radiation was not only in- homogeneous, but it was also in most cases emitted together with an intense 7 radiation, which further complicated its detection, since for the fl radiation no such simple and reliable detection device as the scintillation method for e particles existed. In addition, i t was difficult to distinguish between single and multiple scattering - i.e., whether the final deviation was caused by a single collision or by a number of them 125 - as well as between electronic and nuclear scattering; These problems made it difficult to obtain definite results on electronic single scatterin9 of fl particles, to which the MOLLER formula applies.

There existed, however, an instrument that avoided most of these troubles: WILSON'S cloud chamber. Its first prototypes had been made by CHARLES T. R. WILSON at the Cavendish around the turn of the century, though only in the twenties did the chamber become an effective and commonly used detecting

124- MOLLER interview, 1971 (ALP), p. 13. 125 This distinction proved decisive in RUTHERFORD's approach to the nuclear

atom, and it was an essential element of the conception he opposed to THOMSON (cfi HEILBRON 1967).

234 X. ROQUI~

device. 126 Its basic features are readily conveyed: it consists in a vapor- saturated container, whose volume may be suddenly increased by means of a mobile piston or an elastic membrane, the expansion causing the vapor to condense on the ions produced by the particles crossing the chamber. When appropriately illuminated, the resulting tracks can be photographed.

In 1922, WALTHER BOTHE observed the single scattering of fl rays by electrons by means of a cloud chamber "built, even in its most unessential details, according to Wilson's prescriptions". 12v In the fl ray tracks photographed the rare "particularly violent" collisions between a fl particle and an atomic electron were easily recognized as a branched track, the incident particle having imparted so high a fraction of its energy to the impacted one that the tracks of both particles after the collision were comparable. BOTI~E analyzed twenty photographs, which totaled 10 m of track, in which he clearly observed eight such collisions and three doubtful ones. Information about the dynamics of the collision was to be gained through "statistical investigation of the frequency of branching of different degrees". 12s The number of photographs clearly did not suffice for statistical analysis, but BOa'HE nevertheless considered that it allowed for a preliminary estimate.

The comparison with theory was based on a simplified model of the collision, according to which the electrons interact electrostatically and the incident particle is assumed to undergo no important change in direction in the collision. The impacted electron moves in this case after the collision nearly at right angles with the incident particle, its velocity

2e z v = - - - (41)

mvod '

where Vo is the velocity of the fl particle, m its rest mass, and d the impact parameter. This calculation was based on the premisses J. J. THOMSON had used in 1912 to calculate the ionization produced by a moving electric charge. THOMSON'S classical analysis, however simple, was highly effective, and BOIqR

126 See GALISON & ASSMUS (1989). In this article, the technical development of the chamber is traced through WILSON'S experiments of 1911. The chamber seems to have remained a problematic and somewhat awkward apparatus well into the thirties, judging from the numerous questions that experienced users were faced with. L. MEITNER acted frequently as consultant on the chamber, exchanging correspondence about it with O. KLEMPERER (in 1925), P. KUNZE (in 1932), G. DE HEVESV (between 1932 and 1934), G. HERZOG (in 1936), and G. J. SIzOO (in 1937), among others. On September 21, 1932, for example, DE HEVESY wrote to MEITNER: "Sie sind ein Engel, dass Sie mir eine so sch6ne Skizze eines Wilsonsapparates zugesandt haben und ich bin Ihnen dafiir aus- serordentlich dankbar" (MEITNER Papers, Churchill Archives Centre, Churchill College, Cambridge). Even in 1938, A. RUARK had to explain to G. B. COLLINS that "[building a cloud chamber] is not a large job, if one simply copies the design of others who are getting good results" (RUARK to COLLINS, January 28, 1938; RUARK Papers, Hoover Institution Archives, Stanford University).

127 BOTHE (1922), p. 117. 128 Ibid., p. 121.


would later develop it into a theory of the passage of charged particles through matter. 129 The theory allows calculation of the frequency of the collisions in which the incident particle loses kinetic energy Q. When the ionization potential is neglected, the cross section for these collisions reads

da - 2roe4 dQ my 2 Q2 �9 (42)

Having roughly estimated the velocity of the electrons from its range, BOa'HE failed to mention TnoMsoN'S theory, obtaining instead from (41) an expression suitable for his measurements. If n be the number of electrons in air per cm 3, the probability that a /3 particle of velocity Vo produce a secondary electron of velocity > v in 1 cm of its path is 13~

n T z ( 2 e 2 ) 2 ndETz = \ ~ V o V . / "

BOTHE obtained from this formula an expected value of 12 for the number of branches in 10 m track, in "satisfactory" agreement with the experimental result, 8 to 11, "considering the little number of cases observed". T M

In 1923 WILSON himself published a detailed study of the X and fl rays by means of his chamber that went little further in its conclusions. Among other phenomena, WILSON studied the "branching tracks" of fl radiation. Having deduced the velocity of the electrons from 'its range, like BOTHE, WILSON nonetheless compared his results with THOMSON'S theory, noticing that though correct in order of magnitude, they were lower than predicted by the theory, la2 In his conclusions he talked only of a "general agreement" with theory. The minimal significance of BOTHE'S and WILSON'S results was not so much due to the method used, which would later afford more precise and conclusive com- parisons with theory, as to the low number of photographs analyzed and the poor reliability of the velocity measurements.

Not until 1929 is there another experimental reference to electronic scattering of fl rays. That year, MALCOLM C. HENDERSON published an article based on his doctoral dissertation, where he had studied the scattering of fl particles

1.29 THOMSON (1912), BOHR (1913), (1915). 13o Dividing this expression by n and introducing Q=�89 2, shows that the kinetic

energy of the secondary electron, the cross section for the production of a secondary electron of energy higher than Q, is

2roe 4 1 o - -

m y 2 Q '

which follows directly from (42). 13J. BOTHE (1922), p. 121. 132 WILSON (1923), p. 201: "The chance of occurrence of a branch track exceeding

a given length may, on Thomson's theory, be calculated if we know the energy of a fl-particle of given range". WILSON did not detail the calculations he had done in order to compare THOMSON'S theory with the experimental data.

236 X. ROQU~

in light gases by means of an ionization chamber. 133 HENDERSON intended to investigate the effect that the electron spin would have in electron-electron interaction, and compared the scattering in hydrogen and helium with that in heavier elements, like nitrogen and argon. Electronic scattering was more prominent in light elements, and HENDERSON thus hoped to be able to disclose any additional scattering which might be ascribed to the electron's new "field of f o r c e " . 13 4

The ionization chamber, unlike WILSON'S chamber, does not register individual encounters, but only the total ionization current produced inside it. Nuclear and electronic scattering could not be distinguished, and the chamber was therefore not suited to the separate investigation of the effects of one of them. 135 HENDERSON nevertheless applied it to the study of the electronic scattering of RaE (Bi 21~ fl rays, a convenient high-energy fl emitter supposedly free from annoying ~ radiation. HENDERSON measured the ionization current in different gases, for those values of the pressure such as the quotient between both magnitudes remained constant - which he considered to be a hint of the preponderance of single scattering. This quotient then measured the relative "scattering power" of the different gases.

The experimental results were compared with the theoretical predictions on the basis of DARWIN'S classical theory of scattering, allowing for a relativistic correcting factor for nuclear scattering. ~36 HENDERSON'S data indicated that electronic scattering was three times its expected theoretical value. This additional scattering was not considered high enough to demonstrate the existence of a magnetic moment; as HENDERSON firgued on the basis of the relative magnitude of electrostatic and magnetostatic forces, which he alone had taken into account. 137 HENDERSON concluded that it seemed "unlikely that the electron can have a magnetic moment as large as one Bohr magneton"? 3s

133 HENDERSON, The Scattering of fl Particles and the Heating Effects of Radium and Thorium, unpublished PhD dissertation, Cambridge University Library (June 1928).

134 HENDERSON (1929). 135 Nuclear scattering in heavy elements could be analized by means of the ioniz-

ation chamber, a correcting factor accounting for the lesser contribution of electronic scattering. J. CHADWICK, P. H. MERCIER, and B. SCHONLAND specially applied this method in the twenties (cf RUTHERFORD, CHADWICK 8~; ELLIS 1930, pp. 227-234).

136 See above, p. 215. 137 If the magnetic moment of the electron was a BOHR magneton, magnetic forces

would strongly determine the final deviation of the particle, since the minimal approach distance between the fl particle and the electron was estimated sixty times lower than the distance at which the electric and magnetic forces became equal. An elemental calculation showed that the scattering would then greatly differ from that actually observed. See HENDERSON (1929), p. 855.

138 Ibid., p. 856. While HENDERSON did not offer an alternative explanation of these anomalies, he objected to the "relative complexity" of the magnetic electron compared to the point charge, and characterized in this context DIRAC'S electron as an alternative, rather than as an explanation, to the spinning electron: "Dirac has shown how to avoid


HENDERSON'S method, which stemmed directly from similar experiments done at the Cavendish, was not applied again owing to the low significance of its results. The efficiency of the cloud chamber, on the other hand, noticeably increased with the accidental introduction in 1927 of a reliable method of determining the velocity of the particles photographed: the application of a magnetic field to the operating chamber. 139 The field curved the path of the particles and enabled a direct measure of their velocity. If the plane of the track was at right angles to the direction of the field, the velocity could be simply expressed as a function of the field intensity /-/ and the radius of curvature of the track p,

Hp(e/mc) fl = (43)

x/1 + (Hp)2(e/mc) 2

(fi = v/c, e, m charge and mass of the electron). Only a small correction was needed if the plane of the track was not at right angles.

Almost at the same time as HENDERSON was performing his experiments, two graduate students at the same laboratory, E. J. WILLIAMS • F. R. TERROUX, were applying SKOBELZYN'S innovation to the investigation of the scattering of fast fl particles in oxygen and hydrogen. Their results appeared in January 1930. The observations of both BOTHE and WILSON being confined to fl rays travers- ing air, the authors considered the extension of the observations to hydrogen "of considerable importance for comparison with theory", given the magnitude of electronic scattering in this case. WILLIAMS & TERROUX emphasized the superiority of the cloud chamber over other methods by the possibility it offered of directly distinguishing between electronic and nuclear scattering, thus avoiding "any theoretical assumptions"? 4~

WILLIAMS & TERROUX, like HENDERSON, analyzed the fl rays emitted by the RaE. One of their aims was to obtain data on the frequency of the collisions with a high energy transfer, that is, on the frequency of production of branches. T M They examined some 500 tracks of fi particles, which totaled

the necessity for a magnetic electron by a more complete solution of the fundamental equations" (ibid., p. 856). He expressed his confidence that the calculation of the effect on the basis of DIRAC's theory would explain his own results.

139 In 1927, the Russian physicist DIMIT~I SNOaELZYN introduced the magnetic field as an auxiliary means to keep tracks of secondary fl rays produced by the cosmic rays in the wall of the chamber from masking the electrons produced by Compton effect, the only ones which interested him (cf SKOBELZYN 1927, 1983).

140 WILLIAMS & TERROUX (1930), pp. 289 and 291. EVANS JAMES WILLIAMS took his doctoral degree at Cambridge in 1929 with a thesis on the passage of fl particles through matter, a recurrent subject in his articles for the years to follow (WILLIAMS, Passage of fl Particles Through Matter, unpublished PhD dissertation, Cambridge Uni- versity Library, September 1929). FERDINAND RICHARD TERROUX completed his dissertation in 1931 (Applications of the Expansion Chamber to the Study of Fast fl Rays, unpublished PhD dissertation, Cambridge University Library, November 1931).

141 WILLIAMS ~r TERROUX also considered primary ionization and conservation of momentum in the collisions.

238 X. ROQU~

18 m, in which they detected 100 branches. The velocity of the incident particles was determined by the curvature of their tracks, the energy of the branches from their range. They found that the velocity of the incident particles lay between 0.6 and 0.97c (kinetic energy 0.13 and 1.6 MeV respectively), and gathered their results in two tables according to (1) the energy of the branches and (2) the velocity of the incident particles.

The comparison with the theory was based on the ratio between the observed and the theoretical number, which was calculated according to THOMSON'S theory, still the only one available. WILLIAMS & TERROVX start from THOMSON'S expression for the probability of the production of a branch of kinetic energy between Q and Q + dQ (see expression (42)),

2zone 4 1 ~b(Q)- mY 2 Q2. (44)

Owing to the impossibility of distinguishing the fl particle and the secondary electron after the collision, WILLIAMS • TERROUX denote as "branch" that of lower energy. 142 The maximum kinetic energy Q of a branch is then T/2, where T represents the kinetic energy of the incident particle. The theoretical number P(Q) of branches of energy Q is obtained by adding to the previous expression the number of collisions in which the energy lost by the fl particle is T - - Q ,

P(Q) = qS(Q)+(a(T-Q)=--2rme~(~ 2 + _ _ 1 ) ml) 2 ( T - Q)2 " (45)

WILLIAMS d~ TERROUX'S results showed that the number of branches observed was "at least twice the number expected on classical theory", a difference too great to be ascribed to experimental and statistical errors. The same conclusion followed from the analysis they had simultaneously done on primary ionization, and ~ it. also agreed with HENDERSON'S results. Like him, they looked for some evidence of the existence of the electron magnetic moment, and made a quantitative estimate, taking an average value for the attracting force between two magnets and calculating the corresponding number of branches. The results they obtained differed in order of magnitude from the observed ones, and WILLIAMS & TERROUX interestingly concluded that "the differences are so great that although the calculated values are only approximate, we may conclude that the electron regarded as a particle does not behave as if it had a magnetic moment equal ~ a Bohr magneton". 143

142 WILLIAMS & TERROUX seem not to have considered both particles indistinguish- able in principle upon noticing that "when a fl-particle produces a branch, there is no criterion available to determine which is the branch and which is the continuation of the primary track" (WILLIAMS & TERROUX 1930, p. 304). WILLIAMS'S next article, as we shall see, confirms this impression.

la3 WILLIAMS 8Z TERROUX (1930), p. 307. WILLIAMS & TERROUX cited HENDER- SON'S thesis.


While WILLIAMS • TERROUX were preparing their article, MOTT was studying in Manchester the interaction between two electrons taking account of their indistinguishability, a problem he had intended to address at the Cavendish earlier in 1929. Introducing properly symmetrized wave functions, MOTT was able to calculate the effect of non-distinguishability in the collision between two electrons, and between two ~ particles. The corresponding article was submitted to the Proceedings of the Royal Society on November 7, 1929, only a day after that of WILLIAMS & TERROUX. 144

WILLIAMS, who had coincided with M o T at the Cavendish, might have known of MOTT'S work before it was published. Thus, MOTT most probably owed to WILLIAMS the suggestion that the anomalous scattering could be detected, among other methods, "by observing collisions between fast electrons and atoms in which an electron is ejected from the atom, e.g., the forked fl-ray tracks in a Wilson chamber". 145 The only experimental data with which MOTT compared his theoretical prediction belonged in fact to a "soon to be published" article by WILLIAMS, which appeared in a few months) 46 WILLIAMS devoted it exclusively to the observation of branching, stressing the significance of applying PAULI'S exclusion principle to free electrons. He re-examined some photographs of slow fl particles (20 KeV\photoelectrons) he had used in his dissertation according to MoxT's new theory. Slightly simplyfying MOTT'S result, WILLIAMS obtained for the frequency of production of branches

P(Q) = 49(Q) + ( 9 ( T - Q) - ~/c~(Q)(a(T- Q) , (46)

and he then formed the quotient with the corresponding classical expression, equation (45), to get

Pq(Q) - 1 - ~/49(Q)49(T- Q) (47) Pcl(Q) ~/qS(Q) + ~b(T- Q)

In 4 m track, WILLIAMS detected 150 branches, which he classified in two groups according to the branch energy. The corresponding number of branches, 89 and 47, was divided by the clasically expected value, 100 and 78, and the result compared with the value given by expression (47). Both coincided within the experimental errors, which WILLIAMS interpreted as "evidence in support of the new quantum theory and in particular of the extension of the exclusion principle to aperiodic, or open, systems") 47

The existing evidence on single scattering of fl particles by electrons around 1930 ends here. We can conclude that experimental data was scarce and of little

144 MOTT (1930). See also MOTT (1929). 145 Ibid., p. 264. 1,6 It is surprising that WILLIAMS did not comment on MOTT's work in his article

with TERROUX, since both tried to explain the discrepancies with classical theory. This article did not contain the slightest reference to the possibility that a more complete treatment, based on quantum mechanics or DIRAC's theory, might account for these differences.

147 WILLIAMS (1930), p. 465.

240 X. ROQUI~

significance. Thus, an authoritative voice could not but acknowledge that "the experiments on the scattering of fl rays leave much to be desired", especially when compared with those on a rays) 4s At the Cavendish, however, there were experiments in progress which were to afford an early test of the MOLLER formula.

3.2. Champion's experiments

The experimental scene just depicted had not substantially changed by October 14, 1931, when MOLLER received an unsigned letter from the Caven- dish. The correspondent had read MOLLER'S scattering article with the greatest interest, but noted that it contained only the limiting cases of the scattering formula. He had recently taken "a large number of photographs of fast fl-ray tracks in an automatic expansion chamber", and had "already commenced the analysis of the collisions with stationary electrons", which he hoped to complete by the following March. An enclosed diagram showed the distribution of the points already determined, corresponding to fl particles with velocities between 0.8 and 0.9 e. The writer asked MOLLER for the suitable scattering formula, as he was "very anxious to know how the distribution will differ from that predicted by the classical expression". 149

Of course, the letter excited MOLLER'S interest, who on the same day asked MAX DEL~R~3CK if he knew who was photographing fl ray tracks in an automatic expansion chamber at the Cavendish) 5~ DELBRT3CK had attended the congress devoted to nuclear physics that had taken place in Rome between October 11 and 18. Soon after the conference, he answered MOLLER that BLACKETT, whom MOLLER had suspected, had been at Rome. Instead, he suggested a young Italian student of BLACKETT'S, referring most probably to GIUSEPPE OCHIAL~NI) 5~ Yet the correspondent was not OCCH~AUNI, but a young research student at the Cavendish, F. C. CHAMPION.

An able student, double First in the Natural Science Tripos, FRANK CLWE CHAMPION 152 had joined the Cavendish in July, 1929.15a By that time, the cloud

148 RUTHERFORD, CHADWICK & ELLIS (1930), p. 215. 149 CHAMPION to MOLLER, October 10, 1931 (AHQP-59). 15o MOLLER to DELBROCK, October 14, 1931 (AHQP-59). 151 DELBRCrCK to MOLLER (AHQP-59). OCCHIALIN~ had just arrived at the Caven-

dish in order to spend some weeks learning to build cloud chambers. He became BLACKETT'S collaborator and remained there for three years (cf HENDRY 1984, p. 24; ROSSl 1981, p. 41).

152 F. C. CHAMPION was born in Esher, Surrey, on November 2, 1907. He was educated at The Royal Grammar School, Guilford, and St. John's College, Cambridge. This data comes from a biographical sheet held at St. John's, which includes his obituary in The Times (March 5, 1976). In addition, the College holds a Tutorial File on Champion (henceforward as CTF). I am greatly indebted to M. G. UNDERWOOD, the College's archivist, for this information.

153 As recorded at the Department of Physics of Cambridge University. I am grateful to J. DEAKIN, secretary, for this information.

Meller Scattering 241

chamber had become a favourite device at the Cavendish, which had a number of experts in its construction - among them PATRICK BLACKETT, "the leading exponent of cloud chambers in the world". ~s4 During the summer of 1929, CHAMPION became acquainted with a chamber recently designed by BLACKErr, and in his first term at the laboratory he began to adapt it for the investigation of fl rays. ~ss This work was unexpectedly interrupted at the very beginning of 1930.

The Proceedings of the Royal Society for January 1, 1930, included the article by MOTT where indistinguishability was taken into account in deducing the scattering law for the collision between two identical particles. 156 On January 2, according to his 1929-1930 research report, CHAMPION temporarily shelved his work on fl radiation to test Morr's new formula for the scattering of ~ rays. By the end of March he had taken between 3,000 and 4,000 photographs, and prospective analysis "had shown itself to be definitely in favour of MOTT'S theory". 157 The final analysis, that CHAMPrON completed together with BLACKETa" by October, showed excellent agreement between the experimental data and MowT's formula, ts8

CHAMPION spent most of the academic year 1930-1931 adapting the chamber to its use with electrons. In May 1931, he had at his disposal 3,000 tracks (about 400 photographs), one-tenth of the number needed according to his estimations. CI~AMPION was aware of the need to take a large number of photographs in order to secure his statistical method of analysis. Yet at the outset of his experiments he had no precise idea of the theoretical results he aimed to test - hence his immediate reaction to the appearance of Moan"s article. In May 1930, he still vaguely wrote of his work that it consisted "in the investigation of atomic processes" by means of the cloud chamber) 59

By May 1931, having worked for two years at the Cavendish, CHAMPION'S aims were clearer. In his second research report, after comenting on his work on the scattering of a particles, CHAM~'rON wrote: 16~

These particles, however, are not the fundamental particles of matter which are the proton and the electron. Dirac has calculated, using relativity theory, the expected properties of a single electron and deduced theoretically the experimentally known property of electron spin. Breit, Gaunt and others have tried to apply Dirac's methods to the theoretical determination of the laws governing the interaction of two electrons i.e. the 'simplest' of the two-body problems. All these attempts have been unsuccessful predicting physically inconceivable events and agreeing badly with doubtful experimental data obtained from the hyperfine structure of spectral lines. A direct

154 HENDRY (1984), p. 22. 155 "Research Report 1929-1930" (CTF). 156 MOTT (1930). ls7 "Research Report 1929-1930" (CTF). 158 CHANIPION & BLACKETT (1931). 159 "Research Report 1929-1930" (CTF). 160 "Research Report 1930-1931", May 8, 1931 (CTF).

242 X. ROQUE

experimental determination of the behaviour of two electrons on interaction would therefore be invaluable.

The shortcomings of the theories of GAUNT and BRZlT were here exaggerated in order to justify a genuine interest in electron scattering, apart from any theoretical consideration. Theoretical expectations, on the other hand, figured explicitly in CHAMPION'S statement on the other "fundamental problems" he aimed to solve: 161

(1) The only purely terrestrial proof of the theory of relativity as applied to individual electrons. Theoretical work on the problem, based on simple conservation principles, has been carried out by the writer; this will be published later when more copious data is available. On the data already available approximate measurements shew beautiful agreement with the theoretical predictions.

(2) The intensity-velocity distribution in the fi ray spectrum of Radium E, for which there is at present conflicting evidence. (Radium E is the source of the fast electrons in the other problems).

(3) The experimental investigation of nuclear scattering; this will afford tests of some applications of wave mechanical theory.

The inclusion of the second point, the first one that CHAMPION worked on, was probably circumstantial. The spectra of RaE had been thought to be continuous and to have a well-defined upper limit around 5,500 Hp (an energy of 1 MeV) since the first investigations of it, about 1910.162 In April 1931 F. R. TERROUX published the results of his own recent investigations that contradicted previous evidence. 163 Using the same chamber as that of his previous experiments with WILLIAMS, TERROUX took some 80 pairs of photographs of RaE fl rays. The analysis of the 500 tracks he found in them led him to conclude that there existed no upper limit, the spectra instead decreasing gradually up to very high energies, following approximately a Maxwellian distribution. TERROUX estimated that the energy of 4% of the particles exceeded 5,000 Hp.

TERROUX did these experiments at the Cavendish, and CHAMPION would have had a first hand knowledge of TERROUX'S results: From the analysis of 1,000 tracks he confirmed the existence of an upper limit about 5,000 Hp, and observed that, in agreement with previous observations, only the 0.05% of the spectrum had an energy higher than 5,500 Hp.

CHAMPION submitted these results to the Proceedings of the Royal Society on October 15, 1931. ~6. The coincidence of dates - his first letter had reached Copenhagen on October 14 - suggests that he waited until a first analysis of his

161 Ibidem. 162 The energy of a fl particle crossing a magnetic field is given by the product of the

field intensity (H) and the radius of curvature (p) of its path (in Gauss and cm, respectively, for the figures given). See above eq. (43).

163 TERROUX (1931). 164 CHAMPION (1932a).


photographs had been completed before writing to M~OLLER. CHAMPION expressed .himself in the anonymous letter to MOLLER ("I have recently taken a large number of photographs of fast r - ray tracks in an automatic expansion chamber") almost in the same terms as in the article he had just finished ("The writer has recently obtained a large number of photographs of r - ray tracks in an automatic expansion chamber"). By this time he had at his disposal most of his photographs, and devoted the following months to their analysis.

In his research report of 1929-1930, CHAMPION said of his work that it consisted in the application of the cloud chamber, "especially in the modified form due to P. M. S. Blackett", to the investigation of atomic processes, and also that "a new chamber was devised for taking fl ray photographs". 165 In the article on the spectrum of RaE, he noticed in like manner that the chamber he had used was "a modification of that previously employed by Blackett and the writer". 166 Had CHAMPION built a new chamber, or simply altered BLACKETT'S? A 1934 report by BLACKETT on CHAMPION'S dissertation shows that CHAMPION had indeed used the chamber built by BLACKETT: "Since Mr Champion left Cambridge he has built a new automatic cloud chamber following in the main the design of the one he used during his time in the Cavendish, and which was built by me six years ago. He does not appear to have made any material or very original changes in this design". 167

BLACKETT'S chamber (figure 1) was based on a standard model supplied by the Cambridge Scientific Instruments Company. 168 More than a new chamber, it was a new arrangement of the cloud chamber and the photographic cameras, which optimized the number of tracks photographed. The chamber was enclosed inside a wooden box, with tubular light screens, so that it was not required to use the apparatus in a completely darkened room. As BLACKETT described it, "an aluminium casting has two faces at its ends making angles of 45 ~ with the top. To these two faces are attached standard Ensign cinr- c a m e r a s . . . Each lens is carried on a brass bush, the two faces of which are inclined at an angle of 81~ 169 The inclination of the lenses was one of BLACKETT'S innovations with regard to previous designs. It was calculated to increase the photographed area.

On November 2, CHAMPION wrote to MOLLER for the second time. He qualified now more precisely his work as "an experimental test of your formula

t65 "Research Report 1929-1930" (CTF). 166 CHAMPION (1932a), p. 672. CHAMPION refers to CHAMPION 8Z BLACKETT (1931). 167 "Report by Professor P. M. S. Blackett on the dissertation of F. C. Champion"

(CTF, my emphasis). A simultaneous report by J. COCKCROFT stated much the same: "The apparatus used [by Champion] was taken over with slight modifications from previous experiments of Mr. Blackett so that no technical contributions of importance were made by Mr. Champion" ("Report by Dr Cockcroft on the Dissertation of F. C. Champion", CTF).

168 The Company supplied cloud chambers as early as 1913, and from time to time made special apparatus for various scientists (cf CATTERMOLE & WOLFE 1987, p. 78).

169 BLACKETT (1929), p. 622.

244 X. ROQUI~

Figure 1. The arrangement of cloud chamber and cameras designed by BLACKETT in 1929. With this same apparatus, adapted to its use with electrons, CHAMPION carried out his experiments on scattering of fl particles, in the context of which he tested the MOLLER formula. (From BLACKETT 1929; this figure features in the volume as no. 20.

Reproduced with permission of the Royal Society.)

for the scattering", t7~ As mentioned above, MI~LLER answered him immediately, showing himself much interested in CHAMPION'S work, and optimistic with regard to the general scattering formula. 171 On November 8, CHAMPION gave MOLLER some details of his work. The chamber was filled with nitrogen, and he had observed the collisions between the fl particles and the electrons "in the extra-nuclear structure of the nitrogen a tom" - nuclear models were still playing with the existence of electrons in the nucleus. According to the first point in his project, CHAMPION was then searching in his photographs for a direct proof of

t h e conservation of momentum and energy in collisions: 172 "I am preparing a paper for publication at the moment on some accurate measurements of the angles between the directions of mot ion of the two electrons after collision taking into account the relativity change of mass". The resulting article, "On

t7o CHAMPION to M~ILLER, November 2, 1931 (AHQP-59). 171 See above, p. 219. 172 CHAMPION to MOLLER, November 8, 1931 (AHQP-59).


Some Close Collisions of Fast fl-Particles with Electrons, Photographed by the Expansion Method", was ready by mid-February 1932.173

According to classical mechanics, when a particle collides elastically with another particle of the same mass initially at rest, both particles always move after the collision at right angles. In relativistic mechanics, the angle between the direction of motion of the two particles depends on the scattering angle and the velocity of the incident particle, and it diminishes as this velocity ap- proaches that of light. CHAMPION mentioned in his article previous qualitative evidence of this phenomenon, but he stated that "up to the present no quantitative study has been made of the general relation between the whole angle after the collision, the angle of scattering, and the velocity of the incident par- title. ''174 He had measured the scattering angle from the two simultaneous exposures taken of each event, and the velocity of the particles from the curvature of the tracks. In his letter to MOLLER of November 8, CHAMPION included two photographs of the same collision taken with two cameras at right angles. The collision had practically taken place in the plane of one of the photographs, so that "the apparent angle on this photograph is almost the true angle. You will observe that it is not nearly 90~ in fact the measured angle is 73 ~ agreeing exactly with the relativity expression for this velocity of the incident electron and the particular angle of incidence". 175

As he had foreseen, CHAMPION had to analyze a lot of photographs, because the probability of the collisions he was interested in was very low. In his first letter to MI21LLER he spoke only of "a large number" of photographs; through a later article we know that they were about 4,000.176 The criteria that a track should satisfy to allow a proper measurement of the scattering angle and the velocity reduced drastically the number of collisions observed. In 30,000 tracks CHAMPION could only find 50 "close" or appropriate collisions, 14 of them best satisfying the experimental criteria. CHAMPION based his analysis on these collisions, and he found an "excellent" agreement between the theoretical and the observed values.

At the same time, CHAMPION had been analyzing his photographs with the intention of testing the MOLLER formula. In his first letter to MOLLER he had already included some data, 50 points in a diagram 0 (scattering angle)-fl (v/c). The same diagram, extended to 250 points, became the first figure of CHAM- PION'S next article, "The Scattering of Fast r-Particles by Electrons", finished in June 1932, when MOLLER'S article containing the scattering formula had not yet appeared. 177 CHAMPION had previously completed his doctoral dissertation,

173 CHAMPION (1932b). 174 Ibid., p. 630. Among previous observations CHAMPION mentioned those of

BOTHE and WILSON. 175 CHAMPION to MOLLER, November 8, 1931 (AHQP-59). 176 CHAMPION (1932C), p. 691: "The present results are from the analysis of 4,000

photographs, giving about 30,000 tracks of fast r-particles in nitrogen". In the article we are considering, about 30,000 tracks were also investigated.

177 CHAMPION (1932C), submitted on June 25, 1932. Both diagrams are reproduced in KRAGH'S paper. CHAMPION could only quote MOLLER (1931), but he expressed of course his gratitude to MOLLER for communicating his results.

246 X. ROQUg

where he reached the same conclusions as in the article, on the basis of 131 collisions. 178

In the tracks suitable for measurement, which totalled some 650m, CHAMPION observed 250 collisions with scattering angles exceeding 10 ~ , with fl = v/c between 0.8 and 0.9 for the incident particle. Collisions with a lower scattering angle had not been considered, for they might have introduced a comparatively large percentage of error, and also greatly increased the number of measurements.

In order to compare his results with the theoretical predictions, CHA~aPION followed MOLLER'S advice. Upon giving CHAMPION the incorrect version of the general scattering formula, MOLLER had suggested that he introduce x = cos 0cm and 7 as independent variables, instead of 0 and v: "If you then plot your data in a (x, 7) diagram, this formula should directly give the density of the dots". 179 In his article, CHAMPION gave the MOLLER formula in the form (see (40))

/ / e 2 "~2

do.(0) = 4rc~vSv2) 7 + 1 dx \ /

�9 (1 - x2) 2 1 - - X 2 q- 4y 2 \ + ~ ' (48)

where 2 - - (y + 3) sin 20

x = COS 0 e m = 2 + (7 -- 1)sin 2 0 (49)

This expression was graphically integrated over the range of incident energies and the angular intervals that CI~AMPION had considered. A single table in CHAMPION'S article represented his results and the comparison with theoretical predictions (table 1). CI~AMPtON judged the results to be "in good agreement" with the MOLLER formula, both in the discussion of the results and in the summary of his paper. He also concluded that "MOLLER'S formula gives the best account of the scattering of electrons by electrons". ~8~ Nevertheless, he acknow- ledged that for scattering angles greater than 20 ~ the corrected non-relativistic formulae also agreed well with his results.

CHAMPION'S experiments were conceived when the MOLLER formula did not yet exist. As we have seen, CHAMPION did intend to analyze experimentally the interaction between two electrons, but in a broad context of study of the fl radiation, and explicitly disregarding theoretical predictions. In the article we have just examined, the MOLLER formula was praised as "the most satisfactory theoretically" among the formulae proposed for the interaction between two

178 CHAMPION, On Some Applications of an Automatic Expansion Chamber to the Investigation of the Collisions of ct Particles with Helium, and of Fast fl Particles with Electrons, unpublished doctoral dissertation (Cambridge University Library, May 1932). In his thesis, CHAMPION applied the formula that MOLLER had communicated to him on January 25, 1932 (see above, p. 222).

179 MOLLER tO CHAMPION, 7 December 1931 (AHQP-59). 180 CHAMPION (1932C), pp. 694 and 695.


Table 1. The comparison of CHAMPION'S results with the theoretical predictions, according to CHAMPION (1932C), p. 693. The columns contain: 1, the three angular intervals considered:, 2, the number of points observed in each of them; 3, 4 and 5, the values expected according to the theories of MOLLER, classical, and MOTT, respectively; 6, the values obtained by substituting m / ( 1 - fl2)112 for m in MOTT's formula; 7, the values obtained treating the classical formula in the same way; and 8, the prediction of the classical formula with T 2 instead of (~mv2) 2 in the denominator, where T is the kinetic

energy of the incident particle.

1 2 3 4 5 6 7 8

0 ~ Observed MOLLER C M (MOTT) M (1 -- f12) C (1 - f12) C/T 2

30-max. 10 13 57 28 7 15 9 20-30 26 30 148 105 26 37 21 10-20 214 230 761 650 162 190 108

Total . . . . 250 273 966 783 195 242 138

electrons, but it was simply characterized as a formula "based on quantum mechanics". T M When in 1934 J. COCKCROFT and BLACKETT reported on CHAMPION'S dissertation, who was applying for a lectureship at King's College, London, both failed to mention the testing of the MOLLER formula among CHAMPION'S merits. COCKCROFT said of the questions tackled by CHAM- PION: "The problem attacked is one of considerable importance, being a study of the close collisions of atomic particles; a study which allows a direct test of some of the fundamental results of the theory of relativity and of the wave theory of mat te r"J 82 BLACKZTT simply said of CHAMPION'S results that they were "of considerable interest and importance", yet he understandably criticized his lack of originalityJ 83 Therefore, both when conceived and interpreted, CHAM- PION'S experiments were not related to early quantum electrodynamics.

CHAMPION left the Cavendish in September, 1932, to join the University College at Not t ingham as an assistant lecturer in physics. There, besides contin- uing the analysis Of his photographs, which were not to be completed until 1935, ls4 he worked on a monograph on the cloud chamber, and collaborated with the Zoology Depar tment in the application of spectroscopy to problems connected with sexual differentiation in fish. In 1934, he was appointed lecturer at King's College, London, where the rest of his career was to develop. ~85

181 Ibid., pp. 688 and 695. 182 "Report by Dr Cockcroft on the dissertation of F. C. Champion" (CTF). 183 "Report by Professor P. M. S. Blackett on the dissertation of F. C. Champion"

(CTF). 184 CHAMPION (1936), submitted in July, 1935. 185 In 1948 CHAMPION became reader in physics, and in 1959 he was appointed to

the Chair. CHAMPION'S earlier interest in nuclear physics turned later towards the solid state. He Jived in Spain from his retirement, in 1970, until his death in M/daga in 1976 (cf. "Obituary. Professor Champion. Noted physicist" The Times, 5 March 1976, p. 17).

248 X. ROQUI~

3.3. Subsequent experiments prior to 1947

CHAMPION'S experiments remained unique during the thirties, as not until 1941 did another article concerned with a test of the MOLLER formula appear. Further experiments seem to have been motivated by the discrepancies between CHAMPION'S and WILLIAMS' results. WILLIAMS himself tried to clear up the matter in 1933, but his experiments were not completed. This is the brief account of them he gave to ARTHUR E. RUARK in 1940:186

In view of ~the discrepancy with theory of the number of secondaries produced by fa~t electrons observed by Terroux and myself at Cambridge, I made further observations later at Manchester (1933) with the help of an assistant

- Mr. Car0eron. We did not find the earlier large discrepancies but a rate of about 1.2 to 1 of observed to theoretical [values]. This second work was

/ . also done/using RaE electrons and at the time I decided to extend the observatiops to faster electrons but that was never done and the results with RaE were[not published. The details are in a PhD thesis presented by Cameron.

RUARK wa~ the head of the Physics Department at North Carolina Univer- sity from 193~. Early in October 1940, he had asked WILLIAMS for reprints of some of his articles,~ as he had "been doing work on nuclear radiative scattering and on energ~ distribution of side branches produced in electron-electron collisions", t87 At/least since the beginning of that year, RUARK had spent much of his time "o~ c loud chamber work with electrons", and two of his students were "working on the production of secondaries". ~ss This research was supported by a grant from the American Philosophical Society that enabled him to direct two consecutive analyses of the collision between two relativistic electrons with the explicit aim of testing the MOLLER formula.

The work of RUARK and his students, as it stood in September 1940, is precisely characterized in a long letter he sent to BREIa'. He started by describ- ing their experiments: ~89

As to the electron-electron collision experiments, I will not try to give you any lengthy account, because the experiments are really complicated. Com- parison with theory is accomplished by the use of formula 76 in Moller's article [MOLLER 1 9 3 2 ] . . . This formula, of course, must be averaged over

186 WILLIAMS to RUARK, November 13, 1940 (RUARK Papers). These observations and WILLIAMS' letter were mentioned in the paper by HORNBECK & HOWELL shortly to be considered (HORNBECK & HOWELL 1941, p. 51).

187 R U A R K t o WILLIAMS, October 3, 1940 ( R U A R K Papers). 288 RUARK to BREIT, March 5, 1940 (BREIT Papers). 189 RUARK to BREIT, September 2, 1940 (BREIT Papers). The same results and

a similar description is to be found in RUARK'S letter to WILLIAMS quoted above. He also observed that his students had not found "the large discrepancies with theory which were observed in the paper of yourself and Terroux".


our actual energy spectrum of primary electrons . . . . Let us consider the cross-section for production of side branches with energy superior to some lower limit To; and let us consider the ratio of the observed number to the number calculated from Moller's formula. Calling this ratio R, the revised results are as follows:

To, lower limit Observed Expected R of energy of secondaries No. of No. of

considered, in KeV collisions collisions

20 107 88.3 1.21 _ 0.12 30 63 56.4 1.12 _ 0.14 40 35 40.9 0.86 • 0.15

These data refer to secondary electrons produced in 190 meters of primary electron track within the limits 0.67 to 1.6 MeV. After the work had progressed for a while, we decided to limit our attention to primaries of higher energy than 1.3 MeV. The work was done by Irl Howell and George Hornbeck.

RUARK added: "The important point at the present stage is not the deviation from the prediction of Moller's formula, but rather the closeness of the numbers to his formula". How is this statement to be interpreted? It might suggest a lack of confidence in the formula at the beginning of the work, cleared up by the success of the experiments - as reinforced by RUARICS final comments: "To summarize: Ours are the only data on this important subject, covering the range above about 1.3 MeV. They agree with Moller's formula much better than we hoped for when the work began". Of course, no statement of this kind appears, as we shall see, in the articles published by RUARK'S students. Yet it might also be related to the experimental difficulties encountered, that RUARK briefly discussed immediately thereafter:

The limits of error given in the above table are merely those due to statistical fluctuations in the occurrence of secondaries. I must explain that even with good viewing and measuring facilities, it is difficult to measure absolute cross-sections with the cloud chamber. There are several systematic or semi-systematic sources of error, of which we are cognizant, but which we cannot easily eliminate from the present body of data. These sources of error can be eliminated to a very large extent in future experimental work, but their importance was not suspected, or realized, when we took the photographs on which the above data are based. I shall not go into details, but the whole thing boils down to eliminating little errors and troubles, each of which acting separately can cause an error from 1 to 5 per cent.

In the next paragraph, RUARK depicted "the present situation":

The present situation is this. Champion had tested formula 74 of Moiler which gives the angular distribution which should correspond to the energy

250 X. RoQu~

distribution we are studying. He got agreement to about 10 per cent for electrons with energy below 1.3 MeV. Williams and Terroux made cross- section measurements by the same method we are using, up to 1.6 MeV. Their results deviated from theory by a factor which was sometimes larger than 2, comparison being made with a formula which is gotten by dropping out the entire second line of Moller's formula 76. This corresponds in fact to the use of the old formula employed by J. J. Thomson and by Bohr prior to 1920. If comparison were made with Moller's formula, the values of R would be even higher, for the experiments with Williams and Terroux, than the values given in their paper.

The final paragraph of RUARK'S letter contained, besides the summarizing statement quoted above, interesting remarks about the prospects for the work:

The experimental work is being continued here by P. E. Shearin and Eugene Pardue, using a method of measurement which is much faster than that employed in most of the work of Hornbeck and Howell. We have found out how to do the job by methods which will permit piling up a much larger total number of events. Within two to three months it should be possible to push the fluctuation errors down to only a few per cent. At the present time the data available do not distinguish sharply between the requirements of Moller's formula and the requirements of the older formulas like that of Thomson and Bohr. The difference between the two is small for secondaries having energies which we can deal with. Obviously, we ought to push out into the region of higher secondary energies; that is, we should study "closer" collisions. However, the infrequency of close collisions limits our efforts in this direction, so we are going to try to do a thorough job on secondaries of relatively low energy. Photographs in a larger chamber are needed before we can attack the problem of the closer collisions effectively. I think that even at the present stage, the observations are interesting, because so far they do not indicate the necessity of including in the theory any terms higher than v2/c z. You will note that for a two million volt electron 1)4/C 4 is about 0.92.

The results advanced by RUARK did not differ much from those published, in April 1941, by HORNBECK & HOWELL. They stressed the importance of measuring the cross section for the scattering of high-energy electrons by electrons, much scarcer than those concerning stopping or ionization, and presented their experiments as motivated by the discrepancies in the existing experimental data. From the data provided by WILLIAMS & TERROUX, HORNBECK & HOWELL deduced a cross section 2.1 + 0.25 times that predicted by MOLLER, "in sharp contradiction with the results of Champion, and with our own", concluding that "more extensive cloud chamber observations are needed, both to clear up the discrepancy and to extend the results to higher primary energy". 19~

19o HORNBECK & HOWELL (1941), pp. 38 and 39.

Molter Scattering 251

HORNBECK & HOWELL'S analyses were based on a large body of photographs made available by CREIGHTON C. JONES, a professor at the same university. The study was done by means of a stereoscopic viewing and measuring instrument developed by JONES & RUARK. 191 The primary electrons were recoil electrons produced at the walls of a nitrogen-filled chamber, by ? rays from a me- sothorium source. Their energy was given by the curvature of their tracks; that of the secondary electrons by their range. As stated in RUARK'S letter, HORNBECK & HOWELL calculated the cross section for the production of a secondary electron of kinetic energy T, higher than To, but lower than Tp/2, where Tp is the kinetic energy of the primary. They integrated the MOLLER formula between To and Tp, and averaged it over the primary energy. 192 Their results, summed up in table 2, complemented those of CHAMPION over the range of kinetic energy of the primaries from 0.4 to 0.9 MeV. "Both investigations", they concluded, "support each other in demonstrating the essential correctness of Moller's theory of secondary energy distribution in the domain of primary energy from 0.4 to 2.6 MeV". 193

Table 2. Results of HORNBECK & HOWELL for the ratio R of the observed and calculated value for the cross section, according to the MOLLER formula. To, lower limit for the energy of secondary electrons

(HORNBECK & HOWELL 1941, p. 33).

To(KeV): 20 30 40 R: 1.18 _ 0.14 1.09 • 0.16 0.84 • 0.15

SHEARIN t% PARDUE'S work did not appear until February 1942, in the Proceedings of the supporting institution, the American Philosophical Society. It was intended as a complement to that of HORNBECK • HOWELL "with the objectives of decreasing the statistical errors and increasing the accuracy of all the measurements involved". 194 SHEARIN & PARDUE improved the agreement between the experimental data and the MOLLER formula modifying the criteria for selection of tracks and measuring the ranges more accurately. As their results (table 3), expressed like those of HORNBECK & HOWELL, showed, "the Moller formula is essentially correct for the primary and secondary energy ranges considered here". 195

These experiments differ from CHAMPION'S in their inception. They are the result of teamwork oriented towards the test of the formula and its comparison

191 JONES & RUARK (1940). 192 Formula (76) of MOLLER'S article gave the scattering differential cross-section in

terms of the kinetic energy Q lost by the incident particle. This is the formula that HORNBECK and HOWELL integrate. It follows from formula (74) in MOLLER'S article (see (40)) by means of the relation Q = mc2/2(~- 1)(1- cos0cm).

193 HORNBECK & HOWELL (1941), p. 51. 194 SHEARIN & PARDUE (1942), p. 243. 195 Ibid., p. 243.

252 X. ROQUI~

Table 3. Results of SHEARIN & PARDUE for the ratio R of the observed and calculated value for the cross section. The table is to be interpreted

as for table 2 (SHEARIN & PARDUE 1942, p. 243).

To(KeV): 20 30 40 R: 1.07 + 0.09 0.99 + 0.11 1.04 _+ 0.12

with the other formulas proposed. In the late forties, their real significance and decisiveness would be questioned. Setting these questions momentarily aside, let us consider how the MOLLER formula was appraised in both works. HORNBECI~ and HOWELL characterized the contributions of BETHE, BREIT and WOLFE, without much precision to add simply that "[Moiler 's] final formulas giving the cross-section for production of a branch with energy in the range T to T + dT, are presumably the most accurate ones available". 196 SHEARIN and PARDUE referred to a "relativistic theory for the electron-electron cross section" due to MOLLER that "includes the effects of exchange and retardation of potentials, to terms in (v/c) 2 inclusive, vp being the velocity of the incident electron". 197 At the end of their article they were more explicit though, interestingly enough, they related MOLLER'S treatment to BREIT'S: "We wish to emphasize the importance of thorough tests of this formula, the only one in which the interaction of two similar fundamental particles has been calculated with perfect symmetry, in accordance with the spirit of Breit's considerations [BREIT (1929)]". 198 More fundamental elements of MOLLER'S work, such as DIRAC'S equation of the electron, not to mention the relating of the formula with quantum electrodynamics, were not explicitly mentioned.

3.4. Postwar interest and conclusive test

As far as I know, no other article appeared concerning an experimental test of the MOLLER formula until 1950. That year, a team from the Institute for Nuclear Studies of the University of Chicago published an analysis of 98 collisions between electrons, obtained "as a by-product of several cloud- chamber investigations carried out in this laboratory". 199 The authors justified the study on the limited experimental evidence of this kind of collisions. They compared MOLLER'S theory with relativistic versions of MOTT'S and RUTHERFORD'S theories, and with the classical theory of RUTHERFORD. The results were once more not conclusive: "Our data discriminate definitely only against the latter theory. When combined with 122 electron-electron collisions observed by Champion, they are consistent only with the first two theories, but

196 HORNBECK & HOWELL (1941), p. 40. 197 SHEARIN & PARDUE (1942), p. 243. 198 Ibid., p. 249. t99 GROETZINGER, LEDER, RIBE. & BERGER (1950), p. 454.


are insufficient to discriminate decisively between them". MOLLER theory was characterized as "the theory of electron-electron scattering now generally accepted": z~176 Had by then the theoretical basis of MOLLER'S formula been so well established, that the authors did not need to mention quantum electrodynamics explicitly? Or was the formula simply accepted the same way it had been deduced, without regard to the general encompassing theory of the electromagnetic interactions?

From this moment on an increasing in interest in electron-electron scattering, especially in the United States, is easily observed. In 1951 a team from Illinois University analyzed this process by means of a beam of 15.7 MeV monokinetic electrons, from a 22 MeV bevatron. T M At the end of the same year, another team from the Radiation Laboratory at Berkeley studied this interaction at higher energies, by means of 200 MeV electrons from the sin- croton of Berkeley. 2~ At the Microwave Laboratory of Stanford, yet another team of experimental physicists analyzed the scattering of 6.1 MeV electrons from a linear accelerator. 2~ This series of experiments reached its peak in 1954 with the publication of the article considered to be the most decisive, where the validity of the MOLLER formula was proved for the range of energies between 0.6 and 1.2 MeV. T M This list, which does not aim to be complete, suffices to make the point. In none of these articles is the MOLLER formula specifically related to quantum electrodynamics. They agree in assuming that the formula is the most satisfactory from a theoretical point of view, but they also do not give any reason why this is so. In the more precise characterization of the formula, it is said to be based on DIRAC'S theory.

Let us finally consider more closely the decisive experiments, as, in fact, they are not the last step but one of the very first instances of the postwar interest in MOLLER scattering. They were developed at one of the new laboratories built after the war, the Laboratory of Nuclear Studies of Cornell University. In September 1950, LORNE ALBERT PAGE defended his doctoral dissertation: A Measurement of Electron-Electron Scattering. 2~ His results did not essentially differ from those that would be published together with ASHKIN and WOODWARD four years later. As PAGE had initiated his graduate studies in the fall of 1945, his experiments date back to the period immediately following the end of the war. 206

200 Ibid., pp. 454 i 455. 201 SCOTT, HANSON ~r LYMAN (1951). 202 BARKAS, DEUTSCH, GILBERT & VIOLET (1952). 203 BARBER, DECKER & CHU (1953). 2o4 ASHKIN, PAGE & WOODWARD (1954). 205 PAGE, unpublished doctoral dissertation, Comell University Libraries, Ithaca. See

also PAGE (1951). 206 L. A. PAGE was born in 1921. In September 1940, he entered Queen's University,

Kingston, Ontario, where he was granted the Bachelor of Science degree in Physics in May 1944. In November 1945, following a term in the Canadian Army, he began graduate studies at CorneU, where he stayed until June 1950 (cf PAGE'S dissertation, "Biographical sketch").

254 X. ROQU~

i \ \ i Ip2

PlX + P2x = PO

T1 + T2 = TO

] To T-JL = poPlX EOKI N = V

180 ~

0 6 i i i i i I i

Inches

Figure 2. The 270 ~ scattering chamber devised by L. A. PAGE to measure electron- electron scattering, according to figure 1 of his dissertation. His description: "Upper right: horizontal section through mid-plane of scattering chamber; lower right: elevation view of the chamber. The magnetic field is vertical. - A typical path for a fl particle from source to scattering foil, F, is sketched. Two electrons, el and e2, might emerge from the foil as shown and be recorded in the Geiger counters, A and B. Pertinent horizontal throws are denoted by To, T1, and T 2 and momenta by Po,Pl, and P2. - - The vector diagram at upper left represents momentum in the plane of collision, for incident kinetic

energy Fk~,= 1 mev, and fractional energy transfer v = 0.35". ~ 0


PAGE'S experiments centered on collisions with large energy transfers, those in which spin effects were more pronounced and the MOLLER formula thus more clearly distinguished from the other scattering formulas. Unlike the other experiments going on, he used radioactive electrons - fl electrons from a source of Sr9~ 9~ - whose energy ranged from 0.6 to 1.7 MeV. The scattered electrons were detected by means of two Geiger counters connected in coincidence, a decisive innovation with respect to previous experiments and those taking place simultaneously, based on the observation of forked tracks in a cloud chamber. The experimental disposition devised by PAGE is shown in figure 2. This same figure would be reproduced four years later, without the momentum diagram, in the joint article with ASHKIN & WOODWARD. 207

PAGE briefly characterized the MOLLER formula in his dissertation as based on "the Dirac theory of the electron". He added: 2~ "Careful measurements of the differential cross-section in the relativistic energy region as a function of incident energy should provide verification or denial of the MOLLER formula, and by implication the present theory of the electron". A direct re~ference to quantum electrodynamics is nowhere to be found, though, as in the case of ScoTT, HANSON & LYMAN mentioned above, we may ask whether it was by ,that time necessary. PAGE was fully aware of late theoretical developments, which ~he mentioned "for completeness", in which emission of radiation was taken into account. 2~

PAGE'S experiments not only proved the MOLLER formula. PAGE was satisfied in his conclusions that "the formula is rather 'necessary', in that all the essential features of the formula must be included for a reasonable agreement with experiment - thus it is not merely some degenerate form which applies for the range of E [incident energy] and v [fractional energy transfer] investigated here". 21~ At the same time, he discarded the possibilty that a theory more advanced than MOLLER'S could modify MOLLER'S predictions with regard to the present experiments.

Let us finally note that PAGE had an exceptional interlocutor: RICHARD P. FEVNMA~, to whom he was indebted for "helpful discussions of the process being studied". 211 This acknowledgement shows again that the renewed interest in MOLLER scattering grew soon after the end of Second World War, and also that it concerned both its theoretical and experimental aspects.

Concluding remarks

MOLLER deduced his scattering formula without regard to early quantum electrodynamics, on the basis of more secure theoretical developments, notably

207 See ASHKIN, PAGE & WOODWARD (1954), p. 358. 208 PAGE's dissertation, p. 1 (my emphasis).

209 PAGE cited SCHWINGER (1949) and KATZENSTEIN (1950), and a doctoral dissertation written in his department by LOMANITZ, Second Order Effects in the Electron- Electron Interaction, 1950 (PAGE's dissertation, p. 25).

21o Ibid., p. 20. 211 Ibid., p. 58.

256 X. ROQUI~

a notion of correspondence. In fact, MOLLER shows little concern with this theory in his articles, where he barely mentions it. Though the MOLLER formula might also follow from quantum electrodynamics, as indeed BETIJE and FERMI promptly showed, MOLLER scattering remained an essentially neglected application of the theory during the thirties, to judge by the scarce texts devoted to it before renormalization. This indifference extends to the experimental testing of the formula: the often-cited experiments of CHAMPION were conceived without knowledge of the formula, and they remained unique during the thirties. In addition, only two more significant attempts were made to test the formula before renormalization.

By contrast, around 1950 MOLLER scattering was being reconsidered from the theoretical, as well as from the experimental, point of view. At that moment, a number of experiments were taking place to test the formula, mainly in the United States, among them the one still considered as the classical test of the MOLLER formula at low energies. Furthermore, FEYNMAN shortly before had given to MOLLER scattering a prominent role in his approach to quantum electrodynamics, and radiative corrections were being calculated. Henceforward, as a result of this renewed interest. MOLLER scattering was to find its way into texts devoted to the theory. As a main conclusion, therefore, I shall stress that MOLLER scattering only became an application of quantum electrodynamics after renormalization.

This change of status may largely be explained by the changing status of theory itself. Sound theoretical difficulties, which were acutely perceived, marked early quantum electrodynamics. The lack of confidence in the fundamentals of the theory, no doubt obscured the relevance of its applications. We have seen that to PAUL1, the divergence in the proper energy of the electron prevented "a consistent relativistic treatment of the many-body problem" and much diminished the significance of a mere approximate treatment. 212 OPPENHEIMER and BREIT expressed themselves in similar terms, the case of BREIT being especially significant, as his treatment of electron-electron interaction, which directly related to HEISENBERG • PAULI'S quantum electrodynamics, could not account for the interaction between free electrons, as he repeatedly acknow- ledged. The degree of confidence in the theory accounts at the same time for the reappraisal of the formula after the war, as great achievements and expectations marked the beginning of renormalized quantum electrodynamics. In this context, approximate treatments and processes once thought to be of secondary interest were looked at anew. Electron-electron interaction, in addition, figured prominently in FEYNMAN'S quantum electrodynamics. In the second of the articles where FEYNMAN opposed his "over-all space-time" approach to the usual Hamiltonian one, the interaction between free electrons exemplified his considerations. 213 The "fundamental equation for electrodynamics" which

212 See above p. 231. 213 FEYNMAN (1949), p. 769: "We begin by discussing the solution in space and time

of the Schr6dinger equation for particles interacting instantaneously. The results are immediately generalizable to delayed interactions of relativistic electrons and we represent in that way the laws of quantum electrodynamics". See SCHWEBER (1986).


FEYNMAN proposed "describes the effect of exchange of one quantum (therefore first order in e 2) between two electrons. It will serve as a prototype enabling us to write down the corresponding quantities involving the exchange of two or more quanta between two electrons or the interaction of an electron with itself". 214 The relation of MOLLER'S formula with the new version of the theory was clearly stated a few lines later: "The calculation, from [our fundamental equation for electrodynamics], of the transition element between positive energy free electron states give the Moller scattering, when account is taken of the Pauli principle". 215

On the other hand, the increasing importance that the study of cosmic radiation acquired during the thirties, should also be taken into account. This was a major research field at the time, which demanded the efforts of theorists, and attracted a number of able experimental physicists. The application of his calculations to the elucidation of the nature of the elusive cosmic rays was, as we have seen, one of MOLLER'S major aims when he began working on his thesis. Though in the end he exclusively considered r-radiation, this aspect of his work most interested other physicists. 216 Significant experimental resources were drawn into cosmic rays studies. In 1931, H. V. NEHER, a young research student in Caltech who had completed a doctoral dissertation on the nuclear scattering of high-energy electrons, intended to carry on his experiments. 217 While this denotes an interest in experimentally probing the different scattering formulas, t h e final outcome of NEHER'S experiments shows equally clearly the priority attached to research on cosmic radiation, as the experiments were "dropped within a year because of the press of cosmic-ray work". 21s CARL ANOERSON also felt this pressure upon him. In 1930, he was "greatly interested" in the experiments by CHt~NG-YAo CHAO on the scattering of y rays, and intended to extend them by means of a cloud chamber, when MILLIKAN com- pelled him to build an instrument to measure the energies of the electrons present in the cosmic radiation.219 At the Cavendish, during the early thirties,

214 Ibid., p. 772. 215 Ibid., p. 773. 216 In December 1931, upon answering MOLLER that his results on the collision

between two particles most interested him, MOTT added: "I am especially' interested in the cosmic ray case of energy >> me 2, for small angles" (MOTT to MOLLER, December 15, 1931, AHQP-59; see above p. 222). In his article on the stopping of relativistic electrons, BETHE also stressed the importance of the stopping formula "in interpreiing the experiments on the corpuscular height-radiation" (BETHE 1932, p. 293),

2~7 NEHER'S reports to the December 1930 and June 1931 meetings of the American Physical Society (NEHER 1931a, 1931b), were most probably based on his dissertation. See also WOLFE (1931), p. 601.

218 NEHER (1983). 219 ANDERSON (1983), p. 135. CHAO, a Chinese research student at Caltech, had

recently detected, simultaneously with L. MEITNER and H. HUPFELD in Berlin, and G. T. P. TARRANT in Cambridge, the so-called "Meitner-Hupfeld effect", ~he anomalous scattering of 7 rays (cf BROWN & MOYER 1984).

258 X. ROQUt~

BLACKETT increasingly turned his interest to cosmic rays, and he also adapted his cloud chambers to cope with them.

The exiguous secondary literature has echoed this and stressed cosmic rays as the most significant proving ground for early quantum electrodynamics. 22~ There is no doubt that some of the fundamental processes of quantum electrodynamics, notably bremsstrahlung and pair creation and annihilation, were of great importance in determining the nature of cosmic radiation, and that physicists deeply involved with the theory, such as OPPENHEIMER or HEISENBERC, contributed great efforts towards the comprehension of this phenomenon. Yet cosmic rays were not the only medium for testing the scattering formulas

- indeed, they were all tested in the relativistic domain by means of the radioactive radiations. 221 In the case of the MOLLER formula, that was the only posgibility, because the formula reduces in the ultrarelativistic limit to a simple classical formula. While the BHABHA formula, which appeared in 1936, received even less attention by experimental physicists than the MOLLER formula, the KLEIN-NISHINA formula was much investigated during the early thirties. This was so, however, by means of radioactive 7 rays, and when applied to cosmic rays the formula was taken for granted, and used to deduce the wave-length of the supposedly incoming high-energy 7 radiation.

Early quantum electrodynamics did not lack experimental bases. While cosmic ray particles were puzzling physicists, and causing them to disregard the theory at high energies, the analysis of relativistic collisions might have given it some support. If they turned out to be relatively ignored during the thirties, it was due mainly to the problematic character of the theory, and to the significance attached at that moment to a most exciting and promising research field. 222

Acknowledgements. An earlier version of this paper was written in the spring of 1991, and presented in May 1991 at the Seminari d'Histdria de les Ci6ncies at the Universitat Aut6noma de Barcelona. I am very grateful to MANUEL G. DONCEL for suggesting to me the analysis of MOLLER scattering as a case-study on the experimental basis of quantum electrodynamics, and for helping and encouraging me to write this paper. I benefitted from comments and criticism from ANTONI MALET, KARL V. MEgENN and JORDI CAT. I am also indebted to HELGE KRAGI4 for making available his paper, though

220 CASSIDY (1981), GALISON (1983), (1987). 221 A fact that is easily overlooked. DARRIGOL (1982) states for example: "Les

formules dormant les probabilit6s de difussion dans la th6orie de Dirac . . . furent soumises au verdict de l'exp6rience dans le domaine des 6nergies relativistes, grace ~t la source naturelle de particules tr~s 6nergetiques que constituaient les rayons cosmiques" (p. 14).

121 Though I have not considered them, other factors of a more general nature might be borne in mind, such as "the sudden reentry into physics of graduate students and other researchers who after approximately four years away were anxious to make up for lost time, and the closer relationship between theory and experiment resulting from the experience of the large wartime projects such as building the bomb and developing radar for defense" (BRowN & HODDESON 1983b, p. 28). See also SCHWEBER (1984).


by then I had already finished a preliminary version of my own. I am very grateful to ROSA MARIA ROQUt~ and KEVIN HACKETT for helping with my written English. For permission to quote and use archival material I am greatly indebted to the Niels Bohr Library, Center for History of Physics, at the American Institute of Physics, New York; to the Niels Bohr Archive, Copenhagen; to the Master, Fellows and Scholars of St. John's College, Cambridge; to the Churchill Archives Centre, Churchill College, Cam- bridge; and to the Hoover Institution Archives, Stanford University, Stanford.

This work was partly supported by the Spanish DGICYT under Research Program no. PS88-0020, and by a grant-in-aid from the Friends of the Center for History of Physics, American Institute of Physics.

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Seminari d'Hist6ria de les Ciencies Universitat Aut6noma de Barcelona

E-08193 Betlaterra (Spain)

(Received December 29, 1991)

Møller scattering: a neglected application of early quantum electrodynamics

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