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/dy)2]dxdy, a minimum. According to Dirichlet's principle the existence of the minimum should guarantee the solution of the boundary value problem for the Laplace equation (d2
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was called 'Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen — Mit besonderer Beriicksichtigung der Anwendungsgebiete' ('The Fundamental Topics of Mathematical Sciences in Monographs — With Special Reference to Applications'), appeared in 1921: it was Wilhelm Blaschke's Vorlesungen iiber Differentialgeometrie (Lectures on Differential Geometry, including the geometrical foundations of relativity theory). Three years later the series had already run up to volume seventeen, which was a German translation of Edmund Taylor Whittaker's Treatise on Analytical Dynamics (in the second edition), and included, as No. 12, the first volume of Die Methoden der mathematischen Physik (The Methods of Mathematical Physics), which Courant had written on the basis of the notes of Hilbert's lectures on mathematical topics that played a role in physics (Courant and Hilbert, 1924). 63 In the early years of his professorship in Gottingen, Courant also started a very ambitious project: he thought of establishing a mathematics institute — similar to those of experimental sciences — which would house all existing and future mathematical activities including the Reading Room. And he immediately introduced a new activity, the Anfdngerspraktikum (a mathematical laboratory, similar to the ones existing in physics and chemistry), where the beginning students obtained the practical opportunity of solving problems to reinforce the knowledge gained in the lectures. After several years of thinking and planning, the prospects of establishing the mathematical institute improved: with the help of Harald Bohr, Courant approached the International Education Board for financial support, and by the end of 1926 a sum of $350,000 was pledged for constructing the building and equipping it (see Reid, 1976, p. 108). Three years later the new institute was completed. Among the distinguished speakers at the dedication on 2 December 1929 were such former Gottingen luminaries as Hermann Weyl and Theodore von Karman. Hilbert, happy in the knowledge that his late colleague Felix Klein's dream had been realized, remarked: 'There will never be another institute like this! For to Since January 1913 Springer Verlag had been publishing Die Naturwissenschaften, the weekly journal for science, medicine and technology, edited by Arnold Berliner (first in collaboration with Curt Thesing, then from 1914 with August Putter and, after 1922, alone), which later (in 1924) became the official journal of the Gesellschaft Deutscher Naturforscher und Arzte and of the Kaiser Wilhelm-Gesellschaft zur Forderung der Wissenschaften. After 1922 the journal was supplemented by a series of annual volumes, entitled 'Ergebnisse der exakten Wissenschaften,' which was also edited by Berliner and published by the Springer Verlag. Courant got to know Ferdinand Springer through Berliner. 63 C o u r a n t ' s series was continued in the following decades; thus John von Neumann's Mathematische Grundlagen der Quantenmechanik appeared as No. 38 of the series (von Neumann, 1932). In 1925 the Gottingen physicists Max Born and James Pranck started, again with the Springer Verlag, a similar series in physics, entitled 'Struktur der Materie in Einzeldarstellungen'; the first few volumes of this series were: Ernst Back and Alfred Lande's Zeemaneffekt und Multipletstruktur der Spektrallinien (Back and Lande, 1925), Max B o m ' s Vorlesungen iiber Atommechanik (Born, 1925), and James Franck and Pascual Jordan's Anregung von Quantenspriingen durch Stofie (Franck and Jordan, 1926), respectively. In 1918 Ferdinand Springer had also started a new mathematical journal, the Mathematische Zeitschrift (which, in the beginning, was edited by Leon Lichtenstein). Many of Courant's papers were published in this journal.
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have another such institute, there would have to be another Courant — and there can never be another Courant' (quoted in Reid, 1976, p. 126). 64 Courant's efforts were directed towards institutionalizing the great tradition of Gottingen mathematics, and passing on to future generations the experiences of such outstanding mathematicians as Felix Klein and David Hilbert, who — at least partially — had also withdrawn from active scientific work. He made sure that young people of outstanding talent were attracted to, and found their place in, the University of Gottingen. He also obtained for himself two paid assistantships and filled them first with Hellmuth Kneser (the son of Adolf Kneser, his former teacher at Breslau) and Carl Ludwig Siegel (who had earlier been in Gottingen, but had gone with Erich Hecke to Hamburg). Courant further won Emil Artin, who had taken his degree with Gustav Herglotz in Leipzig, as a member of the Mathematisches Institut.65 Although the plan to add Hermann Weyl in 1922 among the ranks of mathematics professors in Gottingen failed, because of Weyl's refusal to leave Zurich, the prewar reputation of the University of Gottingen as a great center of mathematical learning was soon re-established. Again students and postdoctoral fellows came to Gottingen in large numbers, both from Germany and abroad: they included Kurt Otto Friedrichs, Otto Neugebauer and Hans Lewy, who came in 1922, then Bartel Leendert van der Waerden from Amsterdam and the Russian Pavel Sergeevic Alexandroff. Later, young mathematicians came with fellowships of the International Education Board or the Rockefeller Foundation: e.g., the M.I.T. mathematician Norbert Wiener and the young John von Neumann. And, of course, from the Praktikum there emerged mathematicians who made great reputations in the future, such as Willy Feller (who had come without any formal education from Yugoslavia) and Franz Rellich, who came from Austria. The great success which he had achieved in the tasks of teaching and administration also stimulated Courant's scientific work. After the war he had resumed his work with a long paper on the eigenvalues of partial differential equations, published in Mathematische Zeitschrift (Courant, 1920).66 During the 1920s he invested much work in writing books, especially the Courant-Hilbert, in which many original contributions of his own were contained. Besides, together with his collaborators Friedrichs and Lewy, he worked on the application of the difference calculus to solve elliptic and hyperbolic differential equations. Altogether he wrote a series of good In 1933, after the National Socialists came to power in Germany, Courant was removed from his professorship; in the fall of t h a t year he went to England and finally to the United States. He accepted a professorship of mathematics at New York University, which he occupied from 1934 to 1958. At New York University he built a new institute of mathematics, comparable to the one in Gottingen. Courant died on 27 January 1972 in New Rochelle, New York. In 1922 the philosophical faculty of the University of Gottingen split into a faculty of natural sciences and a philosophical faculty; at that time Courant obtained the permission to use officially the name 'Mathematisches Institut.' 66 C o u r a n t developed a method of obtaining the eigenvalues of the differential equations as extremum values of certain integrals associated with the differential equations, analogous to the situation in Dirichlet's principle.
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and important papers, but there was no special single field in which he was the outstanding expert. Still, with all his activities, he represented the heart of the Mathematisches Institut, where, in 1925, Gustav Herglotz succeeded Carl Runge (who retired in 1924 and died in 1927).67 And, when Hilbert retired from his chair in 1930 and Hermann Weyl succeeded him, a bright future seemed to lie ahead of Gottingen mathematics. More than anyone else, Weyl — who had worked on and contributed to deep and fundamental problems, such as the theory of Riemannian sheets, the continuum hypothesis, the unification of gravitation and electromagnetism, the foundations of mathematics, and group theory and its application to physics — had to be considered as the heir to the mathematical tradition initiated by Gauss and Riemann at Gottingen. 68 Undoubtedly, the mathematical tradition of Gottingen had continued to nourish in the post-World War I years. But its glory was equalled, if not surpassed, by the simultaneous development of physics. This might be considered surprising, for the physicists after Wilhelm Weber could not compete in reputation with their mathematical collegues. In 1900 Eduard Riecke and Woldemar Voigt were, respectively, fifty-five and fifty years old. They had sought to uphold and extend what they had learned from their teachers: thus Riecke had made an original contribution to the electron theory of metals (in the spirit of Weber), and Voigt had investigated primarily the properties of crystals (in the tradition of Franz Neumann). Voigt had also performed pioneering work on the Zeeman effect. But both Riecke and Voigt seemed to be concerned too much with the past and did not play a leading role in preparing for the physics of the twentieth century. However, at their institutes, they supported men like Max Abraham, Johannes Stark and Walther Ritz, who worked at the forefront of research on electron theory and atomic physics. On the other hand, important impulses for the development of the new physics were given by the Institute of Physical Chemistry, which Walther Nernst had directed 67 G u s t a v Herglotz was born on 2 February 1881 at Wallern in Bohemian Forest. He studied at the Universities of Vienna (1899-1900) and Munich (1900-1902) and received his doctorate in astronomy under Hugo von Seeliger in 1902. Then he went to Gottingen and became a Privatdozent in mathematics and astronomy in 1904. Three years later he was promoted to an extraordinary professorship in astronomy. In 1908 Herglotz was called to the Technische Hochschule, Vienna, as an extraordinary professor of mathematics, and he moved to the University of Leipzig the following year as a full professor. He occupied the chair of applied mathematics at t h e University of Gottingen from 1925 to 1947. He died on 22 March 1953 at Gottingen. Herglotz contributed to many problems of theoretical astronomy (e.g., the three-body problem), theoretical physics (e.g., electron theory, mechanics of continua, gravitation theory), and mathematics (power series, hypergeometric and spherical functions, differential geometry, and analytical and algebraic number theory). 68 H e r m a n n Weyl was born on 9 November 1885 at Elmshorn, Schleswig-Holstein in Prussia. He studied mathematics at the Universities of Munich (1905-1906) and Gottingen (1906-1908); he received his doctorate in 1908 under Hilbert with a thesis on singular integral equations. In 1910 he became a Privatdozent at the University of Gottingen and, just three years later, he was called to the E.T.H. in Zurich as a professor of mathematics, a position which he held until 1930 — when he succeeded Hilbert in Gottingen. He left Gottingen in spring 1933 and in the fall of t h a t year assumed a professorship at the Institute for Advanced Study, Princeton, New Jersey. He retired in 1951 and died on 8 December 1955 in Zurich.
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since 1894 and where he had performed the investigations leading to his heat theorem. After Nernst left Gottingen for Berlin in 1905, the Physico-Chemical Institute provided less stimulus, but now the mathematician Hermann Minkowski filled the gap: he contributed to the progress of physics by his fundamental work on electrodynamics and relativity theory. Then, following Minkowski's unexpected and untimely death, David Hilbert became deeply interested in the problems of physics. He not only did research on selected problems of physics, but also gave lectures on them: for example, he lectured on the molecular theory of heat in the winter semester 1912-1913 and on the theory of electrons in the summer semester 1913. Hilbert's enthusiasm for physics infected all his mathematical colleagues except Edmund Landau. Indeed, it became fashionable among mathematicians to work on problems of atomic physics and radiation theory. 69 In addition, the Wolfskehl lectures acquainted many people in Gottingen with the foremost topics of physical research. Peter Debye, one of the lecturers at the Kinetische GasKongress in April 1913, impressed the Gottingen mathematicians and physicists so much that in summer 1914 he was invited as Professor of Physics to the University of Gottingen. To make Debye's appointment possible, Woldemar Voigt resigned from his chair, retaining the personal title of a professor. Debye immediately began to lecture on various aspects of modern physics: for example, in the winter semester 1914-1915 he gave a course on quantum theory, which he repeated regularly in the following years, and in the summer semester 1915 he lectured on the kinetic theory of dielectric phenomena. Besides, he conducted, together with Hilbert, a Seminar on 'The Structure of Matter' (' Seminar uber die Struktur der Materie,)> which was a continuation of the earlier Hilbert-Minkowski seminars. Debye himself worked, both experimentally and theoretically, on the problems of the structure of matter: in particular, he concerned himself with the interference of X-rays by crystals (Debye, 1915b) and developed, together with Paul Scherrer, a new method — which came to be called the 'Debye-Scherrer' method — for analyzing the structure of crystals (Debye and Scherrer, 1916a,b). This method introduced a definite advantage over the existing methods of research in crystal structures, such as the ones of Laue, Friedrich and Knipping, and of the Braggs, because the substances could be used as powders rather than as good crystals. Thus Debye and Scherrer were able to study the structures of many more materials than had been investigated before. After only six years in Gottingen — though six most successful years — Debye accepted in spring 1920 the call to become Professor of Experimental Physics and Director of the Physics Laboratory at the E.T.H. in Zurich. Hilbert even assigned problems of theoretical physics to his doctoral students. Thus Ludwig Foppl, son of August F6ppl (the professor of technical mechanics at the Technische Hochschule, Munich), worked on the stability of electron configurations in atoms for his thesis (Foppl, 1912); Hans Bolza (1913) and Bernhard Baule (1914) investigated the theory of dilute gases; and Kurt Schellenberg (1915) studied electrolysis. At the same time Hermann Weyl also published his first papers on physical problems, especially the problems of radiation (Weyl, 1912a,b,c).
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In early 1920 a new situation developed with regard to the physics professorships at the University of Gottingen. Riecke and Voigt were no more, and Debye was about to leave for Zurich. Eduard Riecke had died already on 11 June 1915 and Woldemar Voigt on 13 December 1919. During wartime Riecke's position had not been filled; instead Robert Wichard Pohl, a Privatdozent from Berlin, had been appointed in 1917 as an extraordinary professor.70 In summer 1920 Pohl, who had become known for his work on the photoelectric effect, succeeded to Riecke's chair and began a new field of investigation: the photoelectric action in solids. 71 Later he turned to the study of the processes of absorption of light by crystals (especially of alkali halogenides), of luminescence, and of photochemistry. Pohl founded his own school of devoted collaborators and followers in Gottingen (who were called the 'Pohlierten' — the 'Pohlians'); he also introduced a large number of students to physics through his lectures and demonstrations and successful textbooks on experimental physics. In this respect, Robert Pohl continued the tradition of Erxleben, Lichtenberg and Riecke. Although Pohl and his associates at Gottingen worked experimentally on effects connected with quantum theory, their field was not in the center of interest in those years. As Pohl recalled later: 'We were altogether a bit isolated with our investigations. At that time there were these wonderful results of Bohr, and all that could be explained on Bohr's theory was particularly interesting' (Pohl, AHQP Interview, 25 June 1967, p. 7). 72 He had in mind, in particular, the experiments of his Gottingen colleague James Franck, who was appointed professor of physics and one of the two successors of Peter Debye in 1920. Franck owed his appointment to the other (real) successor, Max Born. Born, who had been selected first for the Gottingen professorship, recalled later that he had hesitated initially to accept the position because he lacked the knowledge to run a big experimental laboratory and to give lectures on experimental topics in physics. While discussing matters in Berlin with the appropriate official for universities in the Prussian Ministry of Education, a Director Wende, he recognized that on the document presented to him, besides Debye's chair, two extraordinary professorships were listed — connected with the names of Pohl and Voigt, respectively. Now Voigt had retired in 1914 in favour of Debye, but had retained a 'personal' professorship, which was supposed to expire upon his death. However, due to a clerical error, the notation 'to be cancelled after the death of the occupant' stood after Pohl's name instead. 'I am in general not very quick to grasp a situation, but in this case I did,' Born wrote in his Recollections. 'So 70 R . W . Pohl was born on 10 August 1884 at Hamm, Westphalia. He studied at the Universities of Heidelberg (1903-1904, under Georg Quincke) and Berlin (1904-1906), obtaining his doctorate under Warburg in 1906. He became a Privatdozent at the University of Berlin in 1911 and did his military service from 1916 to 1918. '1 Before Pohl was promoted to a full professorship Max Wien, the experimental physicist at the University of Jena, had received an offer from Gottingen (in late 1919), but he had declined. 72 P o h l ' s work was appreciated later on when semiconductors came to be used for technical purposes. Pohl retired in 1952 from his Gottingen chair. He lived to the age of ninety-two and died on 5 June 1976.
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I quickly pointed out to Director Wende that there was another vacancy, and that this changed the whole situation. For if they could appoint a second experimentalist to be the head of Voigt's former department I would not hesitate to accept the post, considering the directorship of the two laboratories as purely formal. Wende laughed, saying that this was obviously an error of the copyist; but he very soon saw the possibilities. You must understand we were still living in revolutionary times. Like most of the officials in the Ministry of Education, he was a new man, keen to do new things. One of his aspirations was to stimulate science and in particular to foster the old scientific center of Gottingen. He therefore accepted my proposal and asked me to nominate a candidate for the second Extraordinariat' (Born, 1978, p. 200). 73 And, after some thought, he decided to propose his friend James Franck for the second professorship, because 'The experiments which he, in collaboration with [Gustav] Hertz, had done to demonstrate Bohr's quantum theory of atoms I regarded as most important and fundamental. I had known Franck since my student days and loved him as a most honest, reliable and good-humoured fellow. I also knew that he and Pohl had been at the same school in Hamburg and were great pals, which guaranteed a frictionless collaboration between the two experimental departments' (Born, 1978, p. 200). In Gottingen the news about the appointment of the two new physics professors was very welcome. 'Franck plus Born are the best imaginable replacement for Debye,' Hilbert wrote to Courant. 'I am very happy about this arrangement' (see Reid, 1976, p. 82). Hilbert could indeed be very pleased with Born, who had not only made a considerable reputation as a theoretical physicist but was particularly suitable to represent the Gottingen tradition. Max Born had descended from an academic family: his grandfather, Marcus Born, had been the first Jewish physician to be appointed as the district medical officer in Prussia and his father, Gustav Born, was professor of anatomy at the University of Breslau. 74 Max was born in Breslau, the capital of the province of Silesia, on 11 December 1882.75 He grew up in the well-educated atmosphere of science and culture — among his father's friends, for instance, was Paul Ehrlich, the founder of chemotherapy. He attended primary and secondary school in his hometown. 76 '3According to another recollection of Born (in his commentary to the Born-Einstein letters), Wende did not take the responsibility; he consulted his superior, the minister -— Professor Becker — who agreed with Born's request. (See The Born-Einstein Letters, 1971, pp. 26-27.) 74 M a x Born's mother, Margarethe Kaufmann, came from a Breslau family active in the textile business. She loved music and knew some of the celebrated musicians of her time, including Johannes Brahms, Clara Schumann and Pablo Sarasate. Unfortunately she died already in 1886 when Born was four years old. Four years later Max and his sister Kathe had a stepmother, when their father married Bertha Lipstein from a well-to-do Jewish family originating from Russia. 75 Although many Polish Jews came to Breslau to avoid the pogroms, the Borns had been settled in Germany much earlier. The family of Born's father originated from Gorlitz, Lower Silesia, while his mother's family came from Silesia. Other Jewish scientists, who were in Breslau at the same time as Max Born — such as Oskar Minkowski (Hermann Minkowski's brother, later a famous pathologist), O t t o Toeplitz and Ernst Hellinger — were mostly of Eastern European origin. 76 A t the secondary school, an average German Gymnasium, Born studied Latin, Greek and mathematics. His mathematics teacher, Maschke, also taught chemistry and physics. He was a clever experimenter, for as Born recalled: 'At the time Marconi's experiments on wireless communication
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In spring 1901 Born entered the University of Breslau and began to attend lectures in almost everything that came along: e.g., physics, chemistry, zoology, philosophy, logic, mathematics and astronomy. 77 In particular he was attracted to astronomy: he took several courses in it and even learned to handle the astronomical instruments at the old-fashioned Breslau Observatory which stemmed partly from Bessel's time. Since, however, he disliked the endless numerical calculations that were necessary in astronomy, he soon gave up the plan of becoming an astronomer and began to concentrate on mathematics. 78 From among the Breslau professors he would always remember Jakob Rosanes; Rosanes' lectures on linear algebra included an introduction to matrix calculus. 79 In the mathematics courses Born met and became friends with Otto Toeplitz and Ernst Hellinger. 'As a student in Breslau,' Born wrote later, 'I was much under the influence of Toeplitz, who was my senior by one year, and though my interest in algebra was not great he insisted in my learning matrix calculus properly and occasionally refreshed my knowledge when we were together again as young teachers in Gottingen' (Born, 1940, p. 617). 80 He also got to know Clemens Schaefer, a young Privatdozent, who gave courses on Maxwell's electrodynamics. 81 became known, Maschke repeated them in his little lab with me and another boy as assistants' (Born, 1968, p. 16). The University of Breslau was founded in 1811 to replace the one in Prankfurt-an-der-Oder. Formerly there existed a Jesuit college in Breslau, which had a beautiful Baroque building. 78 Physics at Breslau was represented by the old Oskar Emil Meyer (1834-1909), brother of the chemist Julius Lothar Meyer, who had propagated kinetic theory. He did not give very inspiring lectures and was soon replaced — because of illness — by the young mathematician Ernst Neumann (the grandson of the famous Franz Neumann). Chemistry did not attract Born either, for it seemed to consist just of memorizing facts. 79 J a k o b Rosanes was born at Brody, Austria-Hungary, on 16 August 1842. He studied mathematics at the Universities of Berlin and Breslau, obtaining his doctorate in 1865. He became a Privatdozent at Breslau in 1870, extraordinary professor in 1873 and full professor in 1876. Rosanes became the rector of his university for the year 1903-1904, retired in 1911 and died at Breslau on 6 January 1922. He worked on various problems of algebraic geometry and invariant theory; he also wrote a book on chess. 80 Otto Toeplitz was born on 1 August 1881 in Breslau. He studied mathematics at t h e Universities of Breslau and Berlin, and received his Habilitation in 1907 at the University of Gottingen. In 1911 he was appointed extraordinary professor of mathematics at the University of Kiel (promoted to full professorship in 1920). Toeplitz moved to the University of Bonn in 1928 as professor of mathematics and remained there until 1935. Then he was without a position. In 1939 he went to Palestine as scientific advisor to the administration of Hebrew University. He died in Jerusalem on 15 February 1940. Influenced by Hubert's treatment of integral equations, Toeplitz worked in particular on the algebra of systems of infinitely many linear equations. Later he also wrote on the history of mathematics. Clemens Schaefer was born on 24 March 1878 in Remscheid, in the Ruhr region. He received his education at the Universities of Bonn (1897-1898) and Berlin (1898-1900). After taking his doctorate in 1900 at the University of Bonn with a thesis supervised by Emil Warburg, he served as an assistant to Heinrich Rubens. In 1903 he became a Privatdozent at the University of Breslau; seven years later he was promoted to an extraordinary and in 1917 to a full professorship. After an interlude at the University of Marburg (1920-1926), he returned to Breslau as Director of the Physics Institute (1926-1945). After World War II he went to West Germany and joined the University of Cologne. He retired in 1950 and died in Cologne on 9 July 1968. Schaefer contributed to the problems of infrared rays, the structure of molecules and crystals, and acoustics. He wrote textbooks both on experimental and theoretical physics.
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In those days students at German universities used to move often from one place to another, either 'attracted by a celebrated professor or a well-equipped laboratory, in other cases by the amenities and beauties of a city, by its museums, concerts, theatres, or by winter sport, by carnival and gay life in general' (Born, 1955a, p. 43). Thus Born spent the summer semester 1902 at Heidelberg and the summer semester 1903 at Zurich. In Heidelberg he met James Pranck, who became one of his lifelong friends.82 In Heidelberg, Born learned the most from Leo Konigsberger, who dwelt more on the fundamental ideas than the pursuit of rigour in teaching mathematics, while at Zurich he attended Adolf Hurwitz' course on elliptic functions. 83 He used to return regularly to Breslau for the winter semesters, where he would meet Toeplitz and Hellinger again; from them he 'learned that the mecca of German mathematics was Gottingen and three prophets lived there: Felix Klein, David Hilbert and Hermann Minkowski' (Born, 1968, p. 18). So he decided to follow his friends and make the pilgrimage to Gottingen in summer 1904. Soon after his arrival there he came into close contact with Hilbert, who asked him to work out a manuscript of his lectures on function theory for the Lesezimmer. He was also introduced to Minkowski through a letter from his stepmother, who knew the mathematician from Konigsberg. However, the relations with Felix Klein did not develop so smoothly. Born did not attend Klein's lectures regularly, but he participated in the joint Seminar of Klein and Runge on the theory of elasticity in the winter semester 1904-1905. He gave a talk on the problem of stability in elastic media, which was well received. 'Klein was very favourably impressed by it and wrote me a letter that I should do this as a prize paper for the faculty,' Born recalled later. 'This was very rare and a great honour, but I declined it' (Born, Conversations with Mehra). 84 Bom's refusal offended the almighty Klein; he fell into disgrace with him, although eventually he sat down to work on the problem. He submitted the essay, entitled 'Researches on the Stability of the Elastic Line in the Plane and in Space under Various Boundary Conditions,' just as Klein had wished; it received the prize of the Gottingen faculty and was accepted as a doctoral dissertation (Born, 1906).85 8 Franck recalled later: 'At that time I met in the lecture hall [at Heidelberg] the mathematician Max Born, and we became quick friends. Born got a visit from his sister, and I must confess, that his sister contributed considerably to my pleasure at Heidelberg' (Franck, AHQP Interview, First Session p. 13). Konigsberger, born on 15 August 1837 at Posen, West Prussia, was professor of mathematics variously at Greifswald, Heidelberg, Dresden, Vienna, and again Heidelberg. He worked on elliptic functions and differential equations, and wrote a two-volume biography of Hermann von Helmholtz. He died in Heidelberg on 15 December 1921. 84 Originally Born had not intended to give a talk in the Seminar himself. But it happened that, shortly before the due date, he found that he had to replace another speaker. As he recalled: 'I had only a few days to prepare myself. And I saw that there was such a lot of literature — impossible to read in a few days — so I thought I would better do it myself.... I had learned quite recently the theory of variations, so I worked out a new method to determine, with the help of mathematical criteria of minima, the stability of such lines [i.e., of the so-called elastic lines in continuous media]' (Born, Conversations with Mehra). 85 Born had earlier obtained from Hilbert a subject for his doctoral thesis; it was the determination of the roots of Bessel functions, but he made no progress on it. B o m ' s thesis on the
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The difficult personal situation with Klein drove Born into physics. 86 After four months of military service in winter 1906-1907 — Born was soon released, for he suffered from asthma, which had plagued him since his early youth — he left in April 1907 for England and the University of Cambridge in order to learn more about physics. There were two reasons why he chose to go to Cambridge: first, it was regarded in those days as a great center of experimental research; second, Joseph John Thomson, the discoverer of the electron lived and taught there, and electron physics had always aroused a great interest in Gottingen. 87 Born was accepted as an advanced student of Gonville and Caius College. He attended the lectures of Joseph Larmor on electrodynamics and J.J. Thomson's experimental demonstrations; while he found Larmor's lectures old-fashioned and his Irish accent all but incomprehensible, he was very impressed by Thomson. He even took part in an experimental course at the Cavendish Laboratory. Thomson, at that time, also lectured on his model of atomic structure (Thomson, 1904a), and Born would later choose it as the subject of his trial lecture (Probevorlesung) for the Habilitation (Born, 1909d). In England, Born made another acquisition: he bought the two volumes of J. Willard Gibbs' scientific papers, which had just appeared (Gibbs, 1906). 'I bought these books by Gibbs just by chance in a shop,' he recalled, 'and I read them and I was fascinated by them, and I studied them every day in Cambridge' (Born, Conversations with Mehra). Gibbs' works helped to turn him into a physicist; when he returned to Breslau in summer 1907, he went straight to the physics institute rather than to the problem of physics was accepted by the faculty, hence he was not listed officially as having obtained his doctorate under Hilbert. But Hilbert did examine him in mathematics, and Born remembered the following story about it. He had asked Hilbert in advance about the topic on which he would examine him. 'In what area do you feel yourself most poorly prepared,' Hilbert asked. 'Ideal theory,' replied Born. Hilbert said nothing more, and Born assumed that he would be asked no questions in that area. However, when the day of the examination arrived, all of Hilbert's questions were on the theory of ideals. 'Ja, ja,' Hilbert said afterwards, 'I was just interested to find out what you know about things about which you know nothing' (Reid, 1970, p. 105). Since Born wanted to avoid being examined by Felix Klein, he chose astronomy as one of his minors. He had attended Karl Schwarzschild's Seminar on the atmosphere of planets, in which he also learned kinetic theory. 86 I n Gottingen Born attended several courses on physics. He liked Woldemar Voigt's lectures on optics and recalled: 'We made all the elementary optical experiments on diffraction, interference, etc., with our own hands. To every experiment belonged a sheet, where the experiment was explained, and one had to fill out lines about what one was doing and then to write down the measurements and so on. And it was awfully well prepared and I learned a lot there' (Born, Conversations with Mehra). Voigt's course, together with Schwarzschild's lectures on optical instruments provided Born with a thorough knowledge of optics, which he used later when he wrote a book on t h a t subject. On t h e other hand, Born did not like Johannes Stark's course on radioactivity. As he explained later: 'It was all so dogmatic, no proofs which I considered to be necessary, and I left it after a few weeks. And this is the reason I never have done nuclear physics at all. One has to learn it young, otherwise one cannot do it' (Born, Conversations with Mehra). 87 T h e first to be interested in electrons had been Wilhelm Weber. Later Wiechert, Riecke, Voigt, Abraham and Runge had written papers dealing with electron theory. The subject of electron theory had also been treated in the Hilbert-Minkowski Seminar in summer 1905, which Born had attended.
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mathematicians. 88 There he met Rudolf Ladenburg, who had just returned from Cambridge himself. Born found that, 'He was really a physicist, and there [at the physics institute] I learned first the real problems of physics, like the optical coherence of waves and such things that I had never heard before, because Voigt was quite formal' (Born, Conversations with Mehra). In Breslau, besides Clemens Schaefer and Rudolf Ladenburg, Erich Waetzmann was another Privatdozent (he became one in 1907), and in 1908 Fritz Reiche joined as an assistant after completing his doctorate under Planck. Lummer and Pringsheim continued their work on heat radiation also in Breslau, and they gave lectures on it. Born, who learned about this subject for the first time, decided to get involved himself. He went to Lummer's Institute, where the professor gave him 'what was called then a black body — a tube of porcelain material with a heating arrangement and a table with a gas fire and a cooling mantle of water around' (Born, Conversations with Mehra). But, not being 'very gifted for this type of work,' Born just caused a flooding, upon which he gave up the experiments. Instead, together with Ladenburg, Waetzmann and Reiche, he started reading papers on new and interesting subjects. 'And there one day,' Born recalled later, 'we came across the relativity paper of Einstein [1905d]. Reiche knew it already from Planck, and he told us we must read it; and there I became excited, I found this marvellous' (Born, Conversations with Mehra). 89 Born also studied Einstein's other papers on relativity and tried to connect their content with what he had learned in Minkowski's Seminar in 1905. Since many questions arose from this study, he wrote to Minkowski in Gottingen asking him for help. Instead of answering these questions, however, Minkowski invited Born to return to Gottingen and become his assistant and collaborator in relativity theory. This response made Born extremely happy. He met Minkowski again at the Naturforscherversammlung in Cologne in September 1908 and listened to his lecture on 'Space and Time' (Minkowski, 1909). There he decided to accept his offer. After arriving in Gottingen in early December 1908 he saw Minkowski nearly every day and discussed with him the problems of electron theory. He soon worked on a paper, which he completed during the Christmas vacation (Born, 1909a). 90 The unexpected death of Minkowski on 12 January 1909 put an end to a promising collaboration, but it did not ruin Born's career. In summer 1909 — with the help of Hilbert, Runge and Voigt — he became a Privatdozent in physics at the University of Gottingen, and in September he spoke at the 81st Naturforscherversammlung in Salzburg on the dynamics of the electron with Einstein in the audience (Born, 88 I n 1907 Born published his second paper on variational principles in thermodynamics together with Erich Oettinger of Karlsruhe (Born and Oettinger, 1907). In this paper Born cited the papers of Lorentz and Gibbs and generally demonstrated his knowledge of the literature (Born's first paper on variational principles was his thesis: Born, 1906.) 89 I n fact Ladenburg became acquainted with Einstein personally in 1908; he had been sent by Planck to Berne to invite Einstein as a speaker for the next (i.e., 81st) Naturforscherversammlung in Salzburg. 'His [Ladenburg's] account was the first I heard of Einstein the man,' Born recalled. 'Ladenburg was enthusiastic and made us curious of the great unknown' (The Born-Einstein Letters, 1971, p. 1). 90 I n his first paper on relativity theory, Born formulated an action principle. He had spoken on this problem already in the Hilbert-Minkowski Seminar in 1905.
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1909c).91 As Minkowski's last collaborator to work on relativity theory he was charged with editing his unpublished papers on that subject. 92 But he wanted to do more, namely, to extend Minkowski's work. Therefore, in his Habilitation thesis, he discussed the concept of a rigid body in relativistic kinematics and dynamics, a concept which the founders of relativity theory had always used in some sense without giving it a proper definition (Born, 1909b). 93 Although Born's definition of the rigid body in relativity theory encountered certain objections and a controversy arose in the following year — in which Max von Laue and Fritz Noether (Emmy Noether's brother) took part — Gustav Herglotz used it successfully in developing his relativistic mechanics of deformable bodies (Herglotz, 1911). 94 Born's scientific work at Gottingen between 1909 and 1911 grew more or less from the ideas which Minkowski had expressed in his long memoir on the electrodynamics of moving bodies. Thus he established for himself a reputation as an expert on relativity and electron theory, and as such he was invited by Albert A. Michelson — who spent the summer semester 1911 in Gottingen — to visit Chicago in 1912 to give a series of lectures on relativity theory. However, in spite of the good beginning, Born made only slow progress in the problems he treated. One reason may have been the fact that he employed somewhat clumsy mathematical tools; for instance, following Minkowski, he tried to formulate problems with the help of matrices, which were soon replaced by the more elegant concepts of 'four-' and 'sixvectors' (Sommerfeld, 1910b,c).95 In Born's furture work also elegance would often be sacrificed, especially when he sought to arrive at the most general results. Of course, the difficulties that faced the extension of relativity theory after 1910 were extremely serious, and Born lacked the ingenious vision which enabled Einstein to go forward nonetheless to create the theory of general relativity. He rather preferred to take over the physical ideas of others and to formulate them with the help of the rigorous mathematical methods he had learned. When he later saw Einstein's completed theory of general relativity, he gave up working on relativity and electron theory for nearly twenty years. 96 For his Habilitation thesis, Born submitted an extended version of his earlier work on the relativistic electron (Born, 1909b). On Hilbert's advice, Minkowski's widow requested Born to examine and organize Minkowski's manuscripts on physical problems. On the basis of some surviving notes of Minkowski and his previous conversations with him, Born constructed a paper, which was subsequently published in Annalen der Physik (Minkowski, 1910). It is of interest to note t h a t another paper based on Minkowski's surviving manuscripts appeared in 1915; Sommerfeld then edited Minkowski's lecture on relativity theory (Minkowski, 1915). 93 B o r n noticed that the definition of the rigid body presented no difficulties, except when the motion was accelerated. However, when the accelerated body moved in a straight line, a rigid body could still be defined as the one which obeyed the simple dynamical laws of a mass point. Born tried to formulate a relativistic mechanics of the continua himself. In a paper submitted in May 1911 to Physikalische Zeitschrift he introduced the first step (Born, 1911). However, before making further progress he came to know that Herglotz had already done the job. ^ 5 Born continued to use matrix methods later on — as, for example, in his paper dealing with energy momentum conservation in Mie's electrodynamics (Born, 1914a). 96 I n 1916 Born published a pedagogical review article on Einstein's theory of general relativity in the Physikalische Zeitschrift (Born, 1916a), which prompted Einstein to write to him: 'This
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Unlike relativity theory, the other great theoretical conception of t h e early twentieth c e n t u r y — t h e q u a n t u m theory — h a d not yet m a d e a great impact in G o t t i n g e n . 9 7 Born, who h a d learned a b o u t Planck's theory in Breslau, lectured in t h e s u m m e r semester 1911 on h e a t radiation. B u t this did not s t i m u l a t e him t o become personally involved in q u a n t u m theory. A n impulse h a d to come from outside, and it did in the following manner. At Gottingen, Born lived, until h e married in August 1913, in a boarding house on Nikolausberger Weg, and there (in 1912) he m e t Theodore von K a r m a n , t h e assistant of Ludwig P r a n d t l . T h e y soon became friends and s t a r t e d to discuss all new problems in physics and mechanics. And one day they c a m e across Einstein's p a p e r on the specific heat of solids (Einstein, 1906g). As von K a r m a n recalled: I brought Einstein's paper to Born and we both studied it. Unfortunately experiments with different substances soon showed that Einstein's formula [with just one characteristic frequency] was limited. It worked only for higher temperatures, but not for the lowest temperatures. We wondered why. And we finally agreed that the discrepancy was due to the fact that Einstein's approach while basically correct was too simple. (Karman and Edson, 1967, p. 67) 9 8 In order to improve on Einstein's theory, B o r n and von K a r m a n thought a b o u t the s t r u c t u r e of crystals, which they imagined as a three-dimensional lattice of coupled a t o m s . Now crystal lattices were a rather well-known scientific topic in Gottingen, especially a r o u n d Woldemar Voigt, who however imagined crystals usually as continuous media w i t h certain s y m m e t r y properties. ' B u t he mentioned t h a t it [the crystal] could b e reduced to a crystal lattice,' Born recalled, 'and when I s t a r t e d m y work with K a r m a n , we took it for granted t h a t t h e r e are real lattices' (Born, Conversations w i t h Mehra). T h e two friends felt t h a t they were on safe ground w i t h this assumption, for Erwin Madelung had already used molecular lattices two years earlier in order to explain the observed discrete infrared radiation orginating from diatomic crystals (Madelung, 1909; 1910a,b). By investigating systematically t h e motion of a t o m s in three-dimensional crystal lattices, they obtained t h e characteristic s p e c t r u m of t h e crystals and arrived at a satisfactory explanation of t h e morning I received the corrected proofs of your paper for the Physikalische Zeitschrift, which I read with a certain embarrassment but at the same time with a feeling of happiness at being completely understood by one of the best of my colleagues. But, quite apart from the material contents, it was the spirit of positive benevolence radiating from the paper which delighted me — it is a sentiment which all too rarely flourishes in its pure form under the cold light of the scholar's lamp' (Einstein to Born, 27 February 1916). There started a friendship between Einstein and Born, which lasted until Einstein's death. Later Born occasionally lectured on general relativity theory to the general public (see Born, 1920d). After 1933 Born worked again on electron theory and published papers on a nonlinear generalization of Maxwell's electrodynamics. (See, e.g., Born, 1934; Born and lnfeld, 1934.) 9 During the summer semester 1907 Minkowski gave a course on heat radiation, but we do not know whether he also presented Planck's theory in detail. Max Abraham, Planck's former student, had announced a course on heat radiation for the winter semester 1910-1911, but before giving it he went to Milan. 98 The Gottingen people learned about the status of the problem of specific heats from Walther Nernst, who used to visit Gottingen frequently. (See, e.g., Courant, AHQP Interview, p. 5.)
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deviation of the empirical data from Einstein's formula of 1906 (Born and von Karman, 1912). In arriving at the results, Born made use of his great mathematical erudition; he applied especially the theory of infinite sets of algebraic equations developed by his friends Ernst Hellinger (1907) and Otto Toeplitz (1911). Before Born and von Karman published their formula for the specific heats in their second paper on crystal lattices (Born and von Karman, 1913a), they learned about a different theory, which Peter Debye had presented somewhat earlier and which led to similar conclusions (Debye, 1912b). However, they felt, in spite of Debye's priority, that their description was more appropriate to the physical situation." Crystal lattices and their dynamics became Bom's main field of interest in the following years. 100 As he remarked later: 'From a personal point of view the occupation with vibrations in crystal lattices was of great practical importance for me; it opened up for me my own field of research, the kinetic theory of solids, which — though it does not penetrate into the depths of the ultimate principles — provided a large number of problems — not exhausted even today — for special investigations' (Born, 1956, p. 97). Bom's new field of interest certainly lay close to that of Hilbert and others in Gottingen. After all, in April 1913 Hilbert organized the Kinetische Gas-Kongress, in which, for instance, Peter Debye discussed the problems of the specific heats of solids. Born started with an especially ambitious problem, namely, to calculate the properties of diamond from lattice theory (Born, 1914b). The crystal of diamond had been analyzed before by William Henry Bragg and his son (Bragg and Bragg, 1913). 101 To Born it represented a particularly simple structure because each atom in the lattice possessed only four nearest neighbours; hence he had to consider just two elastic constants if he took into account only the nearest-neighbour interaction. He solved the 24 equations of motion for the eight fundamental mass points in the diamond lattice and thus calculated the frequency spectrum and the specific heat of the substance. 102 Born also succeeded in determining the elastic constants of diamond. His theory represented a milestone in the kinetic theory of solids, 99 A t about the same time the information about the X-ray interference patterns of von Laue, Friedrich and Knipping became known in Gottingen, for Sommerfeld discussed these matters during his visit there in summer 1912. Interestingly enough von Laue's results on X-ray interference were not referred to in the papers of Born and von Karman. As Born remarked later: 'I remember that I at the same time as von Laue was thinking about using X-ray diffraction to prove the lattice structure. I worked it out theoretically and I was so busy with that and then suddenly appeared von Laue's p a p e r s . . . . So you see I took these things seriously' (Born, Conversations with Mehra). 100 T h e time around 1912 was a very happy period in B o m ' s life. Born, von Karman, Albrecht Renner (a medical student) and Hans Bolza (a student of Hilbert's) rented a house in Dahlmannstrasse and hired a nurse for housekeeping and preparing meals. The house was called 'ElBoKaReBo,' the abbreviated version of Bo[rn]Ka[rman]Re[nner]Bo[lza]. Paul Ewald and the graduate student Ella Philipson regularly came to the meals. The bachelors' idyll came to an end when Paul Ewald married Ella Philipson, and Max Born (on 2 August 1913) married Hedwig Ehrenburg, the daughter of t h e Leipzig law professor Viktor Ehrenburg. 101 Paul Ewald attended the Birmingham Meeting of the British Association in September 1913, where William Henry Bragg presented the experimental results on diamond. 102 B o r n found that for low temperatures the specific heat of diamond approached a Debye curve, while for higher temperatures it came close to an Einstein curve.
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for he had arrived, by straightforward calculation, at a complete and satisfactory description of the properties of the crystal. 103 Born was confident that with his work he was on the right track of developing 'a systematic theory of crystal lattices, which explained all simple properties of the crystals without invoking new hypotheses' (Born, 1915a, p. 391). And, being really ambitious, he decided to write a book on the theory, which he called 'The Dynamics of Crystal Lattices,' and which he completed the following year (Dynamik der Kristallgitter, Born 1915c). 104 When Born wrote the book on the dynamic crystal lattices, he already had left Gottingen and occupied, since early 1915, an extraordinary professorship at the University of Berlin. 105 He started to lecture on electrodynamics, but after some time he was called up for military service. Being one of the few younger physicists in Germany who did not take part in actual fighting, he worked first with a small group at the Doberitz Camp, close to Berlin, on the problem of wireless communication between airplanes and the ground. 106 After a short training course as an aircraft wireless operator, Born joined the Artillerie-Prufungskommision (Artillery Testing Commission), a new department founded by his friend Rudolf Ladenburg. 10
In contrast to B o m ' s treatment, Debye's — and, to a lesser extent, Einstein's — description of, say, the energy of the crystal could only be considered as being 'phenomenological.' In discussing the theory of the solid state, Born often seemed to be in competition with his Gottingen colleague Peter Debye, for he frequently pointed out where Debye's phenomenological approach failed. 104 T h e physical idea underlying Born's crystal lattices was t h a t point-like material objects — ions, atoms or electrons — occupy the lattice points in elementary cells. He assumed that between these material objects there existed forces, which could be derived from a potential and decrease very rapidly with the distance. Due to the short range of these forces, Born was able to replace — for the purpose of calculating most of the crystal properties (except the ones connected with the surface) — the finite lattice by an infinite one, which simplified the evaluation considerably. By generalizing the methods used in the specific case of diamond to arbitrarily complicated lattices, Born arrived at many important results. For example, he solved the old riddle whether, in the most general lattice, there are 21 or 15 elastic constants, in favour of the higher number. (He showed that the six Cauchy relations, which allowed the reduction to 15 constants only, came about by the explicit use of the continuum theory.) Born further studied the optical properties of crystals (Born, 1915a). Especially, he solved the equation of mass points on which were impressed periodic oscillations of wavelength A (much larger than the lattice parameter S), by expanding the amplitudes in powers of the small parameter 2n5/\. He found in the first order the Fresnel formula describing the refraction of waves in terms of the elastic constants of the crystal; by expanding to second order and restricting himself to a crystal with only one optical axis he obtained a description of double refraction. In early 1914 a second chair of theoretical physics was established at the University of Berlin in order to relieve Max Planck of his teaching burdens. It was intended to invite Max von Laue to take this chair, but the plan failed to materialize for two reasons. On one hand, World War I broke out in August 1914 and most students left the universities to do military service; on the other, von Laue was not available, for at that time the Akademie fur Sozial- und Handelswissenschaften in Prankfurt-am-Main was turned into a regular university having five faculties, including one for science, and von Laue was appointed professor of physics. Hence the plans in Berlin were changed; an extraordinary professorship was established and Born was called to it. 106 B o r n ' s experience of military service was not all that extensive. He had not been able to complete his obligatory year of military service after his doctorate because of a serious attack of asthma. It should be mentioned that in Germany during World War I there did not exist any special programs, at least in the beginning, in which scientists could do technical work for the military. This was different in Great Britain. As Born remarked: 'In Britain, I think, they were much more patriotic' (Born, Conversations with Mehra).
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The task of this department consisted in developing various methods of 'scientific ranging,' optical, acoustical, seismometric, electromagnetic, etc. 1 0 7 In Ladenburg's department he found many physicists, among them his former Gottingen colleague Erwin Madelung and Alfred Lande, who had been his student and later Hilbert's 'physics tutor.' Born helped Ladenburg in hiring more scientists, for 'It was my main interest in the war to save people from being killed' (Born, Conversations with Mehra). 108 Although military service restricted the activities of the Extraordinarius Max Born, he still enjoyed considerable freedom in his office. Since the military authorities possessed little knowledge of physics and mathematics, they could not control his work. Hence Born and Lande started, besides their work on sound ranging, an intense collaboration on some problems of crystal dynamics. Born was also allowed to attend regularly the seminars and colloquia at the University of Berlin as well as the meetings of the German Physical Society.109 In Berlin, physics at that time was very exciting — Albert Einstein, Max Planck, Walther Nernst, Fritz Haber and Heinrich Rubens being the leading personalities. Almost all of these men were deeply involved in the quantum theory and from them Born learned about its problems. Above all, he became friends with Einstein. They met frequently in Einstein's apartment in Haberlandstrasse — which was very close to Born's office in Spichernstrasse — to discuss physics and politics and to play music together. In Berlin Born continued to work on the kinetic theory of matter, especially on problems of the solid state. But he also extended his interest to the theory of liquids and gases. So, for instance, he answered Debye's question of whether he could explain the double refraction in fluids; he discovered that an optical activity would arise in case the constituents of the molecules formed asymmetric tetraeders (Born, 1915b). The theory of fluids and the dispersion of light by molecules also occupied him in the following years. 110 Thus he extended the dispersion theory of Drude and Lorentz by incorporating fluctuation phenomena — such as those treated 107 Ladenburg, whom Born had first met in 1908 at Breslau, had seen action with a cavalry regiment during the war, but had been wounded and sent back to Berlin. There he developed the idea of sound-ranging and was allowed by the military administration to establish a department and recruit appropriate people. 1 Born did not always succeed in this endeavour. Later on he recalled with unhappiness the case of one of the best students in mathematics from Gottingen, Herbert Herkner, who served in an infantry regiment: 'I tried and tried to get him back and then at last I got permission and sent a telegramme to his regiment. And the night before it arrived he was killed' (Born, Conversations with Mehra). He wrote an obituary of Herkner — a rare mark of honour for a student — which appeared in Naturwissenschaften (Born, 1918a). 10 Evidently, Born's lecturing duties during the war were not heavy; the students were simply not around. So, after the course on electrodynamics in summer 1915 was interrupted, he announced courses on the kinetic theory of matter during the following three semesters (of which he probably gave only one); in the summer semester 1917 he announced a course on thermodynamics; and in the following three semesters (until the winter semester 1918-1919, when he probably delivered the lectures) a course on the dynamics of crystal lattices. 110 B o r n treated, for example, the properties of anisotropic fluids in several publications, using ideas similar to the ones applied by Paul Langevin and Pierre Weiss in ferromagnetism and Peter Debye in his theory of polar molecules (Born, 1916b; 1918b).
446 The Golden Age of Theoretical Physics by Marian von Smoluchowski in his theory of critical opalescence (Smoluchowski, 1908) — to describe the scattering properties of anisotropic molecules (Born, 1917). In the same problem, he also considered Bohr's model of atomic constitution; in particular, he derived the most general formula for the electric moment P of the diatomic molecule, 3
j
where E is the vector of the electric field of the incident radiation having frequency u, and the A j , are fixed vectors depending on the specific molecule and its eigenfrequencies u>j (Born, 1918b). 111 Born became increasingly better acquainted with Bohr's theory of atomic structure from lectures and discussions. For instance, Sommerfeld visited Berlin twice in 1918 to speak about his own researches at the German Physical Society: first, on 26 April, at a meeting of the Society in honour of Planck's sixtieth birthday; second, at a meeting of the Society on 26 July (when he also acted as President of the Society, having been elected on 31 May 1918), he spoke on X-ray spectra. He proposed then to remove the theoretical difficulties connected with the orbits occupied by several electrons by assuming that each electron moved on a separate ellipse; that is, he introduced what he called the 'Ellipsenverein,'' a system of ellipses with the atomic nucleus at one focus. Born and Lande exploited this idea of Sommerfeld's systematically in their calculations of crystal properties (Born and Lande, 1918a). Especially, they determined $ , the potential energy of ionic crystals, obtaining the result (-1)
(-5)
# - - * - + -$-.
(3)
which should describe the attractive binding of the oppositely charged nearest (-1)
neighbours( $ < 0) as well as the repulsive action of the more distant ions having the same charge (where r denotes the distance). The repulsive potential term (-5)
<j> (which is larger than zero) depended on the quantum numbers of the Bohr orbits. While Born and Lande first assumed that all the electron orbits associated with a particular ion were in the same plane, they revised this assumption in the following investigations (Born and Lande, 1918b,c). After comparing the theoretical results with the data on the compressibility of crystals, they concluded: The plane electron orbits do not suffice; the atoms are evidently spatial objects. This conclusion appears to us to be as important as the results and researches on the X-ray spectra; in spite of the successes obtained there with planar systems of [electron] rings, we 111 In order to obtain Eq. (2), in which (Aj • E) denotes the scalar product of the vectors Aj > and E, Born averaged over the phases of the electrons in Bohr orbits. Equation (2) yielded Rayleigh's scattering formula for low frequencies w, but yielded deviations from it in the case of higher frequencies.
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must insist on an extension of the theory in the direction mentioned. (Born and Lande, 1918c, p. 216)112 Born and Lande continued their investigations and sought to determine details of the electronic configuration in ionic crystals. For instance, they used the new potential energy expression (-1)
(-9)
r
ra (-9)
obtained from the spatial electron orbits (where $ describes the repulsion due to interaction with next-to-nearest neighbours), in order to derive far-reaching conclusions. Thus Born calculated with it the energy liberated in the course of the formation of alkali-halogenides, finding a reasonable description of the existing data (Born, 1919a). 113 He also discovered a method for determining the so-called 'electron affinity' of halogen atoms, i.e., the amount of energy gained when an electron is added to a halogen atom to form a halogen ion (Born, 1919b). Since no data existed to check his result, Born invented an indirect method: he related the electron affinity to the energy necessary to split alkali-halogenides into alkali and halogen atoms and compared the latter with the experimental results, again with satisfactory agreement. 114 The success of these calculations persuaded Born that the forces in ionic crystals were of purely electrical origin. Thus, starting from the kinetic theory of solids, he had been able to establish a quantitative connection between physical and chemical properties. In an essay for the Naturwissenschaften, entitled 'Die Briicke zwischen Chemie und Physik? ('The Bridge between Chemistry and Physics'), Born expressed the great hope of the physicist to unify the 112
I n calculating the compressibility Born and Lande first made a crucial error, which Born recalled later as follows: We calculated the mutual energy of these [electron] rings — the sums of the energy of a pair — and we worked it out. And we forgot that one has to write one-half of that [in order not t o count every pair twice]. These ring models fitted beautifully and gave us the correct values of the compressibilities. So we thought this was wonderful, and we gave it to Einstein for publication in the Berlin Academy. And the next morning I came to my military department and there was Lande sitting quite depressed and he said, 'You must destroy the paper — it is quite wrong.' And he told me t h a t he had found this mistake I ran to Einstein, and he laughed — I have never heard him laugh so much. And he said: 'This is so marvellous that you have made a mistake; I thought you never make one. You must not destroy it. Of course, we shall not submit it at once — I will given it back to you. But I expect to have it in a week or two again in an improved form.' (Born, Conversations with Mehra) Born followed Einstein's advice and sat down again to work with Lande. Finally they arrived at the solution: in the case of spatially distributed orbits the second repulsive (-9)
11
term in Eq. (3) had to be replaced by a term <3? / r 9 . Born noted that in the case of lithium-halogenides the data seemed to be explained best by (-5)
a power term 3> / r 5 , but in other cases inverse powers of the distance between r - 7 and r - 1 1 resulted. 114 Fritz Haber, who had helped Born in getting the right data, developed a method of representing Bom's procedure (Haber, 1919). It became known to physico-chemists as the Born-Haber cycle.
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physical a n d chemical forces by reducing b o t h t o the interaction between t h e ele m e n t a r y constituents of m a t t e r , i.e., t h e electrons a n d t h e atomic nuclei. He said: The physics of today already possesses pictures of atoms, which certainly approach reality to some extent, and with these numerous mechanical, electrical, magnetic and optical properties of substances can be explained. Now one should not be stopped by chemical properties, but attempt also to reduce them to atomic forces to the extent they are known. In this respect Nernst's theorem provides valuable preliminary preparation by reducing the intricate complex of chemical forces to a number of simple constants [especially the chemical constants, see Sec. 1.6]. It is now the task of the molecular physicist to calculate these constants — which can be determined by the physical chemists through calorimetric and other measurements — from the properties of atoms: with this begins a new and inconceivably big era of thermochemical research. (Born, 1920b, pp. 373-374) Born, who considered himself a molecular physicist, was p r e p a r e d to c o n t r i b u t e further to this important field of research. In spring 1919 Born left Berlin and went to Frankfurt-am-Main t o take u p t h e position of a n ordinary (i.e., full) professor of physics at t h e University of Frankfurt, while his predecessor there, M a x von Laue, moved t o Berlin. 1 1 5 For a time B o r n h a d hesitated to leave Berlin a n d his friends a n d colleagues there, b u t Einstein encouraged him t o go to Frankfurt, telling him t o 'accept unconditionally' a n d arguing t h a t 'one should not refuse such an ideal post, where one is completely independent' (Einstein t o Mrs. Born, 8 February 1918). At Frankfurt B o r n began t o lecture in s u m m e r semester 1919, choosing, at first, courses on such subjects as mechanics a n d q u a n t u m theory. He quickly a d a p t e d himself t o the new place, finding a suitable home to live in with his family; his friend E r n s t Hellinger, t h e n professor of m a t h e m a t i c s a t the University of Frankfurt, lived w i t h t h e m . T h e institute, which he took over, was small b u t comparatively well-equipped. He found able assistants in O t t o Stern, Privatdozent a t Frankfurt since 1914, and Elisabeth B o r m a n n . Alfred Lande, his wartime collaborator, also came over to Frankfurt as Privatdozent. For Born it was no problem to organize a p p r o p r i a t e lecture courses for the students, who had begun to flock to the universities — including Frankfurt — after World War I. 1 1 6 B u t running a n institute after t h e war was not a n easy task, especially to obtain equipment for experimental work. G e r m a n y had 115 The offer to von Laue to come to Berlin had been made already in early 1918. But it took more than a year before the second professorship of theoretical physics, envisaged already in 1914, was established at the University of Berlin. 116 In Berlin, Born had learned a lot from Max Planck. Thus he also started to give systematic courses in Frankfurt: in summer 1919 he lectured on the mechanics of particles, the following winter semester on the mechanics of continua, and in the winter semester 1920-1921 on electrodynamics. Otto Stern supplemented Bom's courses by lecturing in summer 1919 on kinetic theory of gases, in the winter semester 1919-1920 on thermodynamics, and in winter 1920-1921 he took over the course on mechanics of continua (which followed Bom's course on particle mechanics in summer 1920). The aim was to present the main courses on theoretical physics (mechanics, thermodynamics and electrodynamics) every year. Besides the main courses, special courses were also given: e.g., on atomic and quantum theory. Finally, Born conducted, together with Stern and Lande, a Seminar on modern problems of theoretical physics.
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lost the war and, apart from all the losses that had been incurred, enormous and not-yet-fixed reparations had to be made to the Western Allies. The Weimar Republic had therefore to fight against tremendous economic and financial problems and could not support universities on the same scale as the former, prosperous Kaiserreich. Industry was likewise hindered by the reparations and the fact that the Rhein-Ruhr region, the most important industrial region remaining in Germany, continued to be occupied by French troops for years. Rampant inflation devalued German currency, and Born — like any other institute director — was faced with the problem that working within a fixed budget did not cover the steadily rising expenses of research. At that difficult time, however, some assistance came from various sources, especially from the Notgemeinschaft der Deutschen Wissenschaft (Emergency Association of German Science), which was founded on 30 October 1920 and which distributed additional funds to the universities. 117 Born also obtained financial help from Einstein, who 'tried to squeeze some funds out from the Kaiser Wilhelm-Institut [of Physics]' (Einstein to Mrs. Born, 1 September 1919), and who also shared other monies with his friend. In addition, Born capitalized on the public interest that existed at the time in Einstein's relativity theory: he gave lectures for the general public on it and charged entrance fees, which he put into his institute. 118 Finally, he was supported by some private individuals: thus the rich Frankfurt jeweller, G. Oppenheim, who had previously helped to establish the physics chair at the university of Frankfurt, provided Born a diamond for his experiments on elasticity; and the American banker, Henry Goldman, donated considerable sums in dollars which, not being subject to inflation at that time, were the more valuable. The major part of Bom's budget went for experimental research. For instance, Stern developed in Frankfurt the method of atomic beams, produced from metals heated in a vacuum, for the investigation of atomic properties. In his first important work using this method he measured the thermal velocity distribution of atoms (Stern, 1920a,b,c,d). And soon he became involved in looking for an experimental verification of spatial quantization (Stern, 1921). The experiment, which he later performed together with Walther Gerlach — who had joined the University of Frankfurt in 1920 as a Privatdozent — yielded early in 1922 the final result that a beam of silver atoms was split by an inhomogeneous magnetic field into two parts 117 Shortly after the war the former Prussian Minister of Education, Priedrich Schmidt-Ott, discussed with Max Planck and Adolf von Harnack, then President of the Kaiser WilhelmGesellschaft, the possibilities of helping to promote scientific research in Germany and Austria. The Prussian Academy became involved, and on 19 April 1920 Max Planck — on behalf of the Academy — requested Schmidt-Ott to head the Notgemeinschaft der Deutschen Wissenschaft. The aim of this foundation was to examine, together with the academies in Germany and Austria, the situation in scientific research and to support scientists with funds and instrumentation. Committees, consisting of small numbers of experts, were established in each field of research. The Notgemeinschaft received its funds mainly from the Government of the Weimar Republic and the German states. 118 A book grew out of Bom's public lectures, entitled 'Die Relativitatstheorie Einsteins und ihre physikalische Grundlagen,'1 which sold very well (Born, 1920d).
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(Gerlach and Stern, 1922a). Born also performed certain experiments himself: at the Naturforscherversammlung in Bad Nauheim in September 1920 he presented the results of a measurement of the mean free path of silver atoms in air (Born, 1920c). But his main efforts were on the theoretical side: he continued to study the kinetic theory of matter, to investigate the electron affinity of atoms (with Elisabeth Bormann) and the dispersion of light by diatomic molecules (with Gerlach). Thus he accomplished within a year a respectable amount of work, and then he received the call to Gottingen. Again he hesitated initially to accept the invitation. After all, he felt well in Frankfurt, where he had succeeded in establishing a good institute and in acquiring some good friends. Would he also find in Gottingen some wealthy people, as in Frankfurt, who supported his activities? And would he find such capable, collaborators as Otto Stern and Alfred Lande, whom he could not take along? 119 The decision was made as soon as Born succeeded in getting the additional professorship from the Ministry. With James Franck, he felt certain, he would be able to continue the Gottingen tradition of physics in a maimer worthy of the place. 120 Born appeared to be the ideal man for Gottingen. He had grown up at the University under the influence of David Hilbert and Hermann Minkowski. He had executed a large part of Hilbert's programme of 'axiomatizing' physics: for instance, he had established the kinetic theory of solids on the basis of just a few assumptions concerning atoms and the forces between them; he had started to do the same with the theory of molecules, and was beginning to 'axiomatize' chemistry. Moreover, he was very experienced in giving carefully organized lectures on all aspects of theoretical physics; in Frankfurt he had also shown considerable talent in running the experimental institute and getting the necessary funds. Born knew that the same was expected of him now at Gottingen. Thus he wrote to Einstein, still from Frankfurt: 'Franck has now settled in Gottingen. He must have enough freedom there, and so I am busily collecting money for him. So far I have got 68,000 Marks. It is not at all easy to inspire laymen with some interest in our work. I must have more money. Wien got a whole million for re-equipping his Institute in Munich. I believe that what Wien has, Franck should also get' (Born to Einstein, 12 February 1921). In collecting money Born addressed himself in particular to the industrialist Carl Still of Recklinghausen (in the Ruhr region), to whom Richard Courant had introduced him. 121 119 F o r some time Born tried to have Otto Stern appointed as his successor in Frankfurt, but he did not succeed. Stern left Frankfurt in late 1921 to take up a professorship at the University of Rostock, and Lande was called in 1922 as an extraordinary professor to the University of Tubingen. B o m ' s successor in t h e Frankfurt chair was Erwin Madelung. 120 B o r n ' s coming to Gottingen was delayed by the problem of finding a suitable accommodation for his family. This was not an easy enterprise in those days. It took until spring 1921 when the problem was solved and the Boms moved to Gottingen. 121 Carl Still, the son of a Westphalian farmer, had started off as a mechanic and built up a large firm, which built coke ovens and similar installations. He was profoundly interested in science and frequently invited the physicists and mathematicians of Gottingen to his countryseat in Rogatz on the Elbe.
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Unlike Born, James Franck had never been in Gottingen before. He was born on 26 August 1882 at Hamburg, the son of a Jewish banker. 122 After attending the Wilhelmsgymnasium there, he entered the University of Heidelberg in 1901 and began to study geology, mathematics and physics, but mainly chemistry. 123 Though he loved the surroundings of Heidelberg, he did not find the professors at the University very inspiring; in the following year he went to Berlin. 124 In Berlin, however, he quickly became interested in physics under the influence of Emil Warburg. He began to be involved in measuring the mobility of ions and received his doctorate in 1906. Following a few months as an assistant in Frankfurt-am-Main, he returned to Berlin and became Heinrich Rubens' assistant. Franck continued to remain interested in the problem he had treated in his thesis; he also followed the work of British physicists, like Joseph John Thomson, in this field. He collaborated with Robert Wichard Pohl on ion mobilities and the determination of the velocity of X-rays, with Wilhelm Westphal on the charge of ions, with Robert Williams Wood — who came as a guest of Rubens to Berlin in fall 1910 — on the fluorescence of iodine and mercury vapours, with Peter Pringsheim on the electrical and optical properties of the chlorine flame, and with Lise Meitner on radioactive ions. In 1911 he received his Habilitation at the University of Berlin. In the same year Gustav Hertz obtained his doctorate and became Franck's successor as assistant to Rubens. Franck and Hertz then started a series of investigations on the ionization potentials (of the atoms and molecules) of various elements in connection with the quantum hypothesis. Within three years they arrived at their results on the collision of electrons with mercury atoms (Franck and Hertz, 1914a,b), which were considered right away by Bohr as a confirmation of his model of atomic constitution. Interestingly enough, Franck and Hertz did not know Bohr's theory at all and interpreted their results not as an excitation potential, but as the ionization potential of the atoms. Only a couple of years later did Franck turn to Bohr's interpretation. Meanwhile the war had broken out, and Franck voluntarily joined the army; he became an officer. Having fallen seriously ill, he returned to Berlin in 1917 and joined the Kaiser Wilhelm-Institut fur Physikalische Chemie, whose director was Fritz Haber. 125 Franck became head of the Physics Division and worked to 122 James Franck's ancestors could be traced back to the eighteenth century in the region near Hamburg. The banking firm, J. Franck and Company, was founded by James' grandfather Jacob Franck, Jr. His son James Franck married Ingrid Josephson of Gdteborg. 123 A t school James Franck was not very interested in classical languages. Instead he became attracted to physics at the time when Rontgen discovered X-rays. He liked mathematics, especially geometry, and did chemical experiments at home. At school he also learned some English and French. 124 Franck found the lectures on inorganic chemistry very old-fashioned. For example, he never heard anything about Svante Arrhenius' work on electrochemistry. T h e professor of physics was the old Georg Hermann Quincke, who did not give inspiring lectures either. 125 F r i t z Haber was born on 9 December 1868 in Breslau. He studied chemistry and physics at the University of Berlin (1886, under August Wilhelm Hofmann and Hermann von Helmholtz), the University of Heidelberg (1886-1889, under Robert Bunsen) and at t h e Technische Hochschule in Berlin (1889-1891), where he received his doctorate in 1891 with a thesis in organic chemistry under the supervision of Karl Liebermann (1852-1914). The following year he joined the
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re-establish research in t h e post-war years, one of his collaborators being Fritz Reiche. In April 1920 he m e t Niels Bohr, who on the 27th of t h a t m o n t h gave a lecture o n t h e series spectra of elements at t h e G e r m a n Physical Society. Franck was very impressed by B o h r ' s personality and became a great admirer of t h e Danish physicist. B o h r reciprocated his feelings a n d wrote some m o n t h s later: My stay in Berlin, though regrettably short, has been a particularly beautiful and stimulating event for me; and one of the greatest joys I experienced there was meeting you and getting to know you. I have always followed your important investigations with the greatest interest, and it was for me extremely interesting to see your experimental setup and to hear your opinions about the results. I have lately been especially concerned with your new experiments on helium, and, while contemplating the mechanism of the collision between atoms and electrons, I hit upon a problem, about which I would be very grateful to hear your views. (Bohr to Franck, 18 October 1920) W i t h t h e last remarks Bohr referred to t h e results of Franck a n d P a u l Knipping on helium excitation; they h a d observed there a possible excitation potential of 21.9 V (Franck a n d Knipping, 1920, p p . 326-327). Bohr now wanted to know about the probability of such collisions, which according to his theoretical estimate should be very small, a n d he wrote t o Franck: In connection with my interest in the field of experimental physics that you have opened, I thought of the possibility that you might perhaps one day come to Copenhagen to give your support for some time to our work in the new Institute for Theoretical Physics, which is being established here and which is being especially equipped for experimental research on spectroscopic problems. Bohr wished t o have Franck's advice for the installation of experimental a p p a r a t u s a n d he invited him to come for some time t o Copenhagen. He had just obtained funds from private sources t o pay for the stay of a distinguished foreign physicist ('einen bedeutenden auslandischen Physiker1) for a short time. He wrote t o Franck: Therefore I hasten to ask you whether you have the desire and the possibility of giving us the honour and joy of coming to Copenhagen for a few months in the beginning of the new y e a r . . . . I cannot quite express the anticipation with which we would all look forward to your visit, and how much I, in particular, would rejoice to have the opportunity of discussing with you personally the problems in which both of us are so interested. (Bohr to Franck, 18 October 1920) Eidgenosische Technische Hochschule (E.T.H.) in Zurich and investigated problems of chemical technology. Then he returned to Germany, worked for a while in his father's chemical company, and went to the Technische Hochschule in Karlsruhe in 1894. There he became Privatdozent in 1896, Extraordinarius in 1898 and Ordinarius in 1906. In Karlsruhe, his research dealt first with the decomposition of hydrocarbons by heat, then with the oxydation-reduction process and later, after 1905, with the synthesis of ammonia from nitrogen and hydrogen. Finally, together with Carl Bosch of the Badische Anilin und Soda-Fabrik, he developed the method for the large scale production of ammonia. In 1911 Haber went to Berlin as Director of the newly founded Kaiser Wilhelm-Institut fur Physikalische Chemie und Elektrochemie. During World War I he became involved in chemical warfare. After the war he started to rebuild his institute, and it became an internationally known center of research. Haber received the Nobel Prize in Chemistry for 1918. He resigned the directorship of the institute in April 1933 and left Germany in early summer. He died in Basle on 29 January 1934.
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Franck accepted the invitation with pleasure; after 1921 he returned repeatedly to Copenhagen and established a close relationship with the physicists there — especially, of course, with Niels Bohr. In Gottingen the three new professors of physics, Robert Pohl — who directed what was called the First Experimental Institute —, James Franck — who directed the Second Experimental Institute —, and Max Born — who directed the Institute for Theoretical Physics —, soon arranged their respective work and duties. Franck left to Pohl the teaching of physics to beginners — the big physics course — and restricted himself to giving lectures on specialized topics. 'I had the laboratory course as the main thing, with the Praktikum,'' he recalled later. 'And we used a good deal of the instruments of Voigt to make an advanced laboratory course where more complicated things were measured' (Franck, AHQP Interview, 13 July 1962, p. 4). On the other hand, Born, who continued to work in Gottingen on some of the experiments he had started in Frankfurt, soon gave them up and left experimental research to Franck and Pohl. Instead he organized systematic threeyear cycles of courses on theoretical physics, consisting of six series of lectures distributed over six semesters: they were on (1) mechanics of particles and rigid bodies; (2) mechanics of continuous media; (3) thermodynamics; (4) electricity and magnetism; (5) optics; (6) elements of statistical mechanics, atomic structure and quantum theory. Each course consisted of four hours of lectures per week and a tutorial. Born supplemented these main lectures by continuing series of lectures on special topics: thus, in the summer semester 1921 and the following winter semester he gave a series of lectures on the kinetic theory of solids, and in the winter semester 1922 he dealt with magneto- and electro-optics. 126 Later, when Bom's assistants became Privatdozenten, they helped him with the main courses of lectures; then the cycle of main courses was repeated after three semesters. Born, Franck and Pohl, together with Max Reich, the professor of applied electricity, had a joint colloquium, in which Ludwig Prandtl, Emil Wiechert and the astronomer Johannes Hartmann participated regularly, and where David Hilbert and Richard Courant showed up occasionally. The style of the colloquium was very informal. As Born recalled: 'It was customary to interrupt the speaker and to criticize ruthlessly. We had the most lively and amusing debates, and we encouraged even young students to take part, by establishing the principle that silly questions were not only permitted but even welcomed' (Born, 1978, p. 211). 127 The different personalities and styles of the three physics professors contributed importantly to the atmosphere. Maria Goeppert-Mayer characterized them by relating the following incident: 'I was once 126 ' B o r n gave very thorough, but rather difficult lectures. Born's lectures were difficult to understand,' recalled Friedrich Hund. 'He presented much more than we do nowadays, hence his influence on the physicists was smaller. At that time things had fallen apart: the experimentalists did not really learn theoretical physics, and the theoretical physicists did not learn proper experimental physics.... Of course, the situation was better for the advanced students' (Hund, AHQP Interview, First Session, p. 14). The colloquium took place in the small auditorium (Kleiner Horsaat), and 60 to 70 people use to attend it.
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in the experimental laboratory. I came down the steps. Franck, Born, and Pohl were standing talking to each other. They were all friends of my parents. Pohl just nodded; Born said hello; and Pranck stretched out his hand' (Goeppert-Mayer, AHQP Interview, p. 5). The students were more or less identified with their professors, and Pohl referred to them ldie Pohlierten, die Franckierten, die Bornierten,'1 respectively. Naturally, due to their excellent personal relations, Born and Franck worked together more closely. So they not only organized the institute parties together (Franck and Born belonged historically to the same institute), but also jointly conducted the Physikalisches Proseminar, in which selected topics, mainly of quantum physics, were discussed. 128 In research, Franck and Born continued their earlier interests. The investigation of the collision between electrons and atoms or molecules remained (in the early 1920s) the principal field of research of James Franck and his collaborators, who included Walter Grotrian, Gunther Cario and Patrick Maynard Stuart Blackett, a guest from England. With some of his associates, Born carried on investigations on crystal dynamics; Sommerfeld had also asked Born to write a comprehensive article on that subject for the Encyklopadie der mathematischen Wissenschaften (Born, 1923b). During the writing of the encyclopedia article, Born came across a number of problems suitable for doctoral theses, and a number of students — among them Carl Hermann and Gustav Heckmann — obtained their degrees by working on them. Even in his later Gottingen years, Born returned occasionally to crystal dynamics. 129 Indeed, the kinetic theory of solids, the problems of which he had first attacked in 1912 and which he had continued to develop in the following decade, remained dear to him until the end of his career. While Franck showed little interest in questions of the solid state, he appreciated the investigation of molecular problems, which was the other field of Born's research in the early 1920s. A fruitful collaboration developed between Franck and Born, and they even wrote a joint paper on molecular physics (Born and Franck, 1925a, b). And, finally, the field in which Born became involved during the early twenties — Bohr's theory of atomic structure — united the two friends in a common cause and determined the success of the Born-Franck era of Gottingen physics. 130 For example, the Ramsauer effect was discussed in the Proseminar. T h e latter also replaced the Seminar on the Structure of Matter (Struktur der Materie) which Hilbert tried to reestablish with Born. 129 Together with Maria Goeppert-Mayer, Born wrote another review article on crystal dynamics for the Handbuch der Physik (Born and Goeppert-Mayer, 1933). 13 The great era of Gottingen physics ended in 1933. In May of that year Born was deprived of his Gottingen chair and left Germany. Franck also resigned, but he stayed on until the fall, when he went to Niels Bohr's Institute in Copenhagen. In 1935 he went as a professor of physics to Johns Hopkins University, Baltimore, from where he went to the University of Chicago in 1938. He retired in 1947. In 1964 he returned for a visit to Gottingen, accompanied by his second wife, Hertha Sponer; there he died on 21 May 1964. Born, on the other hand, settled in Great Britain. He stayed for a period in Cambridge as Stokes Lecturer (1933-1936), and from there helped German refugee scientists in getting positions. In 1936 Charles Gatton Darwin, who was about to become Master of Christ Church College, Cambridge, invited Born to become his successor as Tait Professor of Natural Philosophy in the University of Edinburgh. Born retired in 1953 and settled in Bad Pyrmont close to Gottingen. He died there on 5 January 1970.
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Hilbert, D. (1924), Mathematische Annalen 92, 1-32; reprinted in Gesammelte Abhandlungen, 3, pp. 258-289. Hilbert, D., J. von Neumann, and L. Nordheim, (1928), Mathematische Annalen 98, 1-30. Jordan, C. (1870), Traite des substitution et des equations algebraique, Paris: GauthierVillars. Karman, T. von, and L. Edson (1967), The Wind and Beyond, Boston-Toronto: Little, Brown and Company. Klein, F. (1872), Vergleichende Betrachtungen, etc., Erlangen: Andreas Deichert. Klein, F. (1917), Nachr. Ges. Wiss. Gottingen, pp. 469-482. Klein, F. (1918a,b), Nach. Ges. Wiss. Gottingen, pp. 171-189, pp. 394-423. Klein, F., and A. Sommerfeld (1897, 1898, 1903, 1910), Uber die Theorie des Kreisels (Heft 1, 2, 3, 4), Leipzig: B.G. Teubner. Lorentz, H.A. (1910b), Phys. Zs. 1 1 , 1234-1247 (six Wolfskehl lectures, delivered 24 to 29 October 1910 at Gottingen, worked out by M. Born). Madelung, E. (1909, 1910a,b), Nach. Ges. Wiss. Gottingen, pp. 100-106, pp. 43-58; Phys. Zs. 1 1 , 898-905. Mehra, J. (1973a), Einstein, Hilbert and the theory of gravitation, in The Physicist's Conception of Nature (J. Mehra, ed.), Dordrecht: D. Reidel; reprinted as a separate book by D. Reidel, Dordrecht-Boston, 1974. Mie, G. (1912a,b,c), Ann. d. Phys. (4) 37, 511-534; 39, 1-40; 40, 1-66. Mie, G. (1917), Phys. Zs. 18, 551-556, 574-580, 596-602 (revised and extended Wolfskehl lectures, Gottingen, 5-8 June 1917). Minkowski, H. (1896), Geometrie der Zahlen, Leipzig: B.G. Teubner. Minkowski, H. (1907), Encykl. d. math. Wiss. VI/1, pp. 558-613. Minkowski, H. (1908), Nachr. Ges. Wiss. Gottingen, pp. 53-111; reprinted in Mathematische Annalen 68, 472-525 (1910). Minkowski, H. (1909), Phys. Zs. 10, 104-111. Minkowski, H. (1910), Mathematische Annalen 68, 526-551. Edited by M. Born. Minkowski, H. (1915), Das Relativitatsprinzip, Ann. d. Phys. (4) 47, 927-938. Edited by A. Sommerfeld. Neumann, J. von (1932), Mathematische Grundlagen der Quantenmechanik, Berlin: J. Springer Verlag. Noether, E. (1918), Nachr. Ges. Wiss. Gottingen, pp. 235-257. Pasch, M. (1882), Vorlesungen uber neuere Geometrie, Leipzig: B.G. Teubner. Reid, C. (1970), Hilbert, New York-Heildelberg-Berlin: Springer-Verlag. Riecke, E. (1898), Ann. d. Phys. (3) 66, 353-389. Riemann, B. (1851), doctoral dissertation, University of Gottingen, Gesammelte Mathematische Werke, pp. 3-43. Riemann, B. (1857), J. reine & angew. Math. 54, 115-155; reprinted in Gesammelte Mathematische Werke, pp. 88-144. Runge, I. (1949), Carl Runge und sein wissenschaftliches Werk, Gottingen: Vandenhoeck & Ruprecht. Schellenberg, K. (1915), Ann. d. Phys. (4) 47, 81-127 (doctoral dissertation, University of Gottingen). Schmidt, E. (1908), Rend. Circ. Mat. Palermo 25, 53-77. Smoluchowski, M. von (1908), Ann. d. Phys. (4) 25, 205-226. Smoluchowski, M. von, (1916), Phys. Zs. 17, 557-571, 585-591 (Wolfskehl lectures given at Gottingen, 20-22 June 1916). Sommerfeld, A. (1910b,c), Ann. d. Phys. 32, 749-776; 33, 649-689.
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Stem, O. (1920a,b,c,d), Z. Phys. 1, 147-153; 2, 49-56; Phys. Zs. 2 1 , 582; Z. Phys. 3, 417-421. Stern, O. (1921), Z. Phys. 7, 249-253. Thomson, J.J. (1904a), Phil. Mag. (6) 7, 237-265. Toeplitz, O. (1911), Mathematische Annalen 70, 351-376. Voigt, W. (1887), Nach. Ges. Wiss. Gottingen, pp. 41-51; reprinted in Phys. Zs. 16, 381-386 (1915). Weber, W., and R. Kohkausch (1856), Ann. d. Phys. (2) 99, 10-25. Weyl, H. (1908), Mathematisch Annalen 66, 273-324. Weyl, H. (1912a), Mathematische Annalen 7 1 , 441-479. Weyl, H. (1912b), J. reine & angew. Math. 1 4 1 , 1-11. Weyl, H. (1912c), J. reine & angew. Math. 1 4 1 , 163-181.
13 T h e Bohr Festival in Gottingen: Bohr's Wolfskehl Lectures and the Theory of the Periodic S y s t e m of Elements*
Since its establishment in 1913, Bohr's theory of atomic structure had received greater attention than any other attempt in this direction. This did not mean, however, that this theory was generally accepted by physicists and that rival views were not discussed at all. Thus, for instance, Joseph John Thomson, in two series of lectures at the Royal Institution in 1918 and 1919, respectively, discussed the problems of atomic structure and origin of spectral lines; he invoked a model of atoms differing widely from Bohr's and resting on the assumption that the forces exerted on the electrons in atoms were not only of the Coulomb type but also included repulsive terms (Thomson, 1918, 1919).l It also did not mean that no opposition arose against Bohr's conceptions. For example, Johannes Stark, in an article entitled 'Zur Kritik der Bohrschen Theorie der Lichtemission' ('On a Criticism of Bohr's Theory of Light Emission'), expounded a strong criticism; he not only emphasized the unclear points in the fundamental assumptions, but also concluded that the theory failed in many respects to describe the data correctly (Stark, 1920). Arnold Sommerfeld defended Bohr's theory against Stark's accusations (Sommerfeld, 1921c). While he agreed with Stark that certain points of Bohr's theory, such as the question concerning the coupling of electrons to radiation or the dependence of the emitted radiation on the final state involved in the transition, required further Lectures delivered at T h e Niels Bohr Institute and Nordita, Copenhagen, 16 September 1975, CERN (European Organization for Nuclear Research), Geneva, 26 September 1975, and Universite libre de Bruxelles, 18 October 1977. Revised and enlarged version published in The Historical Development of Quantum Theory (with Helmut Rechenberg, Springer-Verlag New York, 1982). *In his 1918 lectures, Thomson referred explicitly to a repulsive term proportional to the third inverse power of the distance between the electron and the centre of atom (i.e., nucleus), which he had used several years before (Thomson, 1913b). W i t h this assumption the electron would possess a stable position, i.e., it would be in equilibrium at the position of zero force, and the radiation would come about from oscillations around the stable position. In his 1919 lectures Thomson chose the full potential (i.e., Coulomb attraction plus additional repulsive forces) as being given by ( 1 / r 2 ) • (sin(2c/r)/(2c/r)), where r is the radial distance and c is the velocity of light in vacuo; this potential led to several stable positions. 459
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clarification, he pointed out that 'one should not forget that Bohr's theory is only eight years old' (Sommerfeld, 1921c, p. 419). Continued investigations, he was certain, would resolve many if not all difficulties. As for the presumed failures of Bohr's theory to account for the empirical facts, Sommerfeld contradicted Stark in all cases. For example, he argued that the most recent measurements of the doublet splitting of hydrogen lines by Ernst Gehrcke and Ernst Gustav Lau, which seemed to yield results smaller than the ones predicted by his relativity calculations (Gehrcke and Lau, 1920), could not, because of their insufficient accuracy, be used against the Bohr-Sommerfeld theory. On the other hand, Sommerfeld claimed that all Stark effect data agreed with the theory, while he declared it to be premature to conclude any failure of the theory in describing the anomalous Zeeman effects. Altogether, he strongly opposed Stark's opinion that the success of Bohr's theory was restricted to explaining the Balmer spectrum, the frequencies of the Stark components and the normal Zeeman effect, and remarked: Even if we exclude, as Mr. Staxk does, the wide field of X-ray spectroscopy, which has been organized and clarified only by applying Bohr's theory, and even if we forget about the questions of fine structure, which have been discussed above sufficiently, we must still count in the field of visible spectroscopy such an amount of successes in favour of Bohr's theory that one will not hesitate to call the latter the greatest progress of all times in the understanding of the atom. (Sommerfeld, 1921c, p. 429) The great progress to which Sommerfeld referred included not only the wellknown explanation of the spectra of ionized helium and the qualitative interpretation of the structure of the spectra of non-hydrogen-like atoms (i.e., the occurrence of the principal and subordinate series, etc.), but also the recent advances in the theory of band spectra. Band spectra had already been discussed in connection with quantum theory before Bohr proposed his models of atomic structures, especially by Niels Bjerrum, as an example of quantizing the rotator system (Bjerrum, 1912). Later, Karl Schwarzschild had considered the band spectra as arising from the transitions between different stationary states: a band-line with frequency v,
should be emitted when the electrons in the molecule make a transition liberating the energy huo and the molecular rotation changes the rotational quantum number from m to m' (Schwarzschild, 1916). By inserting into Eq. (1) the data from the so-called cyanide-bands (emitted by molecular nitrogen), he had found, however, that the moment of inertia, A, thus derived was much larger than the one obtained from kinetic gas theory. Torsten Heurlinger of Lund had then investigated the situation more closely; in particular, he had observed that the nitrogen-bands could be separated into several systems of bands, the frequencies of the components being described by the formula v = VQ + c\m + c2m2 , (2)
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where m is an integer. He had then developed, from the point of view of Bohr's theory of atomic and molecular constitution, a description of the band spectra of diatomic molecules (like N2) by associating with each state a series of quantum numbers TO,711,712,... and deriving the frequencies as differences of terms, ip(m,ni,«2i • • •) and rp(m',n^,^,. •.), where m and m' are the rotational quantum numbers of both terms and n i , n 2 , . . . and n'^n^,... the other quantum numbers of the initial and final states, respectively (Heurlinger, 1918, 1919, 1920).2 Now, on returning to Munich after the end of the war, Sommerfeld's former student Wilhelm Lenz entered the field.3 Lenz considered the molecular models more carefully; especially, he took into account the alteration of the moment of inertia of the molecule due to a change in the radial quantum number of the electron ring, and he was indeed able to justify Heurlinger's Eq. (2) including the term proportional to m 2 (Lenz, 1919).4 However, in determining the quantum numbers of the ring from the band spectra and their Zeeman effects in the cases of the nitrogen and the hydrogen molecules, Lenz noticed a difficulty: the data required the quantum number n of the ring to be zero, in contradiction to Bohr's models, which required n = 2 in the case of hydrogen and a still higher value for nitrogen. 5 When Lenz left Munich in 1920, he had already trained Adolf Kratzer, his successor at Sommerfeld's Institute. 6 Kratzer extended the theory of diatomic molecules by including anharmonic forces between the nuclei, which changed the oscillation frequencies (Kratzer, 1920a). He would devote his efforts in future basically to the investigation of band spectra and become one of the leading experts in this field.7 The theory of band spectra represented only one of the subjects discussed at Sommerfeld's Institute in Munich. An increasing number of new students, including Wolfgang Pauli, Gregor Wentzel, Werner Heisenberg, Otto Laporte and, a little later, Walter Heitler, Karl Bechert and Albrecht Unsold, helped Sommerfeld in ex2 B y adding the selection rules, m ' = m, m ± 1, Heurlinger found t h a t for the term differences, V>(m, m , . . . ) — ip(m ± 1, T»I, . . . ) , Bq. (2) could indeed be derived. 3 W . Lenz was born on 8 February 1888 at Frankfurt-am-Main. He studied physics at the Universities of Gottingen (1906-1908) and Munich (1908-1911), obtaining his doctorate in 1911. He then stayed on in Munich (after 1914 as a Privatdozent). In 1920 he was called to the University of Rostock as an extraordinary Professor; the following year he moved to the University of Hamburg as ordinary (i.e., full) professor of theoretical physics. He retired in 1956 and died in Hamburg on 30 April 1957. Without this change of the radial quantum number, only the term linear in m can be explained. (See Lenz, 1919, p. 634, Eq. (4).) In another paper on the theory of band spectra (Lenz, 1920b), Lenz discussed the spectra of iodine, which had been obtained recently by Robert Williams Wood (1918). 6 A . Katzer was born on 16 October 1893 at Giinzburg. He studied physics at the Technische Hochschule, Munich (1912-1914), and — after two years service in the army — at the University of Munich from 1916 to 1920, when he received his doctorate under Sommerfeld. Then he was sent to Gottingen as Hilbert's assistant for physics (1920-1921). In 1921, after his return to Munich, he became a Privatdozent. A year later he accepted the invitation to the chair of theoretical physics at the University of Miinster. In a second paper in 1920, Kratzer studied the influence of nuclear masses (the isotope effect) on band spectra (Kratzer, 1920b). His detailed analysis of the cyanide-bands resulted in the introduction of half-integral quantum numbers to account for the rotation (Kratzer, 1922).
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ploring various parts of atomic theory: Pauli obtained his doctorate with a thesis on the hydrogen molecule-ion, Wentzel worked on X-ray spectra, Heisenberg on the anomalous Zeeman effect and Laporte on complex spectra. The flourishing Sommerfeld school sent its representatives to other places — where, with Sommerfeld's effective assistance, they obtained professorships and began to develop atomic theory themselves: thus Peter Debye in Zurich, Wilhelm Lenz in Hamburg, Adolf Kratzer in Minister and Erwin Fues in Stuttgart. Besides Munich, atomic theory also played an important role in Berlin and at those universities where people from Berlin went: thus in Breslau, Rudolf Ladenburg and Fritz Reiche pursued it. The publications of the results emerging from these numerous and intensive investigations assumed a prominent place in the German scientific journals. The Zeitschrift fur Physik, a new journal, played a major role in reporting about the developments in atomic theory; it was established in 1920 to publish the growing number of contributions that was submitted to the German Physical Society for inclusion in its Verhandlungen.s Soon not only the physicists from Germany, Austria and Switzerland, but also from the Netherlands, the Scandinavian countries and the Soviet Union submitted their papers to Zeitschrift fur Physik, making it the leading journal on atomic theory. Next to it the Physikalische Zeitschrift still contained a considerable number of papers on atomic theory, while the number of papers submitted to the more conservative Annalen der Physik in this field declined.9 While innumerable publications made the theory of atomic structure a really fashionable topic at the forefront of physics, the man, who had invented the theory, remained comparatively silent. Since his return to Copenhagen in fall 1916 as a professor of theoretical physics, Niels Bohr had published only two papers dealing with the fundamental aspects of the theory of line spectra (Bohr, 1918a,b) and one paper suggesting the existence of a triatomic hydrogen motecute (Bohr, 1919). Besides these papers, he had given several lectures reviewing the progress of his theory at the Physical Society of Copenhagen, whose chairman he was from 1916 to 1919; and he had lectured abroad, for example, at the University of Leyden on 25 April 1919 (on 'Problems of the Atom and the Molecule') and in Berlin at the German Physical Society on 27 April 1920 ('Uber die Serienspektren der Elemente'). To an outsider the years between 1918 and the end of 1920 appeared ° T h e number of papers submitted to the Verhandlungen had increased steadily since 1899. After World War I the production costs for the journal rose tremendously due to inflation and the leading scientists in the German Physical Society decided — against some opposition, e.g., from Philipp Lenard — to reduce the content of the Verhandlungen by restricting its pages to reports on the activities of the Society, obituary notices, etc. The extended original publications were supposed to appear in Zeitschrift fur Physik, a journal edited by the Society (with Karl Scheel as the responsible editor) and printed by Friedrich Vieweg und Sohn, Braunschweig. The first issues of Zeitschrift fiir Physik appeared in February 1920; it soon became the leading physics journal, overtaking the Annalen der Physik. The period between the submission of a paper and its publication was much shorter in the Zeitschrift, and papers on modern topics, especially atomic theory, were more likely to be submitted t o it. It should be mentioned, however, that many detailed experimental investigations on spectroscopy and a number of extended theoretical studies (e.g., doctoral theses) continued to appear in Annalen der Physik.
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to be a quiet period in Niels Bohr's scientific productivity. He seemed to be deeply involved in establishing his own institute, the Institute of Theoretical Physics at Blegdamsvej in Copenhagen. 10 Bohr's Institute, which had been planned with the help of the experimentalist Hans Marius Hansen and funded mainly by the Carlsberg Foundation, was dedicated on 3 March 1921. n Bohr had a few collaborators, the most important among them being Hendrik Anthony Kramers. Hendrik Kramers was born on 17 December 1894 at Rotterdam. He began to study theoretical physics in 1912 under Paul Ehrenfest at the University of Leyden. After passing a predoctoral examination in 1916, he wanted to continue his studies of mathematics and physics at a foreign university. He went to Copenhagen and there wrote a letter to Bohr, in which he introduced himself and requested: 'Of course I should like very much to come in acquaintance with you in the first place, and also with your brother Harald. Therefore I should be very glad if you permit me, to visit you in one of these days. Perhaps you'll be so good to write me a card or telephone to my hotel when I may come to see you' (Kramers to Bohr, 25 August 1916).12 After meeting Bohr, Kramers expressed the wish to become his assistant. Bohr, who was not quite sure about what to do, asked his brother Harald, who answered that 'if the young Dutchman was really so keen, he might as well be given a chance' (Rosenfeld and Riidinger in Rozental, 1967, p. 69). 13 So Kramers stayed on in Copenhagen, and the first problem which Bohr asked him to solve was to calculate the Fourier coefficients of the electron orbits in the hydrogen atom when perturbed by a static (external) electric field. Simultaneously, both Bohr and Kramers began to investigate the structure of the helium atom. At that time Bohr 10 W h e n Bohr assumed his professorship at the University of Copenhagen in September 1916, he had only a single room next to the physics library of the old Polytechnical Institute. A schoolfriend of his, Aage Berleme, started a private initiative to collect money for buying the land for a new physics institute already in 1917. With the help of further private and official contributions the construction of the building began in 1919. 11 Although the building was completed at that time, the equipment of the laboratory for spectroscopic investigations still had to be installed. It would take many years and absorb the energies of Bohr and his collaborators. Nevertheless, the experimental investigations had already been started; for example, James Franck, who had spent several weeks at Copenhagen before March 1921, had set up an apparatus for the observation of collisions between electrons and atoms. George de Hevesy, on the other hand, had done experiments during 1920-1921 at the physical chemistry laboratory of the University of Copenhagen. 12 I n visiting Copenhagen and Niels Bohr, Kramers may have been advised by Paul Ehrenfest, although Ehrenfest in 1916 was not yet in favour of Bohr's atomic theory. Oskar Klein recalled later that Kramers had told him the following in connection with his visit to Copenhagen: 'There was a student meeting at Copenhagen.... It was an aunt of his who invited him to go to t h a t meeting and he went t h e r e . . . . He told Ehrenfest about it, and Ehrenfest just said that he ought to visit Bohr. Kramers very much liked to see people, so he went up to Bohr right at the beginning' (Klein, Conversations with Mehra). 13 Oskar Klein recalled later t h a t Kramers had told him the following story about becoming Bohr's assistant: 'He [Kramers] went to the student meeting after the [first] talk with Bohr, and then he spent all his money. He had no money to go back to Holland, so he went back to Bohr and asked him if he could borrow some money from him. Then they got into further conversation, and Bohr asked him — there might, of course, have been some days in between — to be his assistant' (Klein, Conversations with Mehra).
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was just o n t h e way to developing his correspondence point of view. As Oskar Klein, one of t h e first s t u d e n t s of Bohr a n d K r a m e r s , recalled: Bohr had, of course, the beginnings of it [the correspondence principle] in his first paper [on atomic constitution (Bohr, 1913b)] already, but it hadn't occurred to him that one should be able to approximate that way the probability for transitions and especially also the absence of certain transitions.... Then Bohr saw that was immediately clear from what he called at that time the analogy principle.... I think Kramers told me that it had occurred to Bohr while on a trip he had made walking in Jutland. When he came back from that he told Kramers about these things, and then Kramers immediately tried to calculate Fourier coefficients. (O. Klein, Conversations with Mehra; also AHQP Interview) T h e work on t h e intensities of spectral lines occupied K r a m e r s for a couple of years. He finally s u b m i t t e d a long paper on it in spring 1919 as his doctoral thesis to t h e University of Leyden (Kramers, 1919). 1 4 In his second communication on atomic structure t o t h e Bavarian Academy, in which he dealt with t h e fine s t r u c t u r e of spectral lines, Sommerfeld h a d already published certain considerations on their intensities. T h e r e he had advanced t h e following argument: 'We shall assume t h a t in all cases t h e circular orbit possesses the greater probability, and t h a t a n elliptical orbit is t h e less probable, t h e larger its eccentricity' (Sommerfeld, 1915c, p . 473). 1 5 In later p a p e r s he h a d formulated this idea more quantitatively as a n 'intensity rule' (Sommerfeld, 1917). T h u s / , the relative intensity of t h e fine-structure components of hydrogen e m i t t e d in transitions between elliptical orbits, characterized by the pairs of azimuthal a n d radial q u a n t u m numbers, (m, m!) a n d (n,n'), respectively, should be given by t h e equation n
m
n + n'
m + m' '
(3)
provided t h e transitions were not forbidden. Statistical a r g u m e n t s by Karl Herzfeld a n d Arnold Sommerfeld had appeared t o support Eq. (3): while Herzfeld h a d shown, in particular, t h a t the outer orbits were rendered less probable t h r o u g h t h e action of neighbouring a t o m s on t h e a t o m under consideration (Herzfeld, 1916), Sommerfeld h a d derived Eq. (3) essentially by studying t h e degeneracy of t h e elliptical electron motions (Sommerfeld, 1917). 1 6 Still, m a n y difficulties h a d remained in Sommerfeld's approach; P a u l Epstein, for instance, h a d noted in connection with t h e intensities of t h e S t a r k components of the hydrogen lines t h a t , 'In our a t t e m p t s we were not successful in establishing a general point of view concerning t h e probability for various orbits; we can just report on negative experiences' (Epstein, 1916c, p . 517), and 14
Kramers went to Leyden together with Niels Bohr in April 1919 to receive his doctorate. This assumption was not only consistent with the absence of the pendulum orbit — because of infinite eccentricity it was highly improbable — but also explained the observed intensity ratios of the relativistic doublet components fairly well. 1 In general, the electrons in an atom could move in three dimensions. Each elliptical motion in the plane, therefore, corresponded to a degenerate motion; Sommerfeld determined the degree of degeneracy for each orbit and used it to derive an equation for the intensity similar to Eq. (3). 15
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Sommerfeld had concluded that 'dynamical emergency measures' were necessary to explain Stark's observations (Sommerfeld, 1917, p. 109). For this reason Bohr's approach seemed to be superior, because it made use of the analogy to the (classical) electrodynamic theory of the emission of spectral lines (Bohr, 1918a). In his thesis Kramers applied the correspondence arguments systematically to the problem of the intensity of hydrogen lines, both to the fine structure and to the Stark components (Kramers, 1919). For this purpose he calculated the coefficients of the Fourier series, C Tl ,T2, into which the position coordinates, x, y, and z, of the electron's motion in the atom could be expanded — e.g., z — ETliT:iCTltT2 exp{27ri(r1w;i +T2W2)}, with wi and wi the angular coordinates and T\ and T2 assuming the values 1, 2, 3, etc. — both in the relativistic case as well as the nonrelativistic case, where the atom was under the influence of an external field. Kramers then expressed these Fourier coefficients in terms of Bessel functions, JTl, and J T2 , depending on the arguments T\(j\ and T2CF2, where o\ and 02 denoted combinations of the action variables conjugate to iui, and u>2. In classical electrodynamics the absolute squares of the Fourier coefficients, |CV1)T2|2, described the intensities of the radiation emitted by an electron in a given orbit. Kramers now proposed to take the squares of the average Fourier coefficients, |<5 Tl)T2 | 2 , with CTltTi representing the values of the Fourier coefficients of orbits averaged between the initial and the final orbits, as describing the quantum-theoretical intensities. Thus he computed, for instance, the intensities of the hydrogen line components in the Stark effect and compared them to the data, concluding: 'On the whole it will be seen, that it is possible on Bohr's theory to account in a convincing way for the intensities of the Stark components' (Kramers, 1919, p. 341). Kramers also dealt with the intensities of the fine-structure components, which had been described successfully before by Sommerfeld. He had in mind not only to test Sommerfeld's results from the point of view of the correspondence principle, but also to explore certain limitations of Sommerfeld's theory of fine structure in connection with an earlier proposal of Niels Bohr. Bohr had suggested originally that the fine structure of spectral lines might originate from the influence of small electric forces perturbing the Coulomb attraction of the nucleus on the electron (Bohr, 1914b). After Sommerfeld put forward his theory of relativistic fine structure, Bohr did change his point of view; nevertheless he remained convinced that perturbing electric fields in atoms might have some influence on the fine structure. Therefore he asked Kramers to consider the problem in detail, and Kramers treated the effect of electric fields both in his thesis and, more completely, in a separate paper submitted to Zeitschrift fur Physik, which was received on 1 October 1920 (Kramers, 1920b). He used, in particular, Bohr's perturbation method; that is, he calculated the influence of small electric fields on the fine structure of the hydrogen lines with the help of the perturbation theory of periodic systems (Bohr, 1918a). Thus he obtained the following results. First, for small electric fields the corrections to the relativistic energy terms were proportional to the square of the electric field strength; as a consequence, each fine-structure component splits into
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polarized Stark components. 17 Second, if the electric field strength were increased, the electron would leave the plane of the unperturbed orbit and a third quantum number would have to be introduced in order to describe the energy terms; as a consequence, a complicated structure arose, mixing the Stark components and the relativistic doublets. Third, if the electric field strength was further increased, the Stark effect would dominate the line structure; again the electrons would move in a plane, a plane perpendicular to the electric field which did not contain the nucleus, and the energy terms could be described by two quantum numbers, the principal quantum number and a quantum number characterizing the Stark effect.18 These results, Kramers thought, threw some light on the understanding of the series spectra of elements of higher atomic numbers. According to Sommerfeld these spectra could be represented by the spectra emitted by a one-electron atom, whose Coulomb field is perturbed by an additional central field (Sommerfeld, 1915c, p. 478). Now Kramers argued: However, one has to remember that the comparison between the perturbing action of the inner electrons exerted on the motion of the external electron in an atom, on one hand, and the action of a central field [of force], on the other, can only be viewed as an incomplete analogy; rather one has to expect that, in investigating the influence of an electric field on the atom, the action of the internal electrons will in various respects be more complicated than the action of a simple central field. (Kramers, 1920b, p. 223) In the following years Kramers became Bohr's closest collaborator. With his mastery of mathematical formalism and his skill in solving tricky problems — such as displayed in his doctoral thesis — he was able to carry out the laborious calculations necessary to test Bohr's ideas on atomic structure. He not only performed the detailed evaluation of the energy states of the helium atom (Kramers, 1923a), but also worked on band spectra (Kramers 1923b; Kramers and Pauli, 1923) and on continuous X-ray spectra (Kramers, 1923d). As Bohr's official assistant since 1919, Kramers exhibited many other talents: especially, he was fluent in several languages (in 1917, a year after his arrival in Copenhagen, he began to give talks in Danish) and he was willing to take care of lecture courses and to advise students. The early students included Oskar Klein. Klein had met Kramers first in fall 1917 at Svante Arrhenius' Institute in Stockholm, where Kramers gave a talk on atomic theory and the adiabatic principle. Kramers impressed Klein so much that he decided to continue his studies under Niels Bohr. Thus Klein arrived at Copenhagen in May 1918 and stayed there, with some interruptions, until fall 1922. 19 'I learned a great deal 1 'Each of the i7a-components, for example, splits into three components, which are due to transitions in which the radial quantum number changes by 0 and ±2 units, respectively. 18 Kramers noted (already in his thesis) that an electric field having the strength of 300 V/cm would give rise to the Stark effect of the hydrogen, which is of the same order as the relativistic fine-structure effect (Kramers, 1919, pp. 376-377). 19 Oskar Benjamin Klein was born in Stockholm on 15 September 1894. From 1912 he studied at the University of Stockholm and the Nobel Institute for Physical Chemistry with Svante Arrhenius, obtaining his Licentiat in 1918. After obtaining his doctorate in 1922 from the University of Stockholm, Klein served as Docent there; in 1922 he went to the University of Michigan at Ann
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from Kramers,' he recalled later. 'Prom Bohr I largely heard generalities as I did not have much occasion to see him at this time. Kramers was my teacher in these new things, the Hamilton-Jacobi equations, the correspondence principle and the beginnings of the Copenhagen philosophy' (O. Klein, Conversations with Mehra). 20 Kramers helped and collaborated with most newcomers to Bohr's Institute in the early 1920s including, for instance, Wolfgang Pauli and Werner Heisenberg. He always came to know Bohr's ideas first, and in many cases he had to formulate them in proper mathematics — as happened, for example, in dealing with the fundamental considerations of a new radiation theory (Bohr, Kramers and Slater, 1924). From the latter he would derive a theory of dispersion of light by atoms, his most important contribution to quantum physics (Kramers, 1924a,b). 21 Bohr found his collaboration with Kramers on atomic theory most satisfactory, and he frequently quoted Kramers' papers (Kramers, 1919, 1920b), especially in his talk at the German Physical Society in Berlin on 27 April 1920 (Bohr, 1920, p. 447). In Berlin he gave a review of the origin of line spectra, in which he discussed the results of the erstwhile models of atomic structure in the light of the correspondence principle. There he met a large number of German physicists, who were interested in atomic and quantum theory, including Born, Pranck, Kossel, Ladenburg and Lande, and he established many fruitful contacts. A few months later Ladenburg, after giving a lecture on atomic constitution at the Bunsen Congress in Halle, asked Bohr a question concerning the occupation of the outer shells in noble gases. May I now ask [he wrote] what you think of Kossel's idea that the halogens F, CI, Br, I in their simple compounds (say with the alkalis) by capture of a single electron, just as the alkalis by losing an electron, assume the particularly stable configuration of the noble gases with eight electrons in the outer shell...? Bom's calculation of the electron affinity of the halogens and sulphur appears to me a strong support for this conception.... Or do you still prefer your old view of two electrons in the outer shell of noble gases? (Ladenburg to Bohr, 18 June 1920) Bohr replied to Ladenburg on 16 July 1920, confessing that he did 'not consider any conception sufficiently assured to make it possible to take a definite standpoint' and that the considerations in his earlier papers should 'only be regarded as a tentative orientation.' He saw difficulties in explaining the observed chemical properties from physical arguments, claiming that the question 'depends not only on the geometrical Arbor (instructor, 1923-1924; assistant professor, 1924-1925). Prom 1926 to 1933 he was a lecturer at the University of Copenhagen; then he accepted a call to the University of Stockholm as professor and director of the Institute of Mechanics. Klien retired in 1968, and died on 5 February 1977 at Stockholm. Klein worked on problems of quantum theory (e.g., the crosssed-field problem), nonrelativistic and relativistic quantum mechanics, meson theories and general relativity. 20 W h e n Kramers stayed on for several months in the Netherlands after obtaining his doctorate in spring 1919, Klein came into closer contact with Bohr. At t h a t time Bohr used to dictate to him his papers and letters. Bohr also set Klein to work on certain problems of quantum theory, the first one being a treatment of the ionized hydrogen molecule. 21 Kramers stayed on in Copenhagen until spring 1926, when he accepted the chair of theoretical physics at the University of Utrecht. In 1934 he was appointed Paul Ehrenfest's successor in Leyden, a chair which he occupied until his death on 24 April 1952 at Oegstgeest.
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character of the configuration but above all on its stability properties.' And, as a consequence of the latter, one had to give up his original ring configurations and be 'forced to reckon with far more complicated motions of the electrons in atoms.' A couple of weeks later, Bohr wrote more explicitly: 'I value the work of Kossel and others [concerning the relation of chemical properties to electronic constitution] very highly; however, I feel myself strongly cautioned by the conviction that ultimately the whole question of explaining the chemical stability cannot be regarded as primarily a geometrical problem but must rather be conceived a dynamical one' (Bohr to Ladenburg, 29 September 1920). Then, in October 1920, Alfred Lande gave a lecture at Bohr's invitation to the Physical Society of Copenhagen, in which he discussed his ideas concerning the spatial symmetry of atoms. Lande's geometrical arguments did not convince Bohr, but they persuaded him to become more deeply involved in the problem of atomic constitution. In a lecture delivered on 15 December 1920 — again before the Physical Society of Copenhagen — Bohr referred to the ideas of Born and Lande on the spatial structure of atoms, especially to Lande's recent theory of the structure of diamond (Lande, 1921a), in which Lande had assumed that the electrons in the normal state of carbon moved on very eccentric orbits. Bohr then remarked: After getting acquainted with this work through a lecture of Lande..., it has occurred to me that it might be possible from simple points of view to give a rational explanation of this at a first glance strange assumption. In fact, in the following I shall try to show that it seems possible, by pursuing these viewpoints further, to throw light on a great number of the problems mentioned in the foregoing and to create a hitherto lacking basis for an understanding of the peculiar stability of the elements, of which their specific properties so strongly bear witness. (Bohr, 1977, p. 58) He then discussed the consequences of the assumption of elliptical orbits for the ground-state electrons in the case of helium, sodium and carbon, as well as a few other elements. He expressed the hope to work out, in the following months, a detailed paper on the new theory of the constitution of atoms. However, hindered as he was by many administrative duties and poor health during most of the year 1921, he did not succeed. Instead, he submitted two letters to Nature, in which he sketched just a few points of his theory. 22 The stimulation to write a letter to Nature came from another letter which the British physicist Norman R. Campbell had published in that journal; Campbell's letter, dated 16 November 1920, had appeared in Nature's issue of 25 November 1920. In it Campbell had compared the Bohr-Sommerfeld theory of atomic structure and the Lewis-Langmuir (or the Born-Lande) theory of molecular structure and had argued that 'they are not really inconsistent' (Campbell, 1920, p. 408). 22 Due to ill health Bohr did not attend the third Solvay Conference in Brussels in April 1921. Instead Paul Ehrenfest presented Bohr's report, entitled 'L'application de la theorie des quanta aux problemes atomiqu.es' ('The Application of Quantum Theory to Atomic Problems,' Bohr, 1923e). In this talk Bohr did not go into any details of atomic structure, but concentrated on general points of view.
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Bohr, however, felt that Campbell had not described the situation correctly, and he wrote: 'Dr. Campbell puts forward the interesting suggestion that the apparent inconsistency under consideration may not be real, but rather appear as a consequence of the formal character of the principles of quantum theory, which might involve that the pictures of atomic constitution used in explanation of different phenomena may have a totally different aspect, and nevertheless refer to the same reality' (Bohr, 1921a, p. 104). Campbell had claimed, for instance, that the electrons in the Bohr-Sommerfeld theory do not really move on the orbits and, in justification, had cited the correspondence principle. Bohr was completely against such an interpretation. On the contrary [he wrote] if we admit the soundness of the quantum theory of spectra, the principle of correspondence would seem to afford perhaps the strongest inducement to seek an interpretation of the other physical and chemical properties of the elements [i.e., those described formerly by the Lewis-Langmuir-Born-Lande theory] on the same line as the interpretation of their series spectra. (Bohr, 1921a, p. 104) He therefore proposed a new view of atomic constitution based entirely on the correspondence principle. In fact, he claimed that the correspondence principle allowed one to establish the definite arrangement of electrons in given atoms in certain groups or shells — a problem which he had been unable to solve satisfactorily by the previous methods. He wrote: Thus by means of a closer examination of the progress of the binding process this principle offers a simple argument for concluding that these electrons are arranged in groups in a way which reflects the periods exhibited by the chemical properties of the elements within the sequence of increasing numbers. In fact, if we consider the binding of a large number of electrons by a nucleus of higher positive charge, this argument suggests that after the first two electrons are bound in one-quantum orbits, the next eight electrons will be bound in two-quanta orbits, the next eighteen in three-quanta orbits, the next thirty-two in fourquanta orbits. (Bohr, 1921a, p. 105) An essential feature of the new view of atomic structure was the fact, which had been suggested already by Lande in connection with his theory of diamond, that 'for each group the electrons within certain sub-groups will penetrate during their revolution into regions that are closer to the nucleus than the mean distances of the electrons belonging to groups of fewer-quanta orbits' (Bohr, 1921a, p. 105). This led to a coupling between the electrons of different shells, which would establish 'the necessary condition for the stability of atomic configurations' (Bohr, 1921a, p. 105). Bohr's letter to Nature, dated 14 February 1921 and published in the issue of 24 March 1921, of which he sent copies to various interested physicists, received an enthusiastic response from many sides. Thus Lande wrote to Bohr that his new ideas were of the greatest significance for the future of atomic theory. 'It seems to me,' he said, 'that until the appearance of your detailed account, it makes no sense at all to work theoretically on atomic theory' (Lande to Bohr, 21 February 1921). Sommerfeld, who agreed with Lande about the importance of Bohr's proposal,
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immediately wanted to make use of it: he asked his student Gregor Wentzel to work out its consequences in the theory of X-ray spectra. Kasimir Fajans wrote from Munich: 'The announcement [in] your letter to 'Nature' of the solution of the riddle of the periodic system has filled us chemists with particularly great expectations' (Fajans to Bohr, 25 June 1921). All these people waited eagerly for the detailed account of the theory which Bohr had promised in his letter to Nature. 'I am especially happy,' Walther Kossel expressed their hope, 'that we may now expect your complete exposition of this subject very soon' (Kossel to Bohr, 15 August 1921). Instead of the expected memoir, Bohr wrote a second letter to Nature on 16 September 1921. It appeared in Nature's issue of 13 October (Bohr, 1921d), and in it he clarified some points of his previous letter; in particular, he modified certain results by assigning slightly different quantum numbers to the electrons in the outer shells. However, Bohr gave the first complete presentation of the theory of the periodic system in his Wolfskehl lectures, which he delivered in June 1922 at Gottingen. 23 Since its beginning in 1913, Bohr's theory of the atomic constitution had been followed in Gottingen with considerable interest. Especially, the mathematicians David Hilbert and Richard Courant had immediately taken a positive attitude towards it. 24 Then Peter Debye, Woldemar Voigt's successor in Gottingen, had worked on some aspects of Bohr's theory, especially on a model of the hydrogen molecule (Debye, 1915a), the Zeeman effect (Debye, 1916a,b), and X-ray spectra (Debye, 1917), and so had his student Paul Scherrer (Scherrer, 1916b). 25 When Debye left Gottingen, Max Born and James Franck worked actively on Bohr's theory. Hilbert also became involved in it, and got his student Hellmuth Kneser to treat certain problems connected with this theory (Kneser, 1921). In order to become better acquainted with the most up-to-date progress of the subject, Hilbert got an idea: he proposed to continue the programme of the Wolfskehl lectures and to invite Bohr to speak on the theory of atomic structure. 26 On 10 November 1920 23
Bohr had first talked publicly on matters connected with his theory of the periodic system on 15 December 1920 at the Physical Society of Copenhagen. (A handwritten manuscript of this talk and fragments of some notes concerning it are in the Bohr Archives; they were published under the title 'Some Considerations of Atomic Structure' in Bohr's Collected Works, Volume 4, 1977, pp. 43-69). T h e next opportunity arose in connection with the third Solvay Conference in April 1921. Bohr had planned to attend the Conference and to give a report on 'The Application of Quantum Theory to Atomic Problems.' However, being unable to go to Brussels, he sent the manuscript of the first part of his talk to Paul Ehrenfest, who presented it at the Conference (Bohr, 1923e). Bohr had promised to send the second part dealing with the details of atomic models, but he never did so; the notes dealing with it were published in his Collected Works, Volume 4 (Bohr, 1977, pp. 91-174). Evidently, these notes constituted a preliminary version of what Bohr later discussed at Gottingen in June 1922. T h e attitudes of Hilbert and Courant towards Bohr's theory of atomic constitution have been described by Constance Reid in her biographies of the two men. (See Reid, 1970, p. 135; 1976, p. 45.) 25 I n this context, it should be mentioned that the old Eduard Riecke, in his last paper, discussed Bohr's theory of hydrogen and helium (Riecke, 1915). 26 As we have mentioned earlier, the funds available from the annual interest on Wolfskehl endowment had been augmented by extra funds from the Prussian Ministry of Education; this was to
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the Wolfskehl Commission, consisting of Felix Klein, David Hilbert, Carl Runge, Emil Wiechert, Ludwig Prandtl, Edmund Landau, Johannes Hartmann, Richard Courant and Robert Pohl, sent a letter to Niels Bohr proposing a date in the following year. Bohr accepted, but because of ill health he had to postpone the visit to June 1922. 27 In the period from 12 to 22 June 1922 Bohr delivered seven lectures on the theory of atomic structure. He had carefully prepared these lectures, knowing that many scientists would come to Gottingen to listen to his latest results and ideas. 28 The Wolfskehl lectures, as a rule, were announced publicly and interested people were invited from all German universities. So the audience did not consist only of the scientists from Gottingen: the mathematicians Richard Courant, David Hilbert and Carl Runge, the physicists James Franck, Max Born and Robert Pohl and their students — for example, Gustav Heckmann, Carl Hermann, Erich Huckel, Friedrich Hund, Pascual Jordan and Rudolph Minkowski. Arnold Sommerfeld from Munich brought with him Werner Heisenberg; Wilhelm Lenz and his assistant Wolfgang Pauli travelled from Hamburg to Gottingen; Alfred Lande, Walther Gerlach and Erwin Madelung came from Frankfurt; and Paul Ehrenfest came from Leyden. Bohr himself was accompanied by Oskar Klein and Wilhelm Oseen.29 Altogether about one hundred people attended the lectures and participated in the events of what soon came to be called the 'Bohr Festival' QBohr Festspiele'). Thirty-five years later Friedrich Hund recalled Bohr's lectures as follows: Bohr did not speak very clearly, and we junior people were not allowed to sit in the front rows reserved for important guests; thus we strained to hear with our ears turned forward [towards the speaker], fighting our hunger for supper. Of course, we had read a little in Sommerfeld's Atombau und Spektrallinien, also Debye in 1920 had given a course of lectures on quantum theory (in an unheated lecture hall); but what Bohr presented sounded quite different, and we felt that it was something very important. The glamour that surrounded enable the University of Gottingen to invite a prominent scientist as guest lecturer for up to a semester. This arrangement was renewed after World War I. 2 When Sommerfeld heard from James Franck about Bohr's ill health, he wrote anxiously to Copenhagen: Dear Bohr: Ten years ago when Hilbert was overworked after completing [his theory of] integral equations and had to go to a sanatorium, I wrote to him: 'The mathematical kingdom, which you have established, is already worth converting for (erne Mess Wert).' ['Paris is well worth a Mass.' (Henry IV of France)] I want to tell you t h e same, for I have heard from Franck that you have gone on strike with your capacity to do work. Your mathematical-physical kingdom will last even longer and contain more citizens than Hilbert's integral-equations empire. Do not think at all that your present work — stoppage is something serious or special. I find it only natural that you have to pay the human tribute for your most recent discoveries, which certainly have demanded an immense concentration of thought. And I would pay it with pleasure, if only such deep insights were given t o me. (Sommerfeld to Bohr, 25 April 1921) 28 For example, Bohr had invited Adalbert Rubinowicz to come to Copenhagen in spring 1922 to help him prepare these lectures. (See Bohr to Rubinowicz, 3 January 1922.) 29 T h e three Scandinavians stayed together in a Pension in Gottingen. (See Klein, AHQP Interview, p. 22.)
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this event [Bohr's lectures] cannot be communicated in words today; for us it was as brilliant as the Handel-Festival of those days in Gottingen. (Hund, 1961, p. I ) 3 0 R u d o l p h Minkowski, assisted by Erich Hiickel, was asked to p r e p a r e notes of t h e lectures (see Minkowski, A H Q P Interview, p . 1). T h e y prepared a typed manuscript, which gave a detailed account of what Niels Bohr presented at Gottingen. In his first lecture on Monday, 12 J u n e 1922, Bohr introduced his audience t o t h e fundamental ideas underlying his theory of atomic structure. ' T h e present s t a t e of physics,' he began, 'is characterized by t h e fact t h a t we n o t only are convinced a b o u t t h e reality of atoms b u t also believe t h a t we possess a detailed knowledge of their building stones. I shall not discuss here the development of our conceptions of these building stones, nor Rutherford's discovery of atomic nuclei, which initiated a new epoch in physics' (Bohr, 1977, p . 343). 3 1 He t h e n sketched t h e picture of t h e nuclear a t o m consisting of a positively charged nucleus having Z positive elementary charges |e|, a r o u n d which a n u m b e r of electrons (with negative charge e) revolve; all building stones, t h e electrons a n d the nucleus, could be regarded as point masses. In describing t h e properties of a t o m s , Bohr said, 'it is not possible t o make any progress with classical electrodynamics' (Bohr, 1977, p . 344). It failed t o account b o t h for t h e observed stability of the a t o m a n d the e m i t t e d radiation; only t h e q u a n t u m theory would explain t h e basic features of the a t o m . T h u s Bohr established a n atomic theory resting on two fundamental assumptions: t h e first one being t h e existence of stationary s t a t e s , i.e., of certain definite mechanical motions in t h e atom, from which t h e electrons cannot emit radiation; t h e second one stating t h a t a change of t h e energy in t h e a t o m h a p p e n s only t h r o u g h a transition from one s t a t i o n a r y point to another, a n d t h a t it is connected with t h e emission (or absorption) of radiation of a uniform frequency, given — u p to a factor 1/h [h being Planck's constant) — by t h e energy difference between t h e states. Bohr d e m o n s t r a t e d t h a t t h e application of these a s s u m p t i o n s led immediately to a satisfactory description of t h e spectra of t h e hydrogen a t o m and t h e helium ion. I n his second lecture on Tuesday, 13 J u n e , Bohr addressed himself t o t h e peculiar situation existing in the q u a n t u m theory of atomic structure. He said: So far, only concepts developed in the classical theories, such as those of the electron and electric and magnetic forces, are available to us for describing the natural phenomena; however, we assume at the same time that the picture of the classical theories is invalid. Now the question arises if there is any possibility at all of uniting the classical concepts with the quantum theory without contradiction. [So far the question had not been really settled.] However, physicists hope that the ideas of both theories possess a certain reality. (Bohr, 1977, p. 351) 30
Hund explained further: 'They [the lectures] took place, I believe, in lecture hall No. 15 of the Auditorienhaus. There were, as far I remember, perhaps 150 seats, and these were certainly occupied I still recall that we were excluded from the first several rows, which were reserved for the members of the German Physical Society...' (Hund, AHQP Interview, 25 June 1963, p. 1). 31 The English translation of the Minkowski-Hiickel notes of Bohr's lectures is included in Bohr's Collected Works, volume 4, 1997, pp. 341-419.
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The contradictory situation led, of course, to 'formidable difficulties' in formulating the quantum principles, and he proposed to use the most cautious procedure, namely, to apply the principles in practical cases and to compare the results with the observations. He then proceeded to develop the basis of the mechanical treatment of atomic systems and the radiation emitted (or absorbed) by them. The atoms, he emphasized, had to be considered as multiply periodic systems, whose motion could be expanded as a Fourier series of harmonic terms involving higher harmonics of the fundamental frequencies u>i, the time derivatives of the angle variables Wi(i = 1 , . . . , / for a system of / degrees of freedom). In the stationary states of the system the canonically conjugate variables Jj were determined by rtih, the integral multiples of Planck's constant, which provided the condition of stability. 32 Bohr then expressed the action variables by phase integrals and discussed their relation to the frequencies. Thus he showed that in the limit of high quantum numbers the frequencies approached their classical values: i.e., for large n' and n" and small \n' — n"|, the relation v = j-fu6J={n'-n")v
(4)
followed. As an example, Bohr calculated the fine structure of a one-electron atom, together with the selection rules. In all these considerations, he pointed out, the influence of the radiation on the stationary orbits was neglected. If this were taken into account, the energy of the states as well as the frequencies were not accurately determined; hence the question arose whether the frequency condition remained valid. Bohr found the resolution of this apparent puzzle to lie in the fact that both the energies and the frequency occurring in a quantum system were determined 'to just the same degree of accuracy' (Bohr, 1977, p. 361). He also claimed that quantum theory must fail to describe the radiation in wireless telegraphy 'because of the immense magnitude of the radiative forces involved' (Bohr, 1977, p. 362). In his third lecture on Wednesday, 14 June 1922, Bohr turned to several applications of the principles of atomic theory, especially of the correspondence and the adiabatic principles. After clarifying the range of validity of the adiabatic principle, Bohr treated the problems of atoms in static magnetic and electric fields, deriving the known results, including Kramers' results on the influence of an electric field on the fine structure of atoms (Kramers, 1920b). With this background preparation Bohr attacked, beginning with his fourth lecture on Monday, 19 June 1922, his main problem: the construction of a theory of the periodic system of elements. That one could do so was not yet generally accepted. Of course, scientists did possess important experimental data concerning the properties of many-electron atoms, their spectra and their chemical behavior. Already 32
B o h r did not hide the difficulty of understanding the stability of atoms against collisions. 'Ordinary mechanics,' he explained, 'is incapable of describing what happens in a collision. T h e reason for this fact is easily perceived. In mechanics, the state of a system determines directly only the states that are inflnitesimally adjacent in time; here, however, the final state is determined beforehand and can be defined only by a consideration of the entire motion' (Bohr, 1977, p. 353).
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more than half a century earlier the chemists Lothar Meyer and Dmitri Ivanovich Mendeleev had proposed the ordering of the elements in a periodic system. Later, new elements had been discovered and the periodic system was improved and completed. Towards the end of the second decade of the twentieth century the periodic system contained nearly ninety elements, from hydrogen to uranium, which were distributed according to increasing atomic numbers into seven periods, each period — except the last one — ending with a chemically inactive noble gas: the first by helium, the second by neon, the third by argon, the fourth by krypton, the fifth by xenon and the sixth by radon (which was usually called niton in those days). After Bohr's theory of the atomic constitution became known, people had tried to explain the periodic system on its basis; thus, for instance, Walther Kossel had discussed the chemical properties of atoms as exhibited in the constitution of molecules (Kossel, 1916a) and Lars Vegard had examined X-ray spectra in detail (Vegard, 1917a,b). 33 Special questions, such as the occurrence of more than eight elements in the so-called long periods had been considered by Rudolf Ladenburg (1920). Bohr had followed these attempts with great interest and, after he had developed the correspondence principle, he felt ready to enter into a discussion of the problem himself. The basis of his practical method consisted in what he called the 'Building-up Principle' (or 'construction principle,' the 'Aufbauprinzip'; see, e.g., Born, 1925, p. 211), that is, the idea that the structure of a given atom can be obtained by considering the successive bindings of electrons to the nucleus. 34 With this method Bohr approached in the early 1920s the explanation of the periodic system, and he presented the results in detail at Gottingen. Of the existing formulations of the periodic system he preferred the one which had been given earlier by his fellow countryman Julius Thomsen because it seemed to be 'particularly well adapted' for his purpose (Bohr, 1977, p. 388). 35 Thomsen had ordered the elements in horizontal groups and In these treatments the periodicity of certain physical properties, such as t h a t of the atomic volume (observed already by Lothar Meyer, 1870) and of the electrical conductivity (Benedicks, 1916), provided valuable help. 34 T h e importance of this principle in Bohr's work was recognized, for instance, by Hendrik Lorentz, who — in the discussion of Bohr's report at the third Solvay Conference — drew attention to the fact that Bohr had not so much discussed 'how the atom is constituted, but how it can be formed' (Lorentz, Rapports et Discussions du Conseil de Physique tenu a Bruxelles de l e r au 6 Avril 1921, p. 257). Bohr agreed with Lorentz' characterization of his method and stated explicitly in his talk at the Physical Society of Copenhagen on 21 October 1921: 'We attack the problem of atomic constitution by asking the question: 'How may an atom be formed by the successive capture and binding of the electrons one by one in the field of force surrounding the nucleus?' (Bohr, 1921e; English translation, Collected Works, Volume 4, 1977, p. 277). Later, he formulated the method of obtaining a picture of the atomic constitution by 'following the process of the building-up of atoms through successive capture of the electrons' by means of what he called the 'postulate of the invariance and permanence of quantum numbers' (Bohr, 1923c, p. 256). This postulate required that, when a new electron is captured, the quantum numbers of the electrons already bound do not change, except t h e ones which describe the orientation relative to the new electron. 35 H a n s Peter Jorgen Julius Thomsen (1826-1909) was professor of chemistry at the Copenhagen Polytechnic (1847-1856), at the Military High School (1856-1866) and at the University of Copenhagen (1866-1891). He worked on such chemical problems as the law of mass action of Guldberg and Waage and the properties of acids. Bohr had learned about Thomsen's periodic system during his university studies.
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118-
Fig. 1. Periodic System of Elements According to Julius Thomsen.
vertical periods and had predicted on its basis the group of noble gases (Thomsen, 1895). Bohr showed, at the beginning of his fifth lecture on 20 June 1922, a table representing Thomsen's scheme (see Fig. 1). It was this scheme which he tried to justify. The hydrogen atom and its properties had been satisfactorily described on the basis of the quantum theory of atomic structure. In all other cases, however, the situation appeared to be completely unclear. The empirical data from atomic spectra provided, of course, the most important hints concerning the atomic structure. It had been known since long that the spectra of all elements could be represented by a Rydberg-Ritz formula of the type R
[n" +
\n'' + 4>k>(n')}2)
(5)
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where the denominator, [n +
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ground state of lithium, Bohr concluded that the two inner electrons were bound as in the helium ground state, i.e., in two circular li-orbits (where l i corresponds to both the principal and the axial quantum number being equal to one), but that the outer electron assumed a 2x-orbit. Since the latter comes very close to the l i orbits of the inner electrons, the s-term of lithium should deviate strongly from the hydrogen ground state; but for higher states, in which the outer electron is further away from the helium-like core, the similarity of hydrogen and lithium states must increase. Bohr derived the doublet structure of the lithium lines from the fact that the core possessed the angular momentum 1 (in units of h/2ir): it would add to the angular momentum of the outer electron to yield the total angular momentum 2 in the case of the ground state, but two possible angular momenta, namely, 2 and 3, in the case of the p-state. Thus the singlet nature of the s-term and the doublet nature of the p-term seemed to follow — although Bohr admitted that 'this argument is quite uncertain' (Bohr, 1977, p. 391). In connection with the explanation of the lithium doublets Bohr also referred to the model which Werner Heisenberg had used in the discussion of the anomalous Zeeman effect of doublet spectra (Heisenberg, 1922a). He called Heisenberg's paper a 'very promising' attempt, but found it at the same time difficult to 'justify Heisenberg's assumptions,' which involved halfintegral quantum numbers (Bohr, 1977, p. 391). Proceeding to the beryllium atom, Bohr pointed out that it did not possess any simple spectra consisting of singlet lines. He assumed that the two inner electrons in beryllium were bound in a manner similar to that of lithium (or helium) atom, only just a bit tighter, and that the two outer electrons moved in elliptical 2iorbits. These orbits were possible because the first outer electron was supposed to revolve very slowly, thereby shielding off the inner system only poorly. 36 In the next atom, boron, there existed three outer electrons, whose description seemed to present considerable difficulty in contrast to the carbon atom having four outer electrons. Bohr claimed that all these electrons 'are bound in 2-orbits which are oriented relative to one another in such a way that their normals are directed towards the corner of a regular tetrahedron with its midpoint in the nucleus' (Bohr, 1977, p. 392). 37 He then discussed the constitution of the succeeding elements by referring to the next stable atom, neon, of whose eight electrons four were supposed to move in 2i-orbits and the other four in 22-orbits. 38 The structure of neon was also supposed to be extremely symmetric, i.e., 'the orbits of these eight outer electrons are arranged in two configurations of tetrahedral symmetry, since then at certain instants two orbital planes would coincide' (Bohr, 1977, p. 394). The reason for the new 22-orbits 36
According to Bohr, the motion of the two electrons did not occur as in Sommerfeld's Ellipsenverein; rather, while one electron is far away from the nucleus, the other comes very close to it, closer than the inner electrons. 37 B o h r stressed the difference of his model of carbon with Alfred Lande's. Lande had replaced the inner system (the helium-like core) by a point charge concentrated in the nucleus, while in Bohr's model the outer electrons penetrated into the inner system. 38 Evidently, nitrogen corresponded to a triply-ionized, oxygen to a doubly-ionized and fluorine to a singly-ionized neon atom.
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had to be found in the fact that more than four electrons could not be bound — due to considerations of stability similar to the ones of Joseph John Thomson — in 22-orbits. Since the electrons were bound in the circular 22-orbits more firmly than the ones in the 2i-orbits, the electronegative properties of nitrogen, oxygen and fluorine followed immediately. The same construction principles, which were applied in the second period, also worked in the third period. Prom the spectrum of sodium one derived — as in the case of the lithium spectrum — that the s-term deviated strongly from the corresponding hydrogen term, while in the higher terms the difference became smaller and smaller. Erwin Schrodinger, starting from Lande's theory of electron shells (Lande, 1920d,e), had attempted to explain the alkali s-terms by assuming that the orbit of the outer electron dives into the region of the innermost electron group. Thus its orbit could be thought to consist in a good approximation 'of pieces of elliptical orbits, which occur when the effective nuclear charge is 1 (at the periphery) and 9 (in the interior), respectively' (Schrodinger, 1921, p. 348). 39 By carrying out a detailed calculation, Schrodinger had arrived at the conclusion that the orbit of the external electron in sodium had to be a 2i-orbit, because orbits with larger principal quantum number than 2 would not be able to penetrate into the interior of the atom. Schrodinger, who knew about Bohr's letter to Nature before its publication — probably because Bohr had sent a copy of it to Ladenburg in Breslau, where Schrodinger was at the time — had sent a reprint of his paper on the diving electron orbits to Copenhagen. In a letter he had pointed out that it seemed 'to fit very well' with Bohr's new views on atomic structure, especially 'on the necessary interaction between the electrons of different 'shells" (Schrodinger to Bohr, 7 February 1921). And Bohr replied: 'Your paper in Zeitschrift fur Physik naturally interested me very much. By the way, I made the same consideration myself long time ago and carried out the corresponding calculations' (Bohr to Schrodinger, 15 June 1921). And he had added that he had presented certain results concerning the lithium atom in his 39
E r w i n Schrodinger was born on 12 August 1887 in Vienna. From 1906 he studied physics at the University of Vienna under Fritz Hasenohrl and Egon von Schweidler, obtaining his doctorate under the latter in 1910. Prom 1911 to 1914 he became assistant to Franz Exner (1849-1926), the Director of t h e Physics Institute, who worked on spectral analysis, colorimetry, atmospheric electricity and Brownian motion. He got his Habilitation in 1914, then served in World War I for the next four years. In 1918 he returned to Vienna as Privatdozent (1918-1920). In 1920, he went to Jena as assistant of Max Wien, but shortly afterwards he became Extraordinarius at the Technische Hochschule, Stuttgart. In the same year he received a call to the University of Breslau as full professor, from where he moved to the University of Zurich in 1921. Six years later Schrodinger succeeded Max Planck in the chair of theoretical physics at the University of of Berlin. With Hitler's coming to power, he left Germany in 1933 and accepted a position at the University of Oxford; then in 1936 he went to the University of Graz. He left Graz in 1938 to join the Institute for Advanced Studies in Dublin (1939-1956). In 1957 a special chair was created for him at the University of Vienna. He died on 4 January 1961 in Vienna. Schrodinger worked on many topics of theoretical (and some of experimental) physics: electricity and magnetism, dispersion of light, X-ray diffraction, problems of specific heat, Brownian motion, colour theory, atomic theory and general relativity. His greatest work was the invention of wave mechanics, the alternative scheme of quantum mechanics.
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lecture at the Physical Society of Copenhagen on 15 December 1920, but had not been able to publish anything. Then, a year later, in his fifth Wolfskehl lecture, Bohr discussed the problem again and arrived at a different conclusion from Schrodinger's with respect to the orbit of the outer electron in sodium. He said: 'However, as with the binding of the third and seventh electron, we must assume that, because of the peculiar stability of the electron configuration already present, the eleventh electron will now be bound in a new type of orbit, namely, in a 3i-orbit. Considerations of the firmness of the binding of this last electron lead us to the same conclusion' (Bohr, 1977, p. 394) 40 Bohr, like Schrodinger, viewed the orbit of the outer electron in the sodium atom to consist of two parts: the part lying in the region of the inner electrons, which differs only little from a 2i-orbit, and the outer part, which must at least be a 3i-orbit. 'By considering the polarizability of the atomic core, as in the case of lithium,' he said, 'we can estimate its value for a definite principal quantum number and compare it with the polarizability of neon as calculated from its dielectric constant. The best argument between these polarizabilities is obtained when the principal quantum number 3 is assigned to the largest p-term' (Bohr, 1977, p. 394). The whole picture of diving or penetrating orbits ('Tauchbahnen'), consisting of an external part and an internal part, led to the term structure, R/(n+atk)2, for the alkali atoms, with o^ denoting a number nearly independent of the principal quantum number. 41 The sum, n + a t , represented what Schrodinger called the 'apparent quantum sum' ('scheinbare Quantensumme') and its difference with the principal quantum number the 'quantum defect' (' Quantendefekt,' Schrodinger, 1921, p. 349). With these considerations the theory of non-hydrogenlike spectra reached a new level of generality. Bohr then quickly worked out the constitution of the elements in the third period. Thus he explained the spectrum of magnesium as being caused by a magnesium core — a system similar to the sodium atom having eleven electrons — plus the twelfth (i.e., the series) electron; he assumed the latter as being bound also in a 3i-orbit. 42 Going on to aluminum, he concluded from the spectroscopic information that 'the term corresponding to the firmest binding is not an s-term, but a p-term, to which we must assign a 32-orbit' (Bohr, 1977, p. 399). With respect to the next element, silicon, Bohr claimed that he could not yet decide whether the four outer electrons were on the 3i-orbits or whether, due to the stronger mutual perturbation of the electrons (which move in closer orbits because of the higher nuclear charge), there were two separate groups consisting of two electrons each (in 3i- and 32-orbits). The decision was even more uncertain in the case of the 40 T h i s assignment was confirmed by A.T. van Urk, who continued Schrodinger's calculations (Urk, 1923). 4 ^ T h e penetrating orbits ('durchdringen', Schrodinger, 1921, p. 348) were later called 'orbits of the second kind' ('Bahnen zweiter Art,' Bohr, 1923c, p. 257) or 'diving orbits' (' Tauchbahnen', see, e.g., Born and Heisenberg, 1924b, p. 391). 42 T h e fact that two types of spectra, singlet and triplet, existed, reminded Bohr of the situation in helium; he argued that such a situation would arise whenever a new electron was added in the same type of orbit as the preceding one.
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elements phosphorus, sulphur and chlorine, while in the case of argon the eight external electrons fell — as in the case of neon — into four 3i-orbits and four 3 2 -orbits. 43 The fourth period presented a new problem to the theory, for it contained in its middle the family of ferrous metals, in which the elements following each other in the periodic system of elements differed only very little in their chemical properties. Bohr concluded from absorption data that the nineteenth electron (in the potassium atom) must be bound in a 4i-state — it had to be an s-state, and the 3i-state was excluded because it did not lie outside the argon-like core. 44 In the calcium atom the twentieth electron had also to be bound in a 4i-state; however, as the nuclear charge increased, the 33-orbits became more stable than the 4i-orbits because, as Bohr argued, they were drawn into the interior of the atom. The possibility to occupy either the 4i- or the 33-orbits then explained the great length of the fourth period. 'In the elements following after scandium,' he concluded, 'the still empty 3-quantic orbits are gradually occupied' (Bohr, 1977, p. 401). As a consequence, for example, the spectra of scandium differ greatly from that of aluminum, in agreement with observation. The noble element krypton at the end of the fourth period should have four outer electrons in 4i- and four electrons in 42-orbits. There remained a difficulty to be explained: why must the fully developed 3-quantic orbits contain 18 electrons? Bohr suggested this number (18) might perhaps be achieved by a perturbation of the 3i- and 32-orbits due to the presence of the 33-orbits; this perturbation might lead to an occupation of all 3-quantic orbits, with each — the 3i-, 32- and 33-orbit — being filled with up to six electrons. Bohr said: 'It is possible to form an idea, albeit rather uncertain, of why there appear just three groups of six electrons each. By means of a stereographic projection, it is easy to see that only in such an arrangement can the coincidence of orbits be avoided' (Bohr, 1977, p. 402). He proposed to test this idea by analyzing the X-ray spectra in detail. Due to the particular situation in the fourth period, the Kossel-Sommerfeld law was also violated: so, for example, the arc and the spark spectra of manganese were very similar. Bohr noticed that his conception had some similarity with an earlier proposal of Rudolf Ladenburg, who had described the constitution of the elements from scandium to nickel by introducing intermediate shells (Ladenburg, 1920). 'However,' he added, 'we do not assume a definite intermediate shell as something lying between the outermost and inner orbits; rather, we have found a real reason for the peculiarity of the fourth period. When we employ the quantum theory to explain the periodic system at all, we must be prepared beforehand that something new must happen here; for, from the theory of the hydrogen atom, it is impossible to exclude the occurrence of 33-orbits' (Bohr, 1977, p. 402). In any case, the picture of the structure of elements from scandium to nickel, with the nonclosed 4 Bohr found t h a t the circular 33-orbits gave diameters of the atom t h a t were too large compared to the d a t a of the kinetic theory. The effective or apparent quantum number of the normal state was equal to 2 in the first approximation.
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subshells, did account for the observed colours of their compounds and, to some extent, for the paramagnetic properties of their ions. 45 In the fifth period, ranging from rubidium to xenon, the situation was very similar to the fourth period: the 37th electron in rubidium had to be bound in a 5i-orbit, and the same for the 38th electron in strontium. Further, there existed the group of elements between yttrium and palladium, in which the 43-orbits had to be filled successively. In xenon, Bohr assumed a symmetric configuration of four electrons in 5i-orbits and four electrons in 52-orbits. A more complicated situation occurred in the sixth period. The period started out, as usual, with an alkaline and alkaline-earth element, i.e., the 55th and 56th electrons could be bound in 61-orbits; the 57th electron in lanthanum was bound in the 53-orbit. But then the group of rare earths exhibited a novel situation: the occurrence of 4j-orbits, which were successively occupied by one electron (Ce), two electrons (Pr), up to eight electrons (Yb). Then came the platinum metals (from lutetium to platinum); in them, according to Bohr, increasingly the 53-orbits had to be occupied, while roughly one electron remained in the outermost 61-orbit. In the following elements, from gold to niton (the radioactive noble gas, now called radon), the 61 and 62shells were filled up until six electrons occupied each orbit type. Bohr concluded that, 'contrary to the customary assumption, the family of rare earths, which begins with praesodymium, is completed with lutetium.' Hence he argued that 'the not yet discovered element with atomic number 72 must have chemical properties similar to those of zirconium and not to those of the rare earths' (Bohr, 1977, p. 405). 46 The discussion of the problem of the lowest state of many-electron atoms became even more involved in the case of the elements of the seventh and last period. On one hand, this period did not exist as a complete one in nature; the next noble gas would have the atomic number 118, while in nature the elements occurred only until uranium with atomic number 92; on the other, the calculations in the seventh period became more delicate due to the existence of an increased number of types of competing orbits. In order to facilitate his task, Bohr referred — much more than he had done earlier — to empirical facts in treating the electronic structure of atoms. Thus, from the spectrum of radium, he concluded that the 87th and the 88th electrons must be bound in 7i-orbits; and, from the absence of a family of elements similar to the rare earths in the sixth period, he concluded that the 54-orbits corresponded to a looser binding of the 89th and further electrons than in orbits of the types 61,62, and 63. He argued that in this way one could, in principle, proceed with the building-up of further atoms having arbitrarily many electrons. At the end of the sixth lecture, Bohr told his audience: 'I hope that ^"Closed shells axe usually achieved in the compounds of other elements, hence their emitted and absorbed radiation frequencies are very high. Optical frequencies can occur only with shells t h a t are not closed. T h e phenomena of paramagnetism were not yet completely understood on t h e basis of atomic theory; neverthless, Bohr argued t h a t nonclosed shells indicated paramagnetism. 46 He noticed that this fact had already been assumed by Julius Thomsen, who had associated zirconium with element Number 72 in his system (see Fig. 1).
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I have succeeded in showing that matters become simpler and simpler the farther we proceed in the periodic system. Formerly, I believed that the difficulties would become increasingly greater the more electrons there are in the atoms. However, if we do not demand too much, it actually seems that we can encounter simpler problems and fewer newer ones.' However, he immediately added a word of caution concerning the reliability of his results. 'I hardly need to emphasize,' he said, 'how incomplete and uncertain everything still is' (Bohr, 1977, p. 406). In his seventh and last lecture on Thursday, 22 June 1922, Bohr turned to a test of his views concerning the structure of atoms. He discussed, in particular, the X-ray spectra and showed that 'our assumptions suffice to explain the stability conditions also for the X-ray spectra.' He considered this fact to be 'perhaps the strongest support for the correctness of our views' (Bohr, 1977, p. 407). Before presenting his own interpretation of the data, Bohr quickly went through the development of the understanding of X-ray spectra since Moseley's pioneering work in 1913. In this review he referred to the work of Kossel, to Sommerfeld's interpretation of fine structure and the following investigations of Sommerfeld and Vegard, as well as to the more recent studies of Smekal, Coster and Wentzel, who had analyzed the complexity of the levels involved in the emission of X-ray spectra. So Dirk Coster had concerned himself since 1920 with the X-ray data obtained in Lund, especially the L- and M-series of heavy elements (Coster, 1921a,b,c). 47 Coster had succeeded in organizing all K-, L- and M-levels, finding that there existed one K-, three L- and five M-levels in each atom (Coster, 1921c), and his organization supported Bohr's new theory of atomic constitution (Coster, 1922a,b). In his scheme, Coster had introduced the division of the levels into two types, calling them a and b, respectively. Gregor Wentzel had suggested a slightly different organization by making use of a third quantum number — besides the azimuthal and radial quantum numbers — which he named the 'fundamental quantum number' (' GrundquantenzahV) and denoted by the letter m (Wentzel, 1921a,b). 48 Together with his teacher Sommerfeld, Wentzel had then separated the X-ray lines into two groups, the 'regular' and the 'irregular' doublets (Sommerfeld and Wentzel, 1921). In a given X-ray spectrum Dirk Coster was born in Amsterdam on 5 October 1889. He studied at the University of Leyden (1913-1916), at the Technician, Delft (1916-1919), at the University of Copenhagen (1919-1920), and at the University of Lund. In 1922 he obtained his doctorate under Ehrenfest at Leyden. Then he returned to Copenhagen for the following year, and in 1923 became Conservator at Teyler's Foundation in Haarlem. A year later he was appointed to a professorship in experimental physics at the University of Groningen. Coster died in Groningen on 12 February 1950. 48 Gregor Wentzel was born on 17 February 1898 at Dusseldorf. He studied physics at the Universities of Freiburg (1916-1919; except 1917-1918, when he did military service), Greifswald (1919-1920) and Munich (1920-1921). He obtained his doctorate in 1921 under Sommerfeld with a thesis on the X-ray spectra. In 1922 he became Privatdozent at the University of Munich, and four years later (in 1926) extraordinary professor at the University of Leipzig. In 1928 he succeeded Schrodinger in the chair of theoretical physics at the University of Zurich. In 1948 he accepted a professorship at t h e University of Chicago. He retired in 1968 and spent his last years in Ascona, Switzerland, where he died on 12 August 1978. Wentzel, who concentrated on the analysis of X-ray spectra in his early career, later made important contributions to wave mechanics, quantum electrodynamics and meson theory.
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the regular doublets were always followed by irregular ones; t h e regular doublets were characterized by a constant wavelength difference, AA, a n d t h e irregular ones by constant differences of the square roots of the frequencies, A-^/ZA 4 9 Bohr essentially agreed with t h e results of t h e recent investigations of X-ray spectra. T h u s he approved of t h e t h i r d q u a n t u m number involved in the description, but he did not wish t o associate it with the motion of t h e electron per se. ' T h e t h i r d q u a n t u m number,' he claimed, 'has to do only with the m u t u a l orientation of the orbits' (Bohr, 1977, p . 410). Then, for example, the fact t h a t the K-level of a n a t o m was single immediately followed, since a n electron orbit could b e oriented t o itself only in one way. 5 0 As for t h e weak X-ray lines, which Wentzel h a d a t t r i b u t e d t o t h e ionized a t o m s (Wentzel, 1922), Bohr — like Coster — preferred t o associate t h e m with t h e normal a t o m : they should occur as a result of j u m p s from a complete shell to a n incomplete one. W i t h t h e survey of X-ray spectroscopy Bohr ended his Wolfskehl lectures, in which he had a t t e m p t e d to guide his audience t h r o u g h t h e latest s t a t u s of his theory of atomic structure. In spite of the optimism evident in his lectures, Bohr did not hide the uncertainty in various specific conclusions, nor did he forget a b o u t t h e i m p o r t a n t role played by empirical facts in the construction of electron orbits. As he emphasized: 'A large amount of experimental material has contributed to the shaping of our view. It is a m a t t e r of taste, I believe, whether to p u t t h e m a i n emphasis on t h e general considerations or o n t h e bare empirical facts' (Bohr, 1977, p . 397). Bohr was t h u s perfectly aware t h a t his fundamental theoretical principles were more or less equivalent to 'bare empirical facts.' T h u s he assumed an a t t i t u d e towards the problems of atomic theory which differed considerably from t h a t of others. As Heisenberg recalled later a b o u t his first impression of B o h r ' s approach: A new feature for me was that one could speak and think about these problems in a very different way from the way which Sommerfeld thought and spoke about them. One of the things which impressed me most was the kind of intuition Bohr had — Bohr knew the whole periodic system. At the same time one could easily [see] from the way he talked about it that he had not proved anything mathematically, that he just knew that this was more or less the connection. When he talked about closed shells at this time, of course, he didn't know why they were closed. [Heisenberg realized that Bohr was aware of this situation.] He took things extremely seriously and at the same time he saw that he could not really prove things. The whole picture was vague, and it was just this vagueness in connection with the enormous force of Bohr's imagination that was attractive for a young man who wanted to do some work for himself. A young man learned that he could do a lot of work himself; that it's not finished at all. At the same time it was almost clear that the whole picture could not be much different from what Bohr said; but still, all the details had to be filled out and [some] essential features were still missing, and we [had to] find out about them. (Heisenberg, Conversations with Mehra; also AHQP Interview) Such irregular doublets had already been shown to exist by Gustav Hertz in the case of the Labsorption edges (Hertz, 1920). Wentzel had discussed them in detail in his thesis (Wentzel, 1921a). 50 Similarly he concluded that the 'uppermost level' of an atom must also be a single term.
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Many questions occurred to people in the audience concerning Bohr's way of presenting atomic theory. As Heisenberg recalled: 'It occurred to me that it was very interesting to raise criticism just to listen to what Bohr would say to this criticism. I [wanted] to see whether or not his answers would be more or less a kind of excuse or whether they would really hit some essential point' (Heisenberg, Conversations with Mehra). In particular, Heisenberg asked a question related to the quadratic Stark effect. In his third lecture, Bohr had cited Kramers' detailed calculations on the multiplet structure of hydrogen lines in weak electric fields (Kramers, 1920b). At the end of his lecture he had remarked that, while the quantum theory of atomic structure still involved many difficulties, Kramers' results should remain valid. Heisenberg, however, did not believe it. He had already thought about this problem and found that the quadratic Stark effect was almost the same phenomenon as the dispersion of light in the limit of infinitely low frequency; he had concluded, therefore, that one could just take the dispersion formula of the hydrogen atom and consider the limit. However, then one would always get resonance at the critical frequency, but this frequency did not show up at all in Kramers' result. Therefore, Heisenberg rose at the end of Bohr's lecture and told him about the above reasoning and that he did not believe in what Bohr had said about Kramers' result concerning the quadratic Stark effect. He noticed that Bohr was rather shaken by his remarks and gave an evasive answer. But Bohr remained worried and, after the lecture, he invited Heisenberg to go for a walk on the Hainberg in Gottingen. In this long walk of about three hours Heisenberg learned about how Bohr thought about the status of the entire theory. 'It was my first conversation with Bohr,' he recalled. 'For the first time, I saw that one of the founders of quantum theory was deeply worried by its difficulties' (Heisenberg, Conversations with Mehra). Enough time was reserved for the discussions of Bohr's lectures and many people, especially Sommerfeld and Pauli, actively participated in them. Sommerfeld, for instance, was still a bit skeptical about the correspondence principle. And Bohr did not know the answer to certain questions. So, when he was asked about the detailed validity of Einstein's equations involving the absorption and emission coefficients, he replied: 'Yes, this is a question which does not depend at all on the application of quantum theory, but rather on the foundations of quantum theory; and about that one knows nothing and can say nothing' (recalled by Jordan, AHQP Interview, 17 June 1963, p. 17). But in general, people were greatly impressed by Bohr: they had learned about his latest ideas and how he had arrived at his explanation of the periodic system. Everybody returned home with a deepened understanding of the problems of atomic structure. As for Bohr, he summarized the impression of his visit in a letter to James Franck by saying: 'My entire stay in Gottingen was a wonderful and instructive experience for me, and I cannot say how happy I was for all the friendship shown me by everybody' (Bohr to Franck, 15 July 1922). To which Franck replied that, 'we considered every minute of your stay as a great gift' (Franck to Bohr, 29 July 1922). The success of the Wolfskehl lectures would soon be felt by Bohr's hosts in Gottingen and by Bohr himself.
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Although Bohr's theory of the periodic system was not the first theoretical approach to this problem, it left the deepest impression on the physicists who worked on atomic structure as well as on those who took some interest in the progress of atomic theory. Thus Albert Einstein wrote to Bohr from his trip to Japan: 'Your recent investigations on the atom have accompanied me on my travel and have further increased my admiration for your mind' (Einstein to Bohr, 11 January 1923). 51 Many years later Einstein commented in some detail upon his impression of Bohr's work; in his Autobiographical Notes he discussed his own endeavors to achieve an appropriate theoretical description of the quantum phenomena which had occupied him since the publication of Planck's radiation theory, and remarked: All my attempts, however, to adapt the theoretical foundation of physics to this [new type of] knowledge failed completely. It was as if the ground had been pulled out from under me, with no firm foundation to be seen anywhere, upon which one could build. That this insecure and contradictory foundation was sufficient to enable a man of Bohr's unique instinct and tact to discover the major laws of spectral lines and of the electron-shells of the atoms together with their significance for chemistry appeared to me like a miracle — and appears to me as a miracle even today. This is the highest form of musicality in the sphere of thought. (Einstein, 1949, pp. 45-47) The obvious success of Bohr's recent considerations also persuaded Arnold Sommerfeld to change his mind towards the correspondence principle, which Bohr employed as his guiding principle in those days. While in the first two editions of Atombau und Spektrallinien he had taken a rather skeptical attitude, for the correspondence principle seemed to arise from ideas foreign to quantum theory, he presented matters differently in the third edition which came out in spring 1922. This change was immediately noticed by Niels Bohr, who thanked Sommerfeld for the copy of the book he had received and remarked: At the same time I would like to express my gratitude for the friendly manner in. which you have treated the work of my collaborators and myself. In the past years I have often felt very lonely in scientific matters, because my attempts to develop the principles of quantum theory systematically to the best of my ability have been received with very little sympathy. For me the question is not just didactic trifles, but a serious effort to obtain such an inner relationship [between the principles] that one might hope to create a secure foundation for further development. I realize how little the matters have been clarified and how clumsy I am in expressing my thoughts in an easily accessible form. My joy has therefore been so much greater, for I beheve I noticed a change in the point of view contained in the latest edition of your book. (Bohr to Sommerfeld, 30 April 1922) Sommerfeld did more than change his attitude in words alone; he also applied the correspondence principle in two papers, written jointly with his student Werner Heisenberg and submitted to Zeitschrift fiir Physik in August 1922. In the first paper Sommerfeld and Heisenberg discussed the relation between the relativistic T h e investigations of Bohr, t o which Einstein referred above, were the ones contained in the book entitled 'Drei Aufsatze iiber Spektren und Atombau,' which appeared in summer 1922 (Bohr, 1922e). These three essays included Bohr's lecture on the structure of atoms and their physical and chemical properties, given at the Copenhagen Physical Society on 18 October 1921 (Bohr, 1921e).
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X-ray doublets and the sharpness of spectral lines (Sommerfeld and Heisenberg, 1922a). Bohr had argued earlier that any quantum-theoretical calculation, yielding a difference of the energy terms smaller than the radiation losses due to classical electrodynamics, was fundamentally wrong (Bohr, 1918b, p. 67). Hence the question had to be asked whether the higher-order relativistic corrections, which appeared to contribute essentially to the L-doublet states of heavy elements, still made sense. Sommerfeld and Heisenberg now calculated the classical energy losses and connected them, via a correspondence argument, to the observed linewidth, and concluded from their result that the higher-order relativistic corrections considered so far were 'perfectly consistent' with Bohr's view concerning the radiation losses (Sommerfeld and Heisenberg, 1922a, p. 393). They made an even deeper application of the correspondence principle in their discussion of the intensity of Zeeman components (Sommerfeld and Heisenberg, 1922b). In this paper they completely followed Bohr's treatment of the magnetic effect on spectral lines, including the decomposition of the position coordinates of the moving electron in a Fourier series and the use of perturbation methods; thus they arrived at a satisfactory description of the relative intensities of doublet and triplet spectra and their Zeeman effects. Sommerfeld's enthusiasm for the correspondence principle faded in the following years as a result of the new fundamental difficulties that emerged in the theory of atomic structure. These difficulties showed up in attempts to explain various phenomena: e.g., complex multiplet structure, the anomalous Zeeman effects, the energy states of the helium atom, and the Compton effect. In all these cases Sommerfeld would assume points of view differing strongly from Bohr's. Unlike Munich, where Bohr's guidance in atomic theory no longer remained so dominant, things developed differently in Gottingen. People there, especially the mathematician Hilbert and the physicists Franck and Born, had a very great respect for Bohr. Since Niels Bohr first visited Gottingen with his brother Harald in July 1914, Hilbert had esteemed him highly as a physicist. Hilbert's regard for Bohr had even grown higher in the following years, when he himself became deeply involved in the problems of foundations of physics. Now, in the early 1920s Bohr seemed to have in hand the key to the problem of the structure of matter, which interested Hilbert particularly. As a consequence of this interest — and of Bohr's Wolfskehl lectures — Hilbert announced that he was to give a course of lectures on the mathematical foundations of quantum theory in the winter semester 1922-1923. In these lectures he discussed Bohr's theory of atomic structure, using variational methods as the main mathematical tool. 52 An even closer association with Bohr had James Franck, who continued in Gottingen his experiments on the collision between electrons and atoms, which had provided the first direct proof of Bohr's postulates of atomic theory. Franck had become acquainted with Bohr in spring 1920 in Berlin, and they had become close friends. Max Born later recalled the A detailed manuscript on Hilbert's lectures on the mathematical foundations of quantum theory (Mathematische Grundlagen der Quantentheorie) is part of Hilbert's archive (Nachlass) at the Library of the University of Gottingen.
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special relationship of Bohr and Prank as follows: Pranck was an ardent admirer of Bohr and believed in him as the highest authority in physics. I sometimes found this rather exasperating. It happened more than once that we had discussed a problem thoroughly and come to a conclusion. When I asked him after a while: 'Have you started to do that experiment?,' he would reply: 'Well, no; I have written first to Bohr and he has not answered yet.' This was at times rather discouraging for me, and even retarded our work to some degree. (Born, 1978, p. 211) Franck's work in the early 1920s concentrated on checking and proving various aspects of Bohr's theory of atomic structure, its fundamental principles as much as detailed applications. So he investigated, for example, the excitation levels of atoms of several elements, including helium and mercury, or the collisions of the second kind, whose existence had been predicted by Oskar Klein and Svein Rosseland. This work of James Franck completed and extended the fundamental investigations on electron collisions with atoms, started a decade earlier in collaboration with Gustav Hertz, for which both would share the 1925 Nobel Prize in Physics. Max Born came to be interested in Bohr's theory in a less direct manner. His first acquaintance with the models of atomic structure had arisen from the discussion of the properties of ionic crystals; these investigations had led him to discuss questions of the constitution of molecules and their formation from atoms. It was characteristic of Born's scientific work that he worked steadfastly for many years on the same subjects. Thus he did not give up in Gottingen his concern with problems of crystal dynamics and of molecules; however, now he began to take as the basis of his investigations Bohr's models of atomic and molecular constitution. For example, with his assistant Erich Hiickel, he extended the quantum theory to molecules consisting of more than two atoms (Born and Hiickel, 1923). 53 They assumed the atoms to be bound in the molecules by potentials, depending on the distance between the atomic centres; these centres performed oscillations against each other, having a small amplitude; by expanding the Hamiltonian in terms of powers of the small parameter associated with these oscillations, and by using appropriate quantum conditions, they obtained an expression for the energy of the multiatomic molecules. 54 A year later Born returned to the theory of molecules in a joint paper with Werner Heisenberg (Born and Heisenberg, 1924a). Even more systematically 53 Erich Armand Arthur Joseph Hiickel was born at Charlottenburg on 9 August 1896. He studied physics at the University of Gottingen (1914-1916, 1919-1921), obtaining his doctorate under Peter Debye; his thesis dealt with the structure of anisotropic fluids. After serving for a year as assistant (to Hilbert and Born) in Gottingen, he joined his teacher Debye at the E.T.H. in Zurich, and became Privatdozent in 1925. From 1928 to 1929 he was a Rockefeller Fellow in London and Copenhagen; the following year he received a grant from the Notgemeinschaft at the University of Leipzig. In 1930 he obtained a position as a lecturer at the Technical University of Stuttgart, and seven years later he became Extraordinarius at the University of Marburg. Erich Hiickel performed pioneering work on the theory of strong electrolytes (with Peter Debye at Zurich) and on the theory of carbon binding in organic substances. He retired in 1962 and died in Marburg on 16 February 1980. 54 For molecules consisting of more than two atoms their energy formula became very complicated; in those cases the axis of rotation could not be considered as fixed (as in the case of diatomic molecules), hence an involved interaction between oscillational and rotational motion emerged.
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than in the work of Born and Hiickel, they treated the following problem: 'What are the motions of a molecule considered as a mechanical system, and how far does the usual theory of band spectra represent these motions?' (Born and Heisenberg, 1924a, p. 2). That is, they looked on molecules as mechanical systems, consisting of nuclei and electrons in dynamical equilibrium, and approached the problem of calculating the quantum-theoretical energy of a molecule by expanding its Hamiltonian in terms of a small parameter A (whose square denoted the ratio of the electron mass to the nuclear mass). In solving the equations of motion, Born and Heisenberg restricted themselves to solutions in which the nuclei did not move farther from their equilibrium positions than by distances of the order of A times the equilibrium distances between two nuclei. Finally they brought in quantum theory by quantizing the appropriate momenta. With these conditions, and by taking into account the additional constraints of molecular constitution, they arrived at the following general results: the mechanical problem could be treated with the help of perturbation theory, the unperturbed system being represented by the motions of electrons around fixed nuclei; the first-order perturbation energy of the molecules vanished, and the second-order term (in A2) arose from the oscillations and rotations of the nuclei; the third-order energy term again disappeared, and the fourth-order term depended quadratically on the quantum numbers of the oscillational motion and in a more complicated manner on the quantum numbers of the rotation of the molecules (i.e., essentially the rotational motion of the nuclei) and of the orbits of the electrons in the molecules. The investigations of Born, Hiickel and Heisenberg on molecular structure were highly theoretical, with almost no practical application. Their style was entirely determined by the methods of Born. As in his work on electron theory or on crystal lattices, Born did not invent any new physical ideas in molecular theory, but made use of existing concepts. His main contribution consisted in carrying out detailed and involved calculations on the basis of given molecular models. These models were provided by Niels Bohr, and it was by means of the conceptual framework he had erected that Bohr guided the theoretical investigations of Born and his associates after 1922. Bohr's influence was also felt in another respect, namely, through his favourite mathematical method in the quantum theory of atomic structure. Since 1913, in discussing atomic systems which could not be described by a periodic motion with a single frequency, he had advocated the use of perturbation theory. In Gottingen the perturbation theory was then developed further in connection with atomic models, and it became the dominant method of approach there, the more so since Born had been acquainted with it through his work on crystal dynamics, performed earlier in collaboration with E. Brody. 55 In 1921 Born and Brody had 55
E. Brody, a Hungarian Jew, obtained his doctorate with a thesis on the chemical constant of monatomic gases; he submitted a paper on the content of his thesis to Zeitschrift fur Physik in June 1921 (Brody, 1921a). Brody went to Germany after World War I and was hired by Born in Frankfurt. Born wrote to Einstein in a letter: 'I am working on my article for the Encylopedia [on crystal dynamics], with Dr. Brody as my private assistant. He is a very clever man. Unfortunately
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entered into a systematic investigation of the thermodynamic properties of solids, with the goal of achieving a complete description from first principles based on their constitution (in terms of atoms and ions). They had succeeded, for example, in obtaining the energy W of a crystal lattice, whose mass points were coupled by a force involving a small anharmonic term of order A, as a power series in the parameter A (Born and Brody, 1921b). 56 They had then used the result of the second-order calculation, i.e.,
W = h ^ vknk + » 5Z VklTlkni> A:
(5)
k,l
where vy. and v\a denoted the characteristic oscillation frequencies in the crystal and nk and ni the integral quantum numbers associated with them, to account for the observed deviation of the specific heats from the value 3R. Thus Born and Brody had derived the equation cv = 3i?(l - 6aRT), (6) R being the universal gas constant and a a numerical factor (which could be smaller or larger than zero), which indeed seemed to describe perfectly the behaviour of the specific heats of solids (at constant volume) as function of the (absolute) temperature r . 5 7 Although Born applied perturbation theory in many other problems of the solid state and obtained valuable results, a number of which were reported in his encyclopedia article (Born, 1923b), this method proved to be even more important for the specific problems of atomic and molecular constitution. So Born and his students used it in particular to discuss the helium atom, the hydrogen molecule-ion and the hydrogen molecule. The rigorous treatment of these problems appeared to be very promising, and one expected quick success in confirming Bohr's theory of the atomic constitution, which achieved a major triumph towards the end of 1922. This triumph had to do with finding one of the missing elements in the periodic system. Ever since the periodic system had been first devised, attempts had been made to fill the gaps in it. While a large number of elements had been discovered in the last decades of the nineteenth and the early years of the twentieth century, some places in the periodic system had still remained unoccupied. Thus the table which Bohr presented at the Wolfskehl lectures in Gottingen (see Fig. 1) did not show the elements with atomic numbers 43, 61, 72, 75, 85 and 87. The properties of all but he knows very little German, and is rather hard of hearing' (Born to Einstein, 12 February 1921). °°In solving the equation of motion, Born and Brody applied Poincare's theory of integral invariants in many-body systems, in which Brody had become interested earlier (Brody, 1921b). In two later papers, Born and Brody corrected a number of errors without, however, changing the final result, Eq. (6). (See Born and Brody, 1922a,b.) Brody stayed on in Gottingen until t h e end of 1922; then he left to take up a professorship in Temesvar, Rumania. In 1925 he took a position with Vereinigte Gliihlampen-und Elektrizitats-A.G. near Budapest. Born learned later on that his former collaborator 'survived all the horrors of the Nazi occupation and the war' (Born, 1978, p. 214).
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one unknown element seemed to be evident. 58 Only the nature of the element No. 72, which was situated near the rare-earth elements, presented certain difficulties. A serious search for it had begun in 1907 when Georges Urbain in Prance and Carl Auer von Welsbach in Austria, after investigating for three years what had been called 'ytterbium probes' at that time, had finally isolated two different substances, which were later baptized ytterbium and lutecium. Both chemists had, however, assumed a third element to exist in the probes and, again after several years of laborious experiments, Urbain had announced the discovery of a new substance; he had given the name 'celtium' to this substance and placed it next to lutecium in the periodic system of elements (Urbain, 1911).59 Several years later a difficulty had arisen with the element celtium: Henry Gwyn Jeffreys Moseley, whom Urbain had visited in early summer 1914 and asked to check the X-ray spectrum of the substance — which he had assumed to contain the element No. 72 — had been unable to obtain the characteristic X-ray lines for the element. Moseley had reported his unsuccessful search in a letter to Rutherford by stating: 'Celtium has proved most disappointing. I can find no X-ray spectrum in it other than those of Lutecium and Neoytterbium [as ytterbium was called in those days], and so number 72 is still vacant' (Moseley to Rutherford, 5 June 1914). Moseley had then argued that the visible spectrum, which Urbain had associated with celtium, had to be ascribed to a mixture of lutecium and ytterbium and that Urbain's substance corresponded to a mixture of lutecium with a few percent neodymium. In spite of that failure, Urbain had continued to believe in the existence of celtium during the following years. The next advance came in 1920, when Maurice de Broglie examined the if-spectra of the very sample from which Urbain had earlier concluded the existence of celtium: his results seemed to indicate that the element number 72 was indeed present (M. de Broglie, 1920). Two years later, in May 1922, Alexandre Dauvillier, the chief assistant at Maurice de Broglie's laboratory, reported about the measurement of the i-spectrum of a sample containing the known elements ytterbium and lutecium; in addition to the lines of the latter elements, he observed two weak lines which he attributed to the element number 72 (Dauvillier, 1922a).60 These lines, situated at 1561.8 and 1319.4 58
T h e elements, numbers 43 and 75, had to be similar to manganese, the element number 61 had to be a rare earth, number 85 should represent a halogen and number 87 an alkali element. 59 G . Urbain, born at Paris on 12 April 1872, received his education at the Hcole Municipale de Physique et Chimie (1891-1894), and obtained his doctorate in 1899 from the University of Paris. From 1894 he served as an assistant and later as professor of chemistry at the Ecole Municipale. In 1906 he became assistant lecturer, in 1908 and professor of chemistry and in 1928 Director of the Chemistry Institute of the University of Paris. Urbain, a member of the Paris Academy of Sciences since 1921, worked mainly on rare-earth elements and on complex inorganic salts. He died in Paris on 5 November 1938. Alexandre Henri George Dauvillier was born on 5 May 1892 at Saint-Lubin-de-Joncherets, Eureet-Loire. He was educated at the University of Paris, where he obtained his doctorate in 1920. Then he joined Maurice de Broglie's laboratory. In 1925 he became lecturer at the Ecole Superieure d'Electriciti (until 1940), and in 1931 also at the Faculte de Medicine de Paris (until 1942). Then he was appointed director of research at the Centre National de Recherche and professor of cosmic physics at the College de France (in 1944). From 1935 onwards, Dauvillier directed the laboratory of cosmic physics at the Paris Observatory in Meudon. Besides X-ray spectra, he worked on
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X-ray units, seemed to fit well into Moseley's diagram for the Lai- and L^-lines for the element in question; hence Urbain concluded: 'Celtium has conclusively won its place among the chemical elements' (Urbain, 1922, p. 1349). Rutherford, who had continued to take interest in the celtium question since 1914, was informed by the French scientists about their findings and he reported their main results in a letter to Nature, which appeared in the issue of 17 June 1922 (Rutherford, 1922). In this letter he basically translated Urbain's paper (giving an account of the historical development of the celtium problem from 1911 to Dauvillier's X-ray analysis of 1922) and added finally the comment: Now that the missing element of number 72 has been identified, there remain only three vacant places of ordinal numbers — 43, 61, 75 — between hydrogen and bismuth in the Moseley classification of elements. With the rapidly increasing perfection of X-ray spectra and the use of powerful installations, it is to be anticipated that the missing elements should soon be identified if they exist in the earth. The law of the X-ray spectra, as found by Moseley, is an infallible guide in fixing the number of an element, even if present in only small proportion in the material under examination. (Rutherford, 1922, p. 781) Bohr saw Rutherford's letter in Nature when he was back in Copenhagen in late June 1922. Its content worried him very much; after all, he had just recently stated (in his sixth Wolfskehl lecture in Gottingen on 21 June 1922) that the element with atomic number 72 should be similar to zirconium (and not a rare-earth element as concluded from Dauvillier's observations), if his conceptions of atomic structure were correct. Now, in a letter to James Franck, he admitted that his statement had been wrong and that the element number 72, 'as shown by Urbain and Dauvillier, contrary to expectation, has turned out to be a rare earth element after all' (Bohr to Franck, 15 July 1922). Rutherford's advocacy of the French results also stimulated him to write an addendum (Nachschrift) to the three essays on 'Spektren und Atombau' (i.e., the German edition, which was about to appear), stating: 'After these essays were printed I learned from an article of Sir Ernest Rutherford in Nature about an investigation of Dauvillier on the X-ray spectra of several rare earths. It seems to follow from it that the element with the atomic number 72 has to be identified with the element celtium, whose existence had been speculated earlier by Urbain' (Bohr, 1922e, p. 147). 61 In order to account for the new situation, Bohr argued further, the classification given earlier must be changed in such a way as to include the elements, numbers 71 and 72, among the rare earths. This could be explained theoretically in the following way: the interaction of the electrons moving on orbits having the principal quantum numbers five and six (i.e., five- and sixquantum orbits) with the electrons on four-quantum orbits might be responsible for the fact that not all four-quantum orbits were occupied, in contrast to what problems of physical chemistry, geophysics and astrophysics. Dauvillier died on 23 December 1979 at Bagneresde-Bigorre. 61 I t appears that Niels Bohr had noticed the papers of Dauvillier and Urbain earlier, but had not attributed much importance t o them; this becomes evident from t h e development leading t o the discovery of hafnium, as reported here.
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seemed to follow from the earlier ideas (of J.J. Thomson and Niels Bohr) on the structure of electron groups in atoms. However, Bohr was not really satisfied with this explanation, and he wrote to Dirk Coster, whom he regarded as an expert on questions of X-ray spectra. Coster had just sent to Bohr a copy of his doctoral dissertation, for which Bohr thanked him in a letter, dated 3 July 1922, and added in a postscript: On my return from Gottingen I found a note by Rutherford in 'Nature' of June 17 which you probably have seen; it mentions some notes by Dauvillier and Urbain concerning the identification of the element with atomic number 72 as celtium. I should be extremely grateful to you if you would write to me about your opinion of the reliability of Dauvillier's identification of the X-ray lines in question. The problem is of the greatest interest, since, as you know, the conceptions of atomic structure seem to demand essentially different chemical properties of an element with atomic number 72 than those exhibited by the rare earths. The question is apparently rather clear, but one must, of course, be always prepared for complications. They may arise from the circumstance that we have to do with a simultaneous development of two inner electron groups. (Bohr to Coster, 3 July 1922) Coster replied to Bohr in some detail in a letter, dated 15 July 1922. 'Dauvillier's paper was not very convincing to me,' he remarked. 'You know my opinion about this author. His papers are a mixture of very good and very bad things and it is always impossible to check his statements.' However, already the next day Coster received the first-hand judgment of an expert about the reliability of the X-ray lines of celtium: Marine Siegbahn, after a visit to Paris, where he had seen Dauvillier's photographic plates, came to The Hague and met Coster on 16 July 1922. Coster wrote to Bohr immediately about Siegbahn's conclusions: He [i.e., Siegbahn] told me that he does not at all trust Dauvillier's work on the element 72. He could not at all see the lines Lai and LQ2 of [the element] 72 (the only lines which Dauvillier claims to have found for that element). When he told this to Dauvillier, the latter said: 'Oh yes, this is perfectly possible, and the reason is that there is no clear weather today!!!' Mr. Siegbahn has authorized me to write to you the following as his opinion: 'If Dauvillier wishes to claim that his probe contained about 0.01 (a hundredth of a percent) of the element 72, then one has to admit that the photographs, which he took, do not at all contradict this — but they also justify it very little.' (Coster to Bohr, 16 July 1922) Coster drafted a letter to Nature, in which he argued against Dauvillier's identification of the element number 72; but Bohr, to whom he sent a copy for approval, suggested not to publish it, for it seemed to be too polemical for the British journal and not based on new observations. Instead he informed Rutherford privately about the doubts of Coster, Siegbahn and himself concerning the existence of celtium. Now Bohr was convinced that one 'can hardly ascribe much significance to Dauvillier's result' (Bohr to Coster, 5 August 1922). And he decided to examine, with Coster, whom he had invited to work in Copenhagen for the following year, the question of X-ray spectra and its relation to atomic structure thoroughly. Coster started to work with Bohr on the renewed analysis of X-ray data immediately after his arrival in Copenhagen in September 1922. Already by the end of
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October they had completed a paper, entitled 'Rontgenspektren und periodisches System der Elemente,' which was received by Zeitschrift fur Physik on 2 November 1922 and published in its issue No. 6 of volume 12 in January 1923 (Bohr and Coster, 1923). In this paper Bohr and Coster discussed all the X-ray data available at that time. 62 They especially included the results obtained by Marine Siegbahn and his students as well as the observations of American scientists, notably William Duane and his associates at Harvard University.63 In a series of papers these authors had reported their measurements of the X-ray diffraction frequencies of many chemical elements; their results showed, for example, that the square root of the AT-absorption frequencies, if plotted against the atomic number, did not fall on a straight line (Duane and Hu, 1919a,b; Duane and Shimizu, 1919a,b), and that the critical if-absorption frequency of a given frequency always exceeded the shortest wavelength of the emitted radiation by fractions of a percent (Duane and Hu, 1919a). Coster had already analyzed these data previously: he had represented each X-ray frequency as the difference of two spectral terms, and he had organized the terms — which, if multiplied by Planck's constant represented the energy levels of atoms (especially the lowest ones) — in a way that was consistent with Bohr's recent views on atomic structure (Coster, 1922a,b). The repetition of the analysis by Bohr and Coster in fall 1922 aimed at a further sharpening of Coster's classification of energy levels, which should enable one to decide clearly about such questions as the electron shell occupation in the element with atomic number 72. Bohr and Coster began their work by ordering the lowest levels of the atoms — as obtained from the X-ray spectra, i.e., the K-\eve\, the three //-levels, the five M-levels, the seven iV-levels, the five O-levels and the three P-levels — according to rising energy: K; Li, Lu, Lm; M\, Mu, Mm, Mw, Mv, etc. 64 Then they associated with each term three quantum numbers, n(ki, k^), where n was the principal quantum number (i.e., n = 1,2,3, etc., for the K-, L-, M-, etc., levels), ki was equal to the azimuthal quantum number, and fei represented a further quantum number, whose value was either equal to k^ or ki + 1. For example, the K-level was described by 1(1,1), the 2
Bohr and Coster emphasized that their compilation of the d a t a was much larger and more reliable than earlier ones, given by Sommerfeld in the third edition of Atombau und Spektrallinien and by Dauvillier (Dauvillier, 1922b). (See Bohr and Coster, 1923, pp. 348-349, footnote 2.) 63 W . Duane was born at Philadelphia on 17 February 1872. He studied at the University of Pennsylvania (1888-1892), Harvard University (1892-1895), the Universities of Gottingen (1895, as Tyndall Fellow) and Berlin (1895-1897), where he took his doctorate under Max Planck. From 1898 to 1907 he was a professor at the University of Colorado; then he spent five years at the radium laboratory of Pierre and Marie Curie in Paris. In 1913 he returned to the United States and became an assistant professor of physics (1913-1917) and a professor of biophysics (1917-1934) at Harvard University. He died on 7 March 1935 in Devon, Pennsylvania. Duane performed important investigations in the fields of radioactivity and X-ray spectroscopy. Thus he discovered with Franklin L. Hunt the so-called Duane-Hunt law, relating the minimum wavelength of X-rays t o the threshold voltage of the cathode rays that excite them (Duane and Hunt, 1915). "4Bohr and Coster were aware of the fact that certain difficulties arose with respect to the uniqueness of the term scheme of a given element. Thus J. Bergengren had observed different if-edges for different allotropical forms of phosphorus (Bergengren, 1920), and Hugo Fricke had noticed a fine structure of the K-edge for many elements (Fricke, 1920).
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three L-levels by 2(1,1) for L\, 2(2,1) for Lu and 2(2,2) for Lin. In the transitions giving rise to X-ray lines the selection rules, An = 1 , 2 , 3 , . . . ,
Afc = l,
AJb2=0,l,
(7)
were obeyed. Bohr and Coster noted that their new organization corresponded formally to the one proposed earlier by Gregor Wentzel — Wentzel's quantum numbers m and n had just to be identified with fci and k2 respectively (Wentzel, 1921a) — but they suggested a different physical interpretation based on the idea of penetrating or diving orbits. According to their view an electron orbit consisted in general of an outer part and an inner part, which penetrates into the interior shells. The different nature of the optical and the X-ray spectra could then be explained as follows: Thus, the typical periodicity of the chemical properties and of the optical spectra depends on the circumstance that, for the outermost electron orbits, the effective quantum numbers, in contrast to the principal quantum numbers, vary only little as one goes from an element to the homologous element in the next period in the system of elements. On the other hand, the striking lack of periodicity of the essential features of X-ray spectra depends on the circumstance that we are here primarily concerned with the conditions of the innermost electrons in the atom, which move in groups that are already completely formed and that repeat themselves unchanged in all subsequent elements. (Bohr and Coster, 1923, p. 357; Bohr, Collected Works, Volume 4, 1977, p. 534) Therefore, in calculating the energy terms one found a remarkable difference if an inner or an outer orbit was considered. For an outer electron it sufficed to take into account a gross screening of the nuclear charge Z (in units of the absolute value of the electron's charge) through the 'screening factor' any, for an inner electron, however, one had also to pay attention to the effect of the outer electrons, which create an additional screening. Finally, Bohr and Coster included the relativity correction and obtained a formula for the most general energy term W, i.e.,
* - ^
+^
(£)'(H).
In Eq. (8), 7 represented the 'total screening factor,' i.e., the total screening effect on the nuclear charge of the inner and outer electrons, while N — 6 represented the effective nuclear charge for the relativity corrections. The screening effect was not supposed to be the same for the nonrelativistic term of the energy and the relativity correction term; in fact, from the data, 6 appeared to be much smaller than 7. From the above formula it was easy to obtain a dependence of the energy term on three quantum numbers, because the screening effects in Eq. (8) actually depended on three quantum numbers. Bohr and Coster did not go into the details of the theoretical problem, but rather confined themselves to studying the dependence of the empirical term data (for given sets of three quantum numbers, n(ki, k2)) on the atomic number: i.e., they plotted the values of the quantity T/R (T being, up
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to the factor h~l, the energy level W) against Z, the atomic number of the elements. Thus they obtained basically regular curves, which increased monotonically with increasing Z. In these plots they rediscovered what Sommerfeld and Wentzel had called 'screening doublets' ('Abschirmungsdublette') as parallel curves (e.g., L\ and i n ) , and the 'relativity doublets' ('Relativitatsdublette') represented by curves diverging rapidly for increasing Z (e.g., L\ and Lu). Bohr and Coster argued that the 'screening doublets' corresponded to transitions between states possessing different screening constants, ov,fc< and an",fe" (which arose from the effect of the inner electrons on the states considered), while the 'relativity doublets' corresponded to transitions between states having different azimuthal quantum numbers, k'% and fc2'. However, they were not able to derive the existence of the two types of doublets from a deeper theoretical reasoning. 65 The most important observation made by Bohr and Coster was that the curves showed discontinuities — i.e., jumps in the slopes of the curves showing T/R against Z, which consisted of connected pieces of straight lines — exactly at those places where, in accordance with Bohr's views, an inner electron shell in atoms became filled up. For example, such discontinuities clearly showed up in the beginning and at the end of the group of rare-earth elements, especially in the AT-levels. That is, between the atomic numbers 56 and 58 the slope of the curves changed suddenly from a larger to a smaller value until all four-quantum orbits were occupied; then the slope again became steeper. Unfortunately, a clear decision could not yet be made from the X-ray data as to the atomic number at which the steeper slope started. Bohr and Coster noted: 'Because of the incompleteness of the measurements for the elements in the vicinity of [Z =] 72, which form the end of the family of rare earths, there exists here some uncertainty about the course of the curves of the levels N\, Nn, Nm, N\y and Ny' (Bohr and Coster, 1923, p. 370; Bohr, Collected Works, Volume 4, 1977, p. 545). So the question concerning the constitution of the element with atomic number 72 and its chemical properties remained unanswered. However, it was soon to be decided in a more direct way by the discovery of the element hafnium. In November 1922 the announcement was made that the Nobel Prize in Physics for the year 1922 was awarded to Niels Bohr. On 11 December 1922, Bohr delivered his Nobel address 'On the Structure of Atoms' (Bohr, 1923b). George de Hevesy recalled later: 'A few minutes before Bohr started his Nobel lecture, Coster announced by telephone the presence of the Hafnium fi\ and fo lines on his photograpic plate' (Hevesy, 1951, p. 683). Bohr happily mentioned the news of the discovery of the new element in Copenhagen in the last part of his lecture. After referring briefly to the story of the element number 72, including Dauvillier and Urbain's claim that it belonged to the rare earths, he remarked: 65
B o h r and Coster drew attention to another organization of the X-ray levels. So they called 'normal' levels those in which both quantum numbers fci and fo coincided; in them the energy calculation could be carried out by assuming the electron shells in the atoms as more or less independent. In 'anomalous' levels, i.e., levels with different k\ and k
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In these circumstances Dr. Coster and Professor Hevesy, who are both for the time working in Copenhagen, took up a short time ago the problem of testing a preparation of zirconbearing minerals by X-ray spectroscopic analysis. These investigations have been able to establish the existence in the minerals investigated of appreciable quantities of an element with atomic number 72, the chemical properties of which show a great similarity to those of zirconium and a decided difference from those of the rare-earths. (Bohr, 1923b, p. 42) Bohr's announcement of t h e discovery of t h e element 72, having t h e properties of zirconium, ended a feverish search in which Dirk Coster a n d George d e Hevesy were t h e m a i n actors. It h a d begun in t h e last week of May 1922, shortly after the publication of Dauvillier's a n d Urbain's notes in Comptes Rendus (Dauvillier, 1922a; Urbain, 1922). Coster h a d j u s t come over from L u n d , a n d Bohr h a d consulted him a n d de Hevesy, who was a n expert o n t h e chemical literature on rare-earth elements. 'I agreed with Coster in t h a t t h e findings of Urbain a n d Dauvillier in n o way disprove t h e correctness of Bohr's prediction as t o t h e n a t u r e of the element,' Hevesy recalled later. He continued: After our talk with Niels Bohr we did not pursue the subject and none of us thought at that date to initiate a search for the element 72. It was a coincidence of fortunate events which led to the discovery of the element, the character of which was predicted by Bohr, in his institute. Among these events Coster's stay in Copenhagen after the completion of his studies with Manne Siegbahn at Lund was the most important one. In collaboration with Coster we intended to use radiolead as anode of an X-ray tube and to investigate the possible effect of intense and prolonged irradiation on the decay rate of radium D, which should reflect itself in a corresponding change in the rate of decay of radium E. While, in search of a suitable high-voltage aggregate, being much interested in the application of Xray spectroscopy to mineral chemistry, I suggested that Coster should look for the element 72. In the beginning he was very reluctant to embark on this investigation, maintaining the view that, if element 72 was present in a mineral, its concentration could be expected to be only very low and, since the sensitivity of X-ray spectroscopy in those days was rather weak, it seemed almost hopeless to discover the missing element by means of this method. Finally, however, Coster yielded to my argument that the aim of our work should be to enable me to learn the technique of X-ray spectroscopy and we could just as well at the same time try to find Bohr's element 72. (Hevesy, 1951, pp. 682-683) 67 After suitable preparations of t h e a p p a r a t u s a n d of t h e probes t o be investigated, Coster began t o take X-ray d a t a by t h e end of November 1922. Working with a concentrate of Norwegian zirconium minerals, he first found t h e Lai- a n d La2-\mes of a new substance, which one could associate w i t h t h e atomic number 72. Still the "According to Oskar Klein, Bohr spoke on this occasion without having his prepared manuscript with him. Klein reported: 'At the obligatory lecture, for which he had chosen to talk about the constitution of atoms, he discovered that he had forgotten his notes and slides at the hotel, so he had to begin without them, while, they were fetched. This, however, was rather an advantage, because it forced him to improvise, as he did in private conversation' (Klein in Rozental, 1967, p. 84). Hevesy became interested in the element 72 from discussions with Friedrich Adolf Paneth (18871958), then professor of chemistry at the University of Hamburg. In 1922 the two of them wrote a textbook on radioactivity, in which they expressed doubts about the existence of Urbain's celtium (Hevesy and Paneth, 1923, p. 109). After the publications of Urbain and Dauvillier, Paneth urged Hevesy strongly to investigate the problem himself. (See Kragh, 1979, p. 184.)
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situation was a bit complicated, for 'by a queer coincidence the second-order zirconium Jf-spectrum coincided with the hafnium Ka-lm.e' (Hevesy, AHQP Interview, 25 May 1962, p. 14). In principle, it was possible to avoid the complication; the second-order zirconium spectrum could be suppressed by keeping the voltage of the X-ray tube below a certain value. 'But we had a very primitive apparatus,' Hevesy recalled, 'and our voltage was going up and down, so it was quite difficult' (Hevesy, AHQP Interview, 25 May 1962, p. 15). However, the uncertainty did not last for long, and 'the final proof of the discovery of hafnium was brought when Coster succeeded in locating the 0i and /?2 lines and soon was able to determine the whole X-ray spectrum of that element' (Hevesy, 1951, p. 683). In obtaining these spectra, Coster used probes in which Hevesy had enriched the content of the new element. 68 After measuring some more lines of the X-ray spectra, Coster informed Bohr on 2 January 1923: 'I am as sure of the existence of hafnium as I was of any other well known element I investigated.' The same day they signed a letter to Nature in which they announced their results and said: 'For the new element we propose the name Hafnium (Hafniae = Copenhagen)' (Coster and Hevesy, 1923, p. 79). 69 Rutherford, who saw Coster and Hevesy's letter to Nature before publication, immediately agreed with it. He first congratulated Hevesy, remarking that the discovery was 'an admirable example of cooperation of theory and experiment' (Rutherford to Hevesy, 8 January 1923). At the same time he wrote to Bohr: I have seen the paper [of Coster and Hevesy] to Nature and I am exceedingly pleased that you have been able to verify your conclusions so rapidly. Urbain sent his letters over to me, obviously with a wish that I would make a note of it in Nature, but I had forgotten at that time that your theory fixed the element as an analogue of zirconium. I cannot understand how Dauvillier made the mistake he obviously has. I am sure this confirmation will put you in good spirits for tackling the work of another year. (Rutherford to Bohr, 8 January 1923) Georges Urbain, on the other hand, was less happy. As Hevesy recalled: 'We had a big fight with Urbain. Urbain said, "Well, I found this element long ago." But he had bad luck. Namely, after hafnium was prepared [in a purer sample], not a single one of perhaps 26 optical lines agreed with those of Urbain' (Hevesy, AHQP Interview, 25 May 1962, p. 15). The optical spectra, referred to by Hevesy, were measured by Hans Marius Hansen and Sven Werner and appeared in the issue of Nature of 10 May 1923 (Hansen and Werner, 1923). The original probes contained of the order of 1% of element 72, but Hevesy worked hard to purify the material. Shortly after the discovery of hafnium, he found a method to separate it from zirconium by fractional crystallization of ammonium-zirconium fluoride (in which also some zirconium was replaced by hafnium) on platinum plates. Due to the large difference in the solubility of the hafnium- and zirconium-ammonium hexafluorides, the method worked extremely well. 69 People in Copenhagen at first disagreed about the name of the new element. Coster and Kramers agreed on hafnium, while Hevesy and Bohr preferred 'danium' (for Denmark). In the proof of the note to Nature, Coster and Hevesy requested to change the name hafnium to danium; but due to some misunderstanding it was not corrected in the actual publication. This situation caused some confusion in the public reports on the discovery of element 72. (See Kragh, 1979, p. 186.)
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T h e discovery of hafnium, with the properties predicted by his theory, constituted a m a j o r t r i u m p h for Niels Bohr. So it was no exaggeration when the spectroscopist Heinrich M a t t h i a s Konen of B o n n remarked t h a t B o h r ' s 'achievements in spectroscopy can only be compared with Darwin's contributions to biology' (Konen t o Bohr, 16 December 1922). At t h e t u r n of t h e year from 1922 t o 1923, the physicists looked forward with enormous enthusiasm towards detailed solutions of the o u t s t a n d i n g problems, such as t h e helium problem a n d the problem of the anomalous Zeeman effects. However, within less t h a n a year, t h e investigation of t h e s e problems revealed a n almost complete failure of Bohr's atomic theory. References Bergengren, J. (1920), Z. Phys. 3, 247-249. Bjerrum, N. (1912), in Festchiff W. Nernst, Halle a. d. Saale: Wilhelm Knapp, pp. 90-98. Bohr, N. (1914b), Phil. Mag. (6) 27, 506-524. Bohr, N. (1918a), Collected Works 3, 67-102. Bohr, N. (1918b), Collected Works 3, 103-166. Bohr, N. (1919), Meddelanden fran K. Vetenskapsakademiens Nobelinstitut 5, No. 28. Bohr, N. (1920), Z. Phys. 2, 423-469. Bohr, N. (1921a), Nature 107, 104-107. Bohr, N. (1921d), Nature 108, 208-209. Bohr, N. (1921e), Collected Works 4, 263-328. Bohr, N. (1922e), The Theory of Spectra and Atomic Constitution, Cambridge: Cambridge University Press, 1922. Bohr, N. (1923b), Nobel Lecture, Collected Works 4, 427-465. Bohr, N. (1923c), Collected Works 4, 611-656. Bohr, N. (1923e), Collected Works 3, 364-380. Bohr, N. (1977), Collected Works 4: The Periodic System (1920-1923). Bohr, N., and D. Coster (1923), Z. Phys. 12, 342-374. Born, M. (1923b), Encykl. d. math. Wiss. V / 3 , 527-781. Born, M. (1925), Vorlesungen iiber Atommechinik, Berlin: J. Springer-Verlag. Born, M. (1978), My Life: Recollections of a Nobel Laureate, New York: Charles Scribner's Sons. Born, M., and E. Brody (1922a,b), Z. Phys. 8, 205-207; 11, 327-352. Born, M., and W. Heisenberg (1924a), Ann. d. Phys. (4) 74, 1-31. Born, M., and W. Heisenberg (1924b), Z. Phys. 23, 388-410. Born, M., and E. Hiickel (1923), Phys. Zs. 24, 1-12. Brody, E. (1921a,b), Z. Phys. 6, 79-83, 224-228. Coster, D. (1921a,b,c), Z. Phys. 4, 178-188; 5, 139-147; 6, 185-203. Coster, D., and G. de Hevesy (1923), Nature 111, 79. Dauvillier, A. (1922a), Comptes rendus (Paris) 174, 1347-1349. De Broglie, M. (1920), Comptes rendus (Paris) 170, 725. Duane, W., and T. Schimzu (1919a,b), Phys. Rev. (2) 13, 289-291; 14, 522-524. Epstein, P.S. (1916c), Ann. d. Phys. (4) 50, 489-521. Fricke, H. (1920), Phys. Rev. (2) 16, 202-215. Gehrcke, E., and E. Lau (1920), Phys. Zs. 2 1 , 634-635. Hansen, H.M., and S. Werner (1923), Nature 111, 322.
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Heisenberg, W. (1922a), Z. Phys. 8, 273-297. Herzfeld, K.F. (1916), Ann. d. Phys. (4) 51, 261-284. Heurlinger, T. (1918), Untersuchungen uber die Struktur von Bandenspektren, doctoral dissertation, University of Lund. Heurlinger, T. (1919), Phys. Zs. 20, 188-190. Heurlinger, T. (1920), Z. Phys. 1, 82-91. Hevesy, G. de (1951), Z. Phys. 1, 82-91. Hund, F. (1961), in Werner Heisenberg und die Physik unserer Ziet (F. Bopp, ed.), Braunschweig: Fr. Vieweg, pp. 1-7. Kossel, W. (1916a), Ann. d. Phys. (4) 49, 229-362. Kragh, H. (1979), Niels Bohr's Second Atomic Theory, Historical Studies in the Physical Sciences 10, 123-186. Kramers, H. (1919), Kgl. Danske Vid. Selsk. Skrifter, 8 Raekke, I I I . 3 ; reprinted in Collected Scientific Papers, pp. 109-133. Kramers, H.A. (1920a), Proc. Kon. Akad. Wetensch. (Amsterdam) 23, 1052-1073. Kramers, H.A. (1920b), Z. Phys. 3, 199-223. Kramers, H.A., and W. Pauli (1923), Z. Phys. 13, 351-367. Kratzer, A. (1920b), Z. Phys. 3, 460-465. Kratzer, A. (1922), Sitz. ber. Bayer. Akad. Wiss. (Miinchen), pp. 107-118. Ladenburg, R. (1920), Naturwiss. 8, 5-11. Lande, A. (1920d,e), Z. Phys. 2, 87-89, 380-404. Lande, A. (1921a), Z. Phys. 4, 410-423. Lenz, W. (1919), Verh. d. Deutsch. Phys. Ges. (2) 21, 632-643. Lenz, W. (1920b), Phys. Zs. 21, 691-694. Reid, C. (1970), Hilbert, New York-Heidelberg-Berlin: Springer-Verlag. Reid, C. (1976), Courant in Gottingen and New York. The Story of an Improbable Mathematician, New York-Heidelberg-Berlin: Springer-Verlag. Riecke, E. (1915), Phys. Zs. 16, 222-227. Rozental, S. (1967), Niels Bohr: His Life and Work as Seen by His Friends and Colleagues, Amsterdam: North-Holland. Rutherford, E. (1922), Nature 109, 781. Schrodinger, E. (1921), Z. Phys. 4, 347-354. Schwarzschild, K. (1916), Sitz. ber. Preuss. Akad. Wiss. (Berlin), pp. 548-568. Sommerfeld, A. (1915c), Sitz. ber. Bayer. Akad. Wiss. (Miinchen), pp. 459-500. Sommerfeld, A. (1917), Sitz. ber. Bayer. Akad. Wiss. (Miinchen), pp. 83-109. Sommerfeld, A. (1921c), Jahrbuch d. Radiaoktivitat & Elektronik 17, 417-429. Sommerfeld, A., and W. Heisenberg (1922a,b), Z. Phys. 10, 393-398; 11, 131-154. Sommerfeld, A., and G. Wentzel (1921), Z. Phys. 7, 86-92. Stark, J. (1920), Jahrbuch d. Radioaktiviat & Elektronik 17, 161-173. Thomsen, J. (1895), Oversigt over det Kgl. Danske Videnskabernes Selskaps Forhandlinger, p. 1 ff. Thomson, J.J. (1918), Engineering 105, 206-208, 234-235, 261-262, 286-288, 317-318, 344-346 (six lectures delivered in February and March 1918 at the Royal Institution). Thomson, J.J. (1919), Engineering 107, 366-367, 410-411, 443-444, 476-478, 511-512, 560-562 (six lectures delivered in March and April 1919 at the Royal Institution). Urbain, G. (1911), Comptes rendus (Paris) 152, 141-143. Urbain, G. (1922), Comptes rendus (Paris) 174, 1349-1351. Urk, T. van (1923), Z. Phys. 13, 268-274.
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Vegard, L. (1917a,b), Verh. d. Deutsch. Phys. Ges. (2) 19, 328-343, 343-353. Wentzel, G. (1921a,b), Z. Phys. 6, 84-99; 8, 85-88. Wentzel, G. (1922), Ann. d. Phys. (4) 66, 437-461. Wood, R.W. (1918), Phil. Mag. (6) 35, 236-252.
14 Satyendra N a t h Bose,* Bose-Einstein Statistics, and t h e Q u a n t u m Theory of an Ideal Gast S.N. Bose was one of India's most eminent scientists. Bose's achievements in scientific research were not as sustained or as numerous as those of his contemporaries like C.V. Raman, 1 Meghnad Saha 2 and K.S. Krishnan, 3 together with whom he became a pioneer of education and research in modern physics in India. But the circumstances of Bose's intellectual development were unusual and he was destined to play an inspiring role in the scientific and cultural life of his country. Bose's novel derivation of Planck's radiation formula, the only significant contribution which he made to physics, came at a turning point between the old quantum theory of Planck, Einstein, Bohr and Sommerfeld and the new quantum mechanics of Heisenberg, Dirac and Schrodinger. Bose sent his paper early in June 1924 to Albert Einstein who recognized its merit, translated it into German, and had it published in the Zeitschrift fur Physik [6]. During the summer of 1924 Einstein also received, from Paul Langevin in Paris, a copy of the doctoral thesis of Louis de Broglie4 dealing with the wave aspects of matter. Bose's work became the point of departure for Einstein's investigation on the quantum theory of monatomic ideal gases and 'gas degeneracy,' leading to his prediction of the condensation phenomenon. 5 Einstein recognized the importance of de Brogli's ideas and also made use of them in his investigation. 5 In turn, these papers of de Broglie and Einstein stimulated Schrodinger6 to develop his wave mechanics. The 'Bose-Einstein statistics' immediately fitted into the framework of quantum mechanics and enshrined Bose's name in physics for ever. Bose lived the legend of this fateful encounter with Einstein throughout the rest of his life. S.N. Bose received affection, hero-worship and veneration from his countrymen in a way that a distinguished man of learning with nobility of character can only in India. Plans had been made to observe Bose's eightieth anniversary year, 1974, by Published in the Biographical ber 1975. T Appendix added in 1982.
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holding symposia and celebrations at various places in India. He died on 4 February 1974. His passing was mourned generally. Satyendranth Bose, generally known as S.N. Bose in the scientific community and affectionately called Satyen Bose in his native Bengal, was born on 1 January 1894 in Calcutta. His father, Surendranath Bose, was a trained accountant who held responsible posts in the Executive Engineering Department of East Indian Railways. Surendranath had an aptitude for mathematical thinking and was interested in several branches of science; he became one of the founders of a modest chemical and pharmaceutical company. He was also interested in philosophical studies, especially the Hindu scriptures such as the Bhagwad-Gita: at the same time he enjoyed the dialectical speculations of Hegel and Marx. Surendranath Bose died on 2 June 1964 at the ripe old age of ninety-six, having witnessed with pride the seventieth birthday celebrations in honour of his famous son, S.N. Bose, during January of that year. Bose's mother, Amodini Devi, had died earlier, in 1939. She often had to struggle against ill health and inadequate resources to maintain a middle-class home. She had received only a nominal school education but she showed considerable ability in managing- domestic affairs. She possessed a remarkable fortitude of will, warm heart, and sense of personal and family dignity. Satyendranath was the eldest child and only son of his parents; he had six younger sisters. He inherited many good qualities and noble aspirations from his parents, and he more than fulfilled their expectations of him. Their happiness was great when the poet Rabindranath Tagore 7 invited Bose to Santiniketan and dedicated to him his Visva-Parichaya, a book giving an elementary account of the cosmic and microcosmic world in Bengali, in recognition of Bose's efforts to popularize science through the mother tongue. Young Satyen Bose attended a neighbourhood elementary school in Calcutta for his studies in the lower forms until he was thirteen years old. In 1905, when Satyen was eleven years old, Lord Curzon declared the partition of Bengal. It had grave political consequences, spurring the Swadeshi [national] movement and arousing Bengalis to fight against the British political and economic domination. The patriotic teachings of Ram Mohan Roy, Bankim Chandra and Vivekananda took on concrete shape and became a living source of inspired sacrifice by the people. This patriotic idealism became a powerful influence in the formative years of young men's lives, including Bose's. Bose entered the Hindu School in 1907. His eyesight was very weak, but his intelligence and memory were keen, and he had a great wish to learn science. The headmaster, Rasamaya Mitra, and the senior teacher of mathematics, Upendralal Bakshi, gave him much encouragement. After passing high school in 1909, Bose entered the Presidency College, Calcutta, where he enrolled in the science courses. The science department of the Presidency College had a distinguished staff. Bose's teachers were P.C. Ray in chemistry, Jagadish Chandra Bose8 in physics, and D.N. Mallik and C.E. Cullis in mathematics. H.M. Percival was the professor of English. Among Bose's contemporaries there
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were several who became familiar names in Indian science, such as Meghnad Saha, 2 N.R. Sen and J.C. Ghosh. P.C. Mahalanobis, 9 the statistician and planning expert, was his senior by one year; N.R. Dhar was senior by two years. This group of students shared the excitement of acquiring scientific knowledge and the patriotic fervour derived from the Swadeshi movement. 'We wanted to put scientific knowledge to use through technology for the benefit of the people or to contribute to science by intensive study.'* Bose took the B.Sc. examination in 1913, and the M.Sc. degree in Mixed Mathematics (more like applied mathematics dealing with astronomy, dynamics, hydrodynamics, and certain mathematical topics) from Calcutta University in 1915. He stood first in both examinations, the second place going to M.N. Saha. He also got married during his final year at college. The future presented a bleak prospect even to brilliant young men like Bose and Saha. They could probably have pursued successful administrative careers in the service of the Government; Saha even considered taking the competitive examination for the Indian Finance Service, but permission to do so was denied him by the Government. They decided to devote themselves to study and research in physics and applied mathematics. But this was also not easy. They had been receiving vague and indirect bits of information about exciting new scientific developments in Europe in atomic physics, relativity and quantum theories, which were not satisfactorily treated in the textbooks available to them at that time. 'Scientific journals were only irregularly available on account of the war than prevailing, although the Philosophical Magazine, in which Bohr had published his papers on atomic theory, was available in the library of the Presidency College.' Modern science was not a matter of books and journals alone; laboratories and apparatus were indispensable and these did not exist. It seemed to them that with their M.Sc. degrees they had entered a blind alley. At this point a new hope in the field of university education in Bengal was provided by Sir Asutosh Mookerji, a Judge of the High Court and Vice-Chancellor of Calcutta University. Sir Asutosh had entered Bengal's educational politics in 1904 as Lord Curzon's ally against the struggle, led by men like Rabindranath Tagore and Aurobindo Ghosh, for a far-reaching reform of national education. For instance, he continued to insist, for a long time, on English as the medium of instruction in preference to the mother tongue of the pupils. In the course of time his views mellowed and the award of the Nobel Prize to Tagore in 1913 (for his lyric poems Gitanjali) provided the turning point. Sir Asutosh now used his considerable influence and shrewdness in transforming the University of Calcutta, founded in 1857 as an official body administering examinations, into an autonomous institution of higher learning. Two wealthy Calcutta lawyers, Tarak Nath Palit and Rash Behari Ghosh, had donated vast amounts of money for the establishment of a national university free from British influence. Sir Asutosh gained control of "The unspecified quotations in this memoir are Professor S.N. Bose's comments in two conversations I had with him. See p. 145.
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these funds for the endowment of new chairs in different fields to be held by Indian scholars. 'Before that time the occupants of most of the university positions in Calcutta used to be third-rate Scotsmen.' As a young man Sir Asutosh Mookerji had himself published some papers on mathematics and written a treatise on conic sections which served as a textbook for some time. 10 He now laid the foundation, on 27 March 1914, of the new University College of Science for post-graduate studies and research. 'The syllabus on physics and mathematics boasted of the existence of a great variety of courses which were not taught at the Presidency College nor were there any professors to teach them.' In 1915 some of the new M.Sc, graduates, including Bose and Saha, approached Sir Asutosh Mookerji with a request to open postgraduate classes in physics and mathematics, and to let them teach. 11 He gave them a sympathetic hearing and recognized the merit of their proposal, but he also knew the difficulties that were involved. He agreed with them on condition that they would prepare themselves first for such an undertaking by advanced study and research. He granted them scholarships and arranged the facilities for procuring scientific journals and working in the laboratories. 'He allowed us access to his own large collection of books on mathematics and physics. I borrowed from him Gibb's book 12 from which I learned about phase space and statistical mechanics.' Bose and Saha had already started to learn French and German in order to be able to read the European scientific literature. Now they embarked upon the study of what then was considered the mainstream of research in physics. Advanced books were not available in the library, and it was difficult to order them from abroad because of the war (1914-18). They were about to give up hope of obtaining them when help came from a strange quarter. They learned that P.J. Briihl, an instructor in the Bengal Engineering College, possessed advanced textbooks on physics. 'Briihl, an Austrian, had received his doctorate in botany. He was afflicted with tuberculosis and was advised to go to a warm and sunny climate. He obtained a scholarship to study the flora in India and came to Calcutta. While in Calcutta he got married, something which was against the terms of the stipend, and he found himself in difficulty. He took a job in the Engineering College, where he became a careful experimenter, taught engineering physics, and generally did rather well. He possessed a good collection of advanced texts and journals on physics in German. He had Planck's Theorie der Warmestrahlung, Laue's Das Relativitatsprinzip, as well as papers on quantum theory and relativity. Since Saha and I had learned some German, we were glad to borrow these things from Briihl. Saha chose to study first thermodynamics, statistical mechanics and spectroscopy, while I decided upon electromagnetism and relativity.' They were going to be autodidacts in the new theories of physics. In 1916 Ganesh Prasad, a distinguished mathematician from Benares, was appointed Rash Behari Ghosh Endowment Professor of Mathematics in the University College of Science. Post-graduate classes in applied mathematics and physics were started, with Bose and Saha as lectureres on probation in the Department of
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Mathematics. 11 In 1917, with the establishment of other departments within the University College, they were both appointed lecturers in physics. 'We took upon ourselves the task of teaching post-graduate students. Saha taught thermodynamics and spectroscopy in the physics department and hydrostatics and lunar theory or some such thing [the figure of the Earth] in the mathematics department. I was more amphibious, teaching both physics and applied mathematics quite regularly. On me fell the task of teaching general physics and giving all the entrants suitable introduction to mathematical physics — teaching them differential equations, harmonic analysis, etc. In general physics I used Thomson and Tait's Treatise on natural philosophy. I also taught elasticity and relativity.' A new phase thus began in the life of S.N. Bose. The great influence on his intellectual development now was the new physics, particularly quantum theory and special relativity. The example of J.C. Bose8 pursuing his researches in electrophysiology was inspiring. In 1918 C.V. Raman took up his appointment as Palit Professor of Physics in Calcutta University, but he continued to do his research work nearby in his laboratory at the Indian Association for the Cultivation of Science. 'He had all the facilities of work there, and his students worked there, too. He could be there whenever he pleased — mornings, evenings, late at night, just about any time he pleased — and his students used to flock around him. He was a magnetic personality but not yet a powerful man. He had to depend upon the authorities and the Indian Association for all the monetary help. As Palit Professor he got his research grant, and he utilized it to the best advantage. There were many young people amongst us who did their research work under his guidance, in optics mainly.' The previous lonely efforts of Jagadish Chandra Bose and P.C. Ray had now become an organized activity of education and research in modern fields, and the University of Calcutta became recognized within a few years as an institution of higher learning int he sciences. Research Work in Calcutta and Dacca When we began to teach we had to do things more seriously. We had to try to do something original.' Bose found a common interest with Saha in statistical mechanics. They published a joint paper in 1918 in the Philosophical Magazine on the influence of the finite volume of molecules on the equation of state [1]. Two years later they published another joint paper on the equation of state in the same journal (4]. During 1919 S.N. Bose read two papers to the Calcutta Mathematical Society: the first [2] dealt with the stress equations of equilibrium in elasticity and showed that the equations satisfied by the stress-coefficients in an isotropic medium could be successfully integrated in the case of a semi-infinite body bounded by a plane, while in the case of a sphere they could be conveniently transformed, thereby admitting of integration in an infinite series of spherical harmonics; the second [3], basing the proof on the dynamical considerations fo the Poinsot motion, showed that Pointsot's horpolhode contains no point of inflection.
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In 1919, shortly after World War I ended, the solar eclipse expedition led by A.S. Eddington verified the deflexion of starlight by the gravitational field of the sun, confirming one of the predictions of Einstein's general relativity theory. In a war-weary world the theory of relativity produced great general excitement, and newspapers and magazines were full of articles on it. Bose and Saha got deeply interested in the relativity theory and published a collection of papers on relativity [26]. Saha translated into English Einstein's 1905 paper on special relativity 13 and Minkowski's 1908 paper on the fundamental equations of electromagnetic phenomena in moving bodies [the principle of relativity]. 14 Bose translated Einstein's 1916 paper on the foundation of the general relativity theory. 15 'Relativity was of course a magnificent structure already. Special relativity was a fascinating thing. Four-dimensional representation, the work of Minkowski, just splendid. We studied all these papers and translated them. Later on other people took up the translation of Einstein's relativity papers. Einstein had given the translation rights to the Methuen people. 16 They wanted to stop the distribution of our work. Einstein was very generous. He said that so long as the book remained in circulation inside India, he had no objection. After the excitement of the first year or so people dropped making inquiries about the book. It is out of print.' Although not a very good translation, it represented the first publication in English of Einstein's famous work. 16a The man who assisted Bose with the translation from German, and in other ways too, was D.M. Bose. 17 'He had been in Cambridge for some time. After he was designated as a professor in Calcutta he went for further work in Berlin, where he was interned during part of the First World War. Then he worked on a-ray scattering and other things. He returned only in 1919 to take up his position. He was a great help to me. He had brought with him many new publications on quantum theory and other things which had been unavailable to us during the war. He even had Planck's Festschrift [1918] which had been published in the Naturwissenschaften. He gave me Planck's Thermodynamik to read, and I greatly appreciated the logical manner in which Planck had deduced the whole of thermodynamics from a limited number of postulates. D.M. Bose had all these new books, and naturally we were excited about them. We were anxious to learn. Saha could read German fairly well without any help, but I tried to go ahead by doing translations on my own and having them checked by D.M. Bose. That's what led me to translate Einstein's paper on general relativity. That's what I did as an exercise to learn German as well as to learn physics. There were no professors, indeed nobody, to guide us except ourselves.' Bose had read Bohr's papers in the Philosophical Magazine.18 From D.M. Bose he obtained Sommerfeld's papers on multiple quantization and the fine structure of spectral lines. 19 In 1920 he published a paper on the deduction of Rydberg's law from the quatnum theory of spectral emission in the Philosophical Magazine [5]. This was the first of a total of five papers which S.N. Bose wrote on some aspect of the quantum theory. He made an attempt to deduce the laws of regularity in
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 507 the spectral series of elements on the basis of Bohr's theory of spectral emission. Starting from Sommerfeld's assumption that the ordinary line spectra of elements are due to the vibration of one outer electron (the valence electron), Bose showed that the field of the nucleus and the remaining (n — 1) electrons may be represented by a potential V = - e 2 / r + (eLcos9)/r2, i.e., the field due to a single charge plus a doublet of strength L. The axis of the doublet being variable, Bose assumed the emission to take place so quickly that in that short time the axis would not change appreciably. Bose applied the quantum conditions in accordance with Sommerfeld's rule, mh — f pidqt. By using Bohr's frequency condition, hv = Wn — Wni, Bose arrived at Rydberg's law of the regularity in spectral series in the case of alkali metals. Bose's friend and colleague Saha had been, in the meantime, making rapid progress in his electrodynamical and astrophysical investigations. He published papers on 'Maxwell stresses' and the dynamics of the electron. 20 He formulated the concept of selective radiation pressure and recognized its role in the relative distribution of the elements in the solar atmosphere. In 1918 Saha was awarded the D.Sc. degree of Calcutta University on the basis of his work in the field of electromagnetic theory and radiation pressure. In 1919 Saha received the Premchand Roychand Scholarship of Calcutta University, which enabled him to spend a couple of years in Europe. He employed his time profitably on research work in A. Fowler's laboratory in London and W. Nernst's laboratory in Berlin. Saha's greatest contribution was the theory of high-temperature ionization and its application to stellar atmospheres. The equation that bears his name was first given in the paper 'On ionization in the solar chromosphere,' 21 published in the Philosophical Magazine for October 1920. In November 1921 Saha returned from Europe and joined the University of Calcutta as Khaira Professor of Physics, a new chair created from the endowment of Kumar Gurprasad Singh of Khaira. In 1923 Saha left Calcutta to take up the appointment of Professor and Head of the Physics Department in the University of Allahabad, where he stayed for fifteen years and created a good school of physics. K.S. Krishnan succeeded Saha at Allahabad in 1938, and Saha returned to Calcutta. 17 In 1921 Dacca University (now in Bangladesh) was started, with the old Dacca College as the nucleus, as a result of the recommendation of the Sadler Commission on the reorganization of university education. D.J. Hartog, who had been a member of Commission, became the first Vice-Chancellor of Dacca University and was entrusted with the task of developing it. 'Hartog selected me as one of the readers. The University College of Science in Calcutta was beginning to be rather crowded, and I thought that Dacca would offer a freer opportunity to carry on studies in all kinds of subjects. So I went to Dacca. My colleague there, W.A. Jenkins, was the professor of physics. He was a clever man but not of any eminence whatsoever, nor had he done any research work. He had been a professor of physics in the old Dacca College, and when the University was created they took him. He was from Cambridge. He belonged to the Indian Educational Service, and was just passable
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in physics. His special qualification was that he was a good football player, and with that merit he was appointed to the professorship of physics. He was a young man of my age, about 27 years. He was a nice, companionable person, and did not interfere with my work. He didn't quarrel; he wanted to be on good terms with Indians.' In the physics department at Dacca University Bose had the usual duties of teaching and guiding students in their work. 'There was a laboratory, the old laboratory of the Dacca College, and I was asked to do experimental work also in order to guide the students, and I had to do some of that. But my interests were in theoretical study principally' He had to study a lot of things in order to be able to teach, as he himself believed in serious and good teaching. Most of his time was taken up with the study of original papers, and he profited a good deal from a critical study of the original papers of the great masters like Planck, Einstein, Bohr and Sommerfeld. Indeed, his research publications grew from the necessity of having to learn and to understand for the purpose of teaching his students. From this experience he developed the motto which he used to impress upon his students: 'Never accept an idea as long as you yourself are not satisfied with its consistency and the logical structure on which the concepts are based. Study the masters. These are the people who have made significant contributions to the subject. Lesser authorities cleverly bypass the difficult points.' 21a Principal Work 'I had studied Planck's derivation of his radiation formula and also his Theorie der Warmestrahlung. I knew about phase space and Boltzmann statistics from Gibbs's book. And I had Boltzmann's Vorlesungen iiber Gastheorie from Bruhl. I knew about Einstein's deduction of Planck's formula and how he had never been satisfied with the derivation of this quantum law; he always kept on coming back to it. Planck's conditions ran counter to the classical ideas. Planck was also aware of the difficulties, but he had never been able to resolve them. He had relations which were derived from Maxwell's electromagnetic theory and others in which discontinuities appeared. He wanted to reconcile his theory with the classical theory. He had the idea of having continuous absorption and discontinous emission. As a teacher who had to make these things clear to his students I was aware of the conflicts involved and had thought about them. J wanted to know how to grapple with the difficulty in my own way. It was not some teacher who asked me to go and solve this little problem. J wanted to know. And that led me to apply statistics. But this came out after a visit of Saha.' It was some time around March 1924 that Bose had a meeting with Saha. 'Saha was a guest visiting Dacca; it was his native land. 22 He came out to see me and stayed with me for a time. I told him about the derivation of Planck's law which we had to teach our boys, with all its contradictions, and how I felt the necessity of getting a derivation without inner difficulties.' In the course of their conversations, 'Saha drew my attention to certain aspects of equilibrium between radiation and
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Cas 509
electron-gas, and what Einstein and Ehrenfest 24 had published in a recent issue of the Zeitschrift filr Physik. Saha told me that Pauli had utilized Einstein's 1917 idea. 25 What seemed to be happening in Pauli's work was that in order to apply the quantum condition you had to know exactly what was going to happen afterwards. So there were certain difficulties and Saha pointed them out to me.' In 1922 Compton had discovered the increase of wavelength of X-rays due to scattering of the incident radiation by free electrons (Compton effect).26 In 1923, Compton 27 and Debye 28 identified the effect as the demonstration of the elastic collision between a light-quantum (or photon 29 ) and an electron. Pauli 23 looked for a quantum-theoretic mechanism for the interaction of radiation with free electrons. He subjected the interaction to the requirement that electrons with the Maxwellian distribution of velocities were in equilibrium with radiation, and that the spectral distribution of radiation obeyed Planck's law. Pauli obtained the expression F = Ap„ + Bp„p„,
(1)
for the distribution function F, where A and B are the Einstein transition probability coefficients. The expression for F consists of two parts, one term depending on the radiation density of the primary frequency alone, and the second, a kind of interference term, depending also on the frequency arising from the Compton effect. The second term seems to imply that 'the probability of something happening depends upon something that has yet to happen. It was to this crazy situation that Saha had drawn my attention.' S.N. Bose was thus inducted into a careful study of the papers of Debye, 30 Einstein, Compton, Pauli, and Einstein and Ehrenfest. From these Bose learned about the status of Planck's radiation law, and his two papers on the statistics of radiation grew out of this study [6, 7]. Planck's
law and the light-quantum
hypothesis
S.N. Bose began by pointing out that 'Planck's formula for the distribution of energy in the radiation from a black body was the starting point of the quantum theory, which has been developing during the last 20 years and has borne a wealth of fruit in every domain of physics. Since its publication 31 in 1901 many methods for deriving this law have been proposed. It is recognized that the basic assumptions of the quantum theory are irreconcilable with the laws of classical electrodynamics. All derivations up to now use the relation pvdv=
8iru2df _ «—E, cJ
.„. (2)
that is, the relation between the radiation density and the mean energy of an oscillator, and they make assumptions about the number of degrees of freedom of the ether, which appear in the above formula (the first factor on the right hand side). This factor, however, can be derived only from classical theory. This is the
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unsatisfactory feature in all derivations and it is therefore no wonder that attempts are being made to obtain a derivation that is free of this logical flaw. 'Einstein has given a remarkably elegant derivation. 2 He recognized the logical defect of all previous derivations and tried to deduce the formula independently of classical theory. Prom very simple assumptions about the exchange between molecules and a radiation field he found the relation Pv
~
e(em-£„)/fcT
_ l •
W
'To make this formula agree with Planck's law he had to use Wien's displacement law and Bohr's correspondence principle. 33 Wien's law is based on classical theory and the correspondence principle assumes that the quantum theory and the classical theory coincide in certain limits.' 34 Bose now declared: 'In all cases it appears to me that the derivations have not been sufficiently justified from a logical point of view. As opposed to these the light quantum hypothesis combined with statistical mechanics {as it was formulated to meet the needs of the quantum theory) [our italics] appears sufficient for the derivation of the law independent of classical theory. In the following I shall sketch the method briefly.'35 Bose's goal was to justify Planck's use of the Boltzmann formula S = klnW,
(4)
and to derive Einstein's result (3) without using Wien's displacement law and the correspondence principle. He considered the radiation, composed of light-quanta or photons of total energy E, enclosed in a volume V. If Ns quanta have the energy hvs (s = 0 , . . . , oo), then the total energy beomes
E = YlN*hv°
=V
IP»dv-
(5)
Now he tried to determine the maximum pobability for a distribution {Ns} subject to the constraint (5). A light-quantum of frequency vs has a momentum hvs/c, and its six-dimensional phase space element for the frequency range dvs is given by / dxdydzdpxdpydpz
= 4nVl — - 1 ,v2 = 4nh3-±Vdvs
hdu.
.
(6)
c
Taking into account the two polarizations of the photon, a light-quantum of frequency v and v + dv has the extension 2 x 4ir(v%/c3)Vdvsh3 in phase space. If, in accordance with Planck's theory, 36 the total phase volume is divided into cells of size h3, then there are 87rv2Vdi>/c3 cells belonging to dv.
Satyendra Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas 511
In how many ways can Ns quanta be distributed among As (= 8irv2c~3Vdvs) cells belonging to the frequency range dv3 ? Let pf, be the number of unoccupied cells, pf the number containing 1 quantum, pi the number containing 2 quanta, and so on. The number of possible distributions is then ASI/(PQ\P[\ •••). Also, N3 = OpQ + lpl +2p|-l is the number of quanta belonging to dvs. The probability of the state defined by all psr is
(?)
^=1 1 - ^ — Vpo!Pi!--" Applying Stirling's formula for large p*, with As =~%2rPr,one obtains
l0gW=Y,A°lnAs-J2T,Prl*Prs
s
(8)
r
Expression (8) must be a maximum under the constraints that the energy and particle number remain constant, that is
3
r
Carrying through the variations Bose obtained the conditions
2I3*Pr(l+lnp;) = 0, s
£)Wii/.=0,
(10a)
r
£ > ; = <>,
6N,=Ylr6p'r.
r
(10b)
r
Prom these h e obtained,
££<^(l+ln^+A s ) + i £ ^ s X>JP;=0, s
r
s
(11)
r
which led him to psr = Bte-rhv''fi
(12)
Since, however, As = ^Bse-^'H3
= Bs(l - e-hv'^)-1,
(13)
r
Bose obtained Bt=A.{\-e-hv'lfi);
(14)
from which the particle number and energy, respectively, follows as AT, = Y,P$r = J M , ( 1 - e-^,/l3y-rhua/0 r
r
= Ate-hv'lpl{l
- e-^'13)
(15)
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The Golden Age of Theoretical Physics
and „
^87r/u/?„ C3
e-WP '
(16)
-^Asln(l-e-'"''^)
(17)
1 — e-rhv,//3
3
Using the result S =k
E
S
and the relation
SS HE
1
(18)
~ r'
Bose obtained P = kT,
(19)
and the Planck formula for the energy density of black-body radiation, E
v-^ 8nhij
1
.
,„.,
s
The important step in the above derivation is the use, in a consistent manner, of Planck's method 37 of dividing the phase space into cells of volume h3. The number of cells in phase space determines the number of sites (Planck's oscillators), and one has just to inquire how often a site is occupied by the photons (or energy quanta) in order to determine the probability of a distribution. Bose drew the most natural consequences from Planck's ideas of 1906.36 In order to understand why Planck himself was not led to Bose's derivation one must realize that Planck had been looking for the justification of quantization not in free but in interacting systems. Bose, after all, considered a free quantum gas: he recognized that the light quanta are particles which must be relativistically; they are massless; their number is not conserved; and, most important, the light quanta must be treated as being indistinguishable. It is because of their indistinguishability that all arrangements of a given number of phase points within a phase cell of volume h3 must be regarded as identical arrangements, whereas in classical statistics they would be regarded as distinct. 36 Bose's derivation was thus a logical completion of Planck's ideas 39 and Debye's proof30 of Planck's radiation law, but in the process he had resolved all earlier contradictions. Immediate
impact
of Bose's
paper
Bose did not realize that in his derivation of Planck's law he had done anything new other than perhaps removing the internal contradictions of earlier derivations to his satisfaction. 'I had no idea that what I had done was really novel. I thought that perhaps it was the way of looking at the thing. I was not a statistician to the extent of really knowing that I was doing something which was really different from what Boltzmann would have done, from Boltzmann statistics. Instead of thinking
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 513
of the light quantum just as a particle, I talked about these states. Somehow this was the same question that Einstein asked when I met him: How had I arrived at this method of deriving Planck's formula? Well, I recognized the contradictions in the attempts of Planck and Einstein, and applied the statistics in my own way, but I did not think it was different from Boltzmann statistics.' Nevertheless, Bose was aware that his derivation was the logical culmination of Einstein's own line of thought. 40 Bose suddenly had the inspiration of sending the paper to Einstein in Berlin, with a request to have it translated into German and published in the Zeitschrift fur Physik,41 He wrote: Physics Department Dacca University Dated, the 4th June, 1924 Respected Sir, I have ventured to send you the accompanying article for your persual and opinion. I am anxious to know what you think of it. You will see that I have tried to deduce the coefficient 87rz/3/c3 in Planck's Law independent of the classical electrodynamics, only assuming that the ultimate elementary regions in the phasespace has the content h3. I do not know sufficient German to translate the paper. If you think the paper worth publication I shall be grateful if you arrange for its publication in Zeitschrift fur Physik. Though a complete stranger to you, I do not feel any hesitation in making such a request. Because we are all your pupils though profiting only by your teachings through your writings. I do not know whether you still remember that somebody from Calcutta asked your permission to translate your papers on Relativity in English. You acceded to the request. The book has since been published. I was the one who translated your paper on Generalised Relativity. Yours faithfully, S.N. Bose.* Einstein thought well of Bose's paper. The point which immediately appealed to him was that Bose had not made use of the wave aspects of light quanta at all. By setting aside the problem of wave-particle duality, Bose had managed to develop a unifying statistical method which Einstein could not only apply to ponderable matter but also analyse in the light of de Broglie's new idea about matter waves.5 Einstein had himself contributed to the foundation of statistical thermodynamics in his early papers, 42 and had obtained deep insight into Boltzmann's methods. He should have been able to interpret Planck's counting method immediately, but he had not succeeded in determining the statistical mechanical basis of Planck's radiation law in his various attempts since 1906. 40 ' 43 Bose had done this. 'Text of S.N. Bose's letter to Albert Einstein, transmitting his paper which Einstein translated and published in Zeitschrift fur Physik under the title 'Plancks Gesetz und Lichtquantenhypothese' [«]•
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Einstein immediately translated Bose's paper himself into German and added the following note at the end: 'Comment of the translator. Bose's derivation of Planck's formula appears to me to be an important step forward. The method used here gives also the quantum theory of an ideal gas, as I shall show elsewhere. [A. Einstein.]' Thus Bose's paper, translated by Einstein under the title Plancks Gesetz und Lichtquantenhypothese, was received by the Zeitschrift fur Physik on 2 July 1924 and was published in its issue of August 1924 [6]. Einstein also acknowledged Bose's letter and paper 'in hand-written post-card stating that he regarded the paper as an important contribution and that he will have it published.' Earlier in 1924 Bose had applied to Dacca University for a two-year leave for study abroad, but by June he had not received a definite reply. 'They were going to discuss it in the Senate Council. Einstein's postcard was a great help to me. I sent it to the Council. As soon as they showed it to Hartog [the Vice-Chancellor] it solved all problems. As a student Hartog had spent some time at the University of Paris and he understood something of what such an experience could do for a young man. That little thing [Einstein's postcard] gave me a sort of passport to the study leave. They gave me leave for two years and rather generous terms. I received a good stipend. 44 They also gave a separation allowance for the family, otherwise I would not have been able to go abroad at all. Also return fares and additional travel allowance. That was very generous. Then I also got a visa from the German Consulate just by showing them Einstein's card. They did not require me to pay the fee for the visa!' In early September 1924, S.N. Bose sailed from Bombay to Europe abroad a steamer of the Llyod Triestino line. After some stops for sightseeing on the way he arrived in Paris around mid-October 1924, and took lodgings at 17 rue du Sommerard, Paris 5. Bose's
second paper on radiation
theory
Just eleven days after sending his first paper and letter to Einstein Bose had sent him another paper with a covering letter. He wrote: Physics Laboratory Dacca University Dacca, India 15.6.24 Respected Master, I send herewith another paper of mine for your kind perusal and opinion. I hope my first paper has reached your hands. The result to which I have arrived seems rather important (to me at any rate). You will see that I have dealt with the
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
515
problem of thermal equilibrium between Radiation and Matter, in a different way, and have arrived at a different law for the probability for elementary processes, which seems to have simplicity in its favour. I have ventured to send you the typewritten paper in English: it being beyond me to express myself in German (which will be intelligible to you). I shall be glad if its publication in Zeitschrift fur Physik or any other German journal can be managed. I myself know not how to manage it. In any case, I shall be grateful if you express any opinion on the papers, and send it to me at the above address. Yours truly S.N. Bose.* Einstein did not send a reply to Bose's second letter but he had not ignored his paper. Soon after arrival in Paris Bose addressed another letter to Einstein: 17 Rue du Sommerard Paris V e 26.10.24 Dear Master, My heart-felt gratitude for taking the trouble of translating the paper yourself and publishing it. I just saw it in print before I left India. I have also sent you about the middle of June a second paper entitled 'Thermal equilibrium in the Radiation Field in presence of Matter.' I am rather anxious to know your opinion about it, as I think it to be rather important. I don't know whether it will be possible also to have this paper published in Zeit. fur Physik. I have been granted study leave by my University for 2 years. I have arrived just a week ago in Paris. I don't know whether it will possible for me to work under you in Germany. I shall be glad however if you grant me the permission to work under you, for it will mean for me the realization of a long-cherished hope. I shall wait for your decision as well as your opinion of my second paper here in Paris. If the 2nd paper has not reached you by any chance, please let me know. I shall send you the copy that I have with me. With respects, Yours sincerly, S.N. Bose.t "Text of S.N. Bose's second letter to Einstein, transmitting his paper which Einstein translated and published in Zeitschrift fur Physik under the title ' Warmegleighgewicht im Strahlungsfeld bei Anwesenheit von Materie' [7]. tText of S.N. Bose's letter to Einstein from Paris, inquiring about his second paper on radiation theory [7].
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Einstein had long since translated the paper to which Bose referred and sent it also to the Zeitschrift fiir Physik, where it had been received on 7 July 1924 and published in the August-September issue under the title 'Warmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie' [7]. This paper was not directly linked to the statistics of light quanta but was connected with Einstein's ideas 25,32 of 1916-17. It dealt with the 'crazy situation' which Saha had discussed with Bose in the early spring of 1924. Bose set himself the goal of demonstrating that the results obtained by Einstein, 25 Pauli 23 and Einstein and Ehrenfest 24 could be derived in a more consistent and conceptually simpler manner. Bose considered a system consisting of radiation and matter in interaction at thermal equilibrium. The thermodynamic probability for an arbitrary state of the system can be obtained by multiplying the probablities for the states of the radiation and the material particles respectively. The radiation probability is given by Irradiation = J ^
~T~\Tr~\
'
\^-J
where Ns quanta belong to the frequency range between vs and vs + dus and the corresponding phase space has AVs = As (= 8wi/2c~3Vdi/) phase cells. Planck's radiation law can be easily obtained from (21). 45 Bose then calculated the thermodynamic probability of an arbitrary state of the material particles. Let N particles be distributed among t elementary phase cells; then nr denotes the occupation number of the cells r, and gr the probability that any particle is in cell r. The weight factors, in general, are equal except for atomic states. Prom the requirement that the probability
,,
W
fVparUcles -
^'g?1^n l | n 2 !
...
(22)
W
be a maximum, the Boltzmann Ansatz is satisfied for the occupation numbers n r = Cgre-BlkT
.
(23)
Bose obtained the total probability of the state of the system consisting of matter and radiation by multiplying the expressions (22) and (21).
s
t
The equilibrium state is obtained from the condition that the probability W, in (24), is a maximum under the conditions that the energy,
E = Y^ Nshu° + XI n t E t ' and the particle number, N = J^rit, remain constant. At equilibrium, the following processes have to be balanced: first, a particle leaves a cell r and goes to another cell s; secondly, a photon of frequency u changes
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 517
into a photon of frequency u'. The first process may be considered as a transition from one state to another, while the accompanying radiative process may be regarded as a scattering after a collision. Taking into account all the events which contribute to the equilibrium state Bose obtained the condition of equilibrium as !ll TT gr^}Nv
Nv
Nu = — TT ' + Av gs *•} Nu, + Au, '
C251 l
;
together with the condition for the conservation of energy, J^hi/-^hv
+ Es-Er = 0.
(26)
Equation (25) can be rewritten as Hz. TT blPu = — TT b'lPv' gr AA ax + bxpv gs AA a[ + b'xpv, '
C271
with Vi=—hv
and ^ = — / " A
In the case of equal statistical weights, (27) becomes nr Y[ hPu Y\ia'i + KP"') = n » H b\pv, f [ ( a i + bxpv),
(28)
which is identical with the equation of Einstein and Ehrenfest. 24 Bose, therefore, claimed that 'the equation [(25)] is more fundamental than the transformed equation of Einstein.' 46 Bose considered the question of the equilibrium between radiating atoms and matter still further. In 1916 Einstein 32 had to assume negative absorption to obtain the Ansatz, nrbipvdt — n3(a[ + b\pv)dt. (29) Bose easily obtained from (25) the equilibrium condition, gr Av + Nv
ga '
This equation could be explained in a very simple way. The quanta of radiation are not equally distributed over the Av cells of the phase space, but p0 cells contain no photons, p\ cells contain 1 photon, and so on. Interaction can occur only in those cells in which at least one molecule (atom) and one photon are present. If there are r quanta in a cell, r + 1 possibilities can occur: no interaction, one quantum exchange, etc. Hence, there are Po + 2pi + 3p2 + • • • =
5^(r + Vl* =A» + N*
( 31 )
518
The Golden Age of Theoretical
Physics
cases, and among these only P i + 2 p 2 + --- = 5 2 r p P = JV1,
(32)
reactions. Of these, only P^2rPr might be successful if absorption is taken into account. A similar factor, a, may be connected with emission. Hence
^ATTK
= an
°'
(33)
which is identical with (30) if gr/3 = g3a, and by substituting for nr/n3 one obtains Planck's law. In order to treat the interaction of electrons and radiation (the Pauli process 23 ), let /3J be the probability of scattering a quantum with frequency vr into a quantum of frequency us in the case of a collision, and let /3* be the inverse coefficient. Then at equilibrium,
^XTK'^^VN.-
(34)
which is the same as (25) if /3r — f3* and gr = g3. The several interaction case (the Einstein-Ehrenfest process 24 ) can also be treated. Instead of one factor, as in (34), there would just arise a product of several factors, ri,si; r 2 , s 2 ; . . . etc. Bose pointed out that from his treatment it followed that the probability of interaction of a particle with a light quantum in a radiation field, given by such factors as f3% in (34), was not just simply proportional to the number of quanta present. If it were so, and one did not consider the induced emission, then one would obtain Wien's law as Pauli had done. Hence one has to alter the basic Ansatz, and Bose remarked: 'The form assumed by Pauli, and generalized by Einstein and Ehrenfest, appears, however, as a quite arbitrary hypothesis, since one cannot conceive of a simple picture of how such an expression is obtained. The other form [(25)] which we have proposed is rather simple and can be justified on the basis of elementary considerations. In deriving the probability coefficients for the interaction (or coupling as Bohr says) it had been assumed that even if a collision takes place nothing happens [i.e. no interactions needs enter] is as probable as the case of any particular coupling. This assumption is fundamental in our derivation Here we have a departure in principle from the classical theory. This hypothesis (it seems to me) is quite similar to the one which is made about the stability of stationary states... which is basic to Bohr's theory of line spectra and can be traced back to a similar cause: the probability innate in material particles for the persistence of their stationary state.' 4 7
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
519
Finally, Bose proved that the condition gr/3 = gsa,
(35)
which is necessary in order to make (33) consistent with the fundamental equation (25), can be derived from Bohr's idea concerning the weight of an atomic level. Thus gs gives the number of ways in which a state r can go into a state s by absorption, and gT denotes the complexions connected with the emission. Einstein's comment: Just as he had done at the end of Bose's previous paper [6] which he communicated to the Zeitschrift fur Physik, Einstein added a note at the close of this paper also. On this occasion, however, he expressed his disagreement. He wrote: 'I do not consider Bose's hypothesis about the probability of elementary radiative processes as being correct for the following reasons. 'According to Bose, the relation [(25)] nr N„ _ ns gr Av + Nv gs holds for the statistical equilibrium between one Bohr state and another. From this it follows that the probabilities for the transitions r —»• s and s - > r o n the left and right hand sides, respectively, of this equation must be proportional. The transition probabilities for a molecule must be related, therefore, as AV/(AU + N„):1 (if, for simplicity, we put the statistical weight of both states as being equal to 1). More cannot be inferred from the knowledge of thermal equilibrium. According to the hypothesis proposed by me, these probabilities should be proportional to Nv (i.e. to the radiation density) and Av + Nu respectively, while according to Bose's hypothesis [they should be proportional] to NU/(AV + Nu) and 1 respectively. 'According to the latter [i.e. Bose's hypothesis], the external radiation can initiate a transition from a state ZT of smaller energy to a state Zs of greater energy, but not to an inverse transition from Zs to Zr, This, however, contradicts a justifiably well-known principle that the classical theory must represent the limiting case of the quantum theory. According to the latter, a radiation field can transfer to a resonator either positive or negative energy (depending on the phase) with equal probability. The probabilities of both transitions must, therefore, depend on the radiation density, N„, in contrast to Bose's hypothesis. The extent to which the classical theory is the limiting case of the quantum theory has been shown by Planck in detail in the last edition of his book on radiation theory. 48 'Secondly, according to Bose's hypothesis, a cold body should posses an absorptive power dependent on the radiation (density decreasing with it). The cold bodies should then absorb "non-Wien" radiation more weakly than one less intense within the range of validity of Wien's radiation law. If true, it would have been detected already in the infrared radiation of hot bodies. A. Einstein.'
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Reply to Bose's letter. In his reply to Bose's letter from Paris Einstein wrote 49 : Berlin W. 30 3 November 1924 Dr. S. Bose 17 rue du Sommerard Paris Dear Colleague: Thank you sincerely for your letter of 26 October. I am glad that I shall have the opportunity soon of making your personal acquaintance. Your papers have already appeared some time ago. Unfortunately the reprints have been sent to me instead of you. You may have them any time. I am not in agreement with your basic principle concerning the probability of interaction between radiation and matter, and have given the reasons in a remark which has appeared together with your paper. Your principle is not compatible with the following two conditions: 1. The absorption coefficient is independent of the radiation density; 2. The behaviour of a resonator in a radiation field should follow from the statistical laws as a limiting case. We may discuss this together in detail when you come here. With kind regards, Yours A. Einstein.* Bose did not write to Einstein in reply to his objections for almost three months. He postponed indefinitely his 'long cherished hope' of visiting Einstein and 'working under him.' He settled down in Paris for the time being and began to pursue other goals. Stay in Paris It seems a bit queer that a young man who had just succeeded in going to Europe on a scholarship which he obtained on the basis of Einstein's recognition of his work as a major advance, whose greatest professed ambition it was to work with Einstein, should not have gone straight to Berlin where Einstein was. But no, he went to Paris. 'I wanted to go abroad directly to Berlin, but I didn't venture to go straight on because I was not sure of my knowledge of German. I came out thinking that perhaps after a few weeks in Paris I should be able to go to Berlin to see Einstein. 'Translation of Einstein's letter in German to S.N. Bose.
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 521
However two things happened: (1) friends; (2) a letter of introduction to Langevin. My friends in Paris, who received me on arrival there, took me to this boarding house where they were staying. Then they all insisted that I should stay there. Well, I found it convenient to be among friends.' In the boarding house at 17 rue du Sommerard not only were numerous Bengalis stayin [a few of whom became prominent later on], but the house served as a place of rendezvous for Indian students in Paris. The Indian circle was presided over by a distinguished French Indologist who was personally acquainted with Paul Langevin, Professeur titulaire at the College de France since 1909, and Directeur de these of Louis de Broglie. 50 'He gave the letter of introduction to Langevin.' 'Because I was a teacher and now had this opportunity of staying in abroad I wanted to try to know as much as possible and utilize it afterwards. In Dacca I had to teach both theoretical and experimental physics. Not much theoretical physics was happening at that time in Paris [!], and I did not know any experimental techniques in modern physics. My motivation then became to learn all about the techniques I could in Paris — radioactivity from Madame Curie and also something of X-ray spectroscopy. As far as the derivation of Planck's law and such things were concerned, that was something I had wanted to understand; when it was sone that was that.' Since Bose was interested in learning the experimental techniques for work in radioactivity Langevin suggested that he should pursue the possibility of working in Madame Curie's laboratory, 'perhaps with Madame Curie herself.' Langevin gave a note of introduction for her, and Bose accordingly sought an interview with Madame Curie. During their meeting she carried on the conversation in English.'She was very nice. I told her that I would remain in Paris about six months and learn French well, but I wasn't able to tell her that I knew sufficient French already and could manage to work in her laboratory. She preferred to have her own ideas and told me that I had better be around a long time, not hurry, and concentrate on the language,' Madame Curie also told Bose about another Indian student who had worked in her laboratory; he had had great difficulty 'because he had not learned French properly.' 51 This completed the interview and also brought to an end Bose's desire of learning the techniques and doing experiments in radioactivity. Since Bose had also expressed an interest in X-ray spectroscopy Langevin had provided an introduction to Maurice de Broglie as well.52 At that time Maurice de Broglie was engaged, together with Alexandre Dauvillier, on a systematic investigation of the Compton effect. After World War I his younger brother, Louis, had assisted Maurice in his experimental work. But in November 1924 Louis de Broglie had competed his doctorate under Paul Langevin, 4,53 and did not visit Maurice's laboratory anymore. 'Maurice de Broglie apparently knew about my work form the Zeitschrift fur Physik when I visited him at the laboratory.' 53a Maurice de Broglie had a fairly large laboratory at 29 rue Chateaubriand in Paris. He also maintained a small private laboratory at his residence in rue Lord Byron, adjoining the main laboratory, where even the attached bathroom had been
522
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converted for scientific use and the four-phase generator to power the X-ray tubes was housed in the stable. Maurice once invited Bose to visit him at his private residence. In the laboratory Bose was placed at the disposal of Dauvillier, Maurice de Broglie's chief assistant. There he learnt some of the techniques of X-ray spectroscopy and crystallography, and also became interested in the theoretical properties of crystals. After working for several months under Dauvillier, Bose vacated in favour of Jean Thibaud. These experiences proved to be of good use to him when K.S. Krishnan and his students engaged themselves on experiments on crystal magnetism in Dacca. In Paris, T utilized my opportunity in seeing other things at the Sorbonne as well.' After almost one year's stay in the French capital Bose went to Berlin. Berlin Bose had written to Einstein in January, 1925, from Paris as follows: 17 Rue du Sommerard Paris V e 27th January '25 Revered Master, I received your kind note of 3rd Nov. in which you mentioned your objections against the elementary law of probability. I have been thinking about your objections all along, and so did not answer immediately. It seems to me that there is a way out of the difficulty, and I have written down my ideas in the form of a paper which I send under a seperate cover. It seems that the hypothesis of negative Einstrahlung stands, which as you have yourself expressed reflects the classical behaviour of a resonator in a fluctuating field. But the additional hypothesis of a spontaneous change, independent of the state of the field, seems to me not necessary. I have tried to look at the radiation field from a new standpoint, and have sought to seperate the propagation of quantum of energy from the propagation of electro-magnetic influence. I seem to feel vaguely that some such separation is necessary if Quantum theory is to be brought in line with the Generalised Relativity Theory. The views about the radiation-field, which I have ventured to put forward, seem to be very much like to what Bohr has recently expressed in May Phil. Mag. 1924.54 But it is only a guess, as I cannot say honestly to have exactly understood all what he means to say, about the virtual fields and virtual oscillators. 55 I am rather anxious to know your opinion about it. I have shown it to Prof. Langevin here and he seems to think it interesting, and worth publishing. I cannot exactly express how grateful I feel for your encouragement, and the interest you have taken in my papers. Your first p. card came at a critical moment,
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
523
and it has more than any other made this sojourn to Europe possible for me. I am thinking of going to Berlin at the end of this winter, where I hope to have your inestimable help and guidance. Yours sincerely, S.N. Bose* The paper in which Bose had hoped to meet Einstein's objections, one which Langevin considered 'worth publishing,' was never published. There did not seem to occur any mention of it afterwards. 56 Bose had announced his intention of going to Berlin 'at the end of this winter,' where he hoped to have Einstein's 'inestimable help and guidance.' If Bose entertained any expectation of collaborating with Einstein on questions of radiation theory and quantum statistics it was already too late by the end of January 1925, when he wrote to him. Such a collaboration could certainly not have arisen when Bose did arrive in Berlin, for reasons of which he was unaware. Events had occurred in which he had taken no part. Quantum
theory of an ideal gas
Immediately on receiving Bose's first paper [6] in June 1924 Einstein saw the possibility of extending Bose's method to establish the quantum theory of a monoatomic ideal gas. He developed the theory in three communications to the Prussian Academy in Berlin on 10 July 1924, 8 January 1925 and 29 January 1925, respectively.5 Einstein extended Bose's method of treating light-quanta to material particles, thereby also solving an old problem which had been raised in 1906 in connection with Nernst's heat theorem, 57 i.e. how do ideal gases behave at very low temperatures, or how does the gas equation deviate from the ideal one under extreme conditions ? Numerous physicists, including Planck and Sommerfeld, had worked on this question, and experiments by Sackur in 1914 seemed to indicate the existence of deviations. 58 Now there was an important point of view which had characterized Einstein's work throughout: he made no difference between massive particles (of ponderable matter) and massless particles (of radiation). They were the 'elementary quanta of matter or radiation' and he treated them on an equal footing in his considerations. 59 This unified point of view helped him in extending Bose's method, and he preserved it throughout his life, as for instance in discussing the singularities in field theory. In his first communication (July 1924)5 Einstein used the concept that the phase cells of elementary probability have a minimal size of h? and Bose's counting method to obtain expressions for the particle distribution, the average energy of the particles and other properties of an ideal gas. Bose's important assumption, which 'Text of S.N. Bose's letter to Einstein from Paris announcing his intention of going t o Berlin during the winter of 1925.
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entered in the theory, was that the distribution of the molecules had to considered among the number of phase cells and not among the states. Einstein contrasted the results of this calculation with the classical theory of ideal gases, the formulae of which appeared as the limiting cases (for high temperatures) of his new theory. He showed that the new theory was consistent with Nernst's heat theorem. Finally, he considered the deviations of the new results from those of the classical theory, e.g. the influence of the quantum on the Maxwellian distribution. In his first communication Einstein developed the theory of ideal gases still further. Among the elementary particles then known — electron, proton and photon — only the photon seemed to obey the statistics which Bose had treated. The photon number is not conserved and their rest-mass is zero, which imparts special characteristics to photons. In the case of a gas of ordinary atoms the particle number is conserved and the particles are non-relativistic. Hence Einstein had to extend Bose's Ansatz. He introduced a non-zero chemical potential (which is zero for photons) in order to conserve the particle number. Einstein drew an important consequence from his theory: an ideal gas cannot be compressed into an arbitrarily small volume, That is, in the case of a gas with non-zero chemical potential there exists a critical temperature below which a finite fraction fo the gas 'degenerates' into a single quantum state; it shows a saturation effect and 'condenses' into a state with no attraction between the particles. In his second communication (8 January 1925)5 Einstein discussed in detail the differences between the counting methods of Boltzmann and Bose and clearly recognized the fact, which had been noticed earlier by Ehrenfest, 60 that the new statistics was at variance with the concept of the statistical independence of molecules. In his application of Bose's approach to the monatomic gas, Einstein realized that 'a far-reaching formal relationship between radiation and gas' can be established. He showed that if the gas particles are subjected to the new statistics, the mean-square fluctuation is given by an expression which is composed of two additive terms, one corresponding to the Maxwell-Boltzmann statistics of non-interacting molecules and the other to the interference fluctuations associated with wave phenomena. In the context of his discussion on the interference term in the energyfluctuation formula, Einstein drew attention to the importance of Louis de Broglie's thesis and discussed his considerations attaching wave properties to ponderable matter as well. Einstein emphasized it by nothing: 'I shall discuss this interpretation in greater detail because I believe that it involves more than a mere analogy.' He pursued the consequences of de Broglie's ideas and sought to intepret the viscosity of hydrogen at low temperatures, which had been measured by Gunther 61 in 1920, by means of the wave properties of hydrogen molecules. Finally Einstein considered the amount of degeneracy of his quantum gas. Whereas the ordinary molecules are too heavy to saturate, Einstein conjectured that an electron gas, if treated according to Bose statistics together with Einstein's extension of the Ansatz, would be highly degenerate because of the small mass of the electrons and effects such as the behaviour of its specific heat could be explained. 62
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 525 He showed that the quantum effect on the equation of state of an ideal gas such as hydrogen was already significant. In his third communication (29 January 1925) 5 Einstein discussed the thermodynamics of gases obeying the new (Bose-Einstein) statistics. In particular he studied the distribution function of the number of molecules of various momenta. Prom considerations concerning adiabatic compression and the action of conservative fields he derived a general form for this distribution function. In the event that Planck's constant drops out Einstein obtained the Maxwellian distribution. Einstein's extension of Bose's Ansatz thus formed a consistent scheme, and the new theory was shown to obey the same thermodynamic laws as the classical theory. The usefulness of the Bose-Einstein theory of ideal gases was to be understood only several years later, but the principal fact of its correctness would be recognized very soon. The foundation of the quantum theory of monoatomic ideal gases was Einstein's last decisive contribution to quantum theory and statistical thermodynamics. 63 The novel derivation of Planck's law, which had inspired Einstein to do this work, an inspiration which Einstein generously acknowledged in all three communications, 5 was Bose's first and only significant contribution to physics. Within six months of the publication of Bose's derivation of Planck's law with the help of his new counting method, Pauli discovered the exclusion principle obeyed by electrons. 62 The identity of particles did not therefore automatically imply BoseEinstein statistics. Early in 1926 Fermi, 64 using Planck's 'classical' method of counting the complexions, published his paper on the statistics of particles obeying Pauli's exclusion principle. A few months later Dirac 65 considered gases of free particles obeying either Bose-Einstein statistics or the statistics deduced from the exclusion principle, and linked both of them to the symmetry properties of the eigenfunctions. In 1927 Dirac 66 sought to develop a quasi-relativistic quantum theory and treated the problem of an assembly of identical systems satisfying the Bose-Einstein statistics, which interact with another different system, obtaining a Hamiltonian function to describe the motion. Dirac applied the theory to the interaction of an assembly of light-quanta with an ordinary atom and obtained Einstein's laws for the emission and absorption of radiation. He also introduced creation and annihilation operators and invented 'second quantization.' A year later Jordan and Wigner 67 developed a similar scheme for Fermi fields. Several years later Pauli 68 showed that integral spin particles obey Bose-Einstein statistics and half-integral spin particles obey Fermi-Dirac statistics. Dirac called these particles 'bosons' and 'fermions' respectively. Pauli 69 made the first application of the Fermi-Dirac statistics to the paramagnetism of an electron gas, and Sommerfeld70 applied it to the metal electrons. In 1938 Fritz London 71 suggested that the behaviour of superfluid helium (4He) should be related to Bose-Einstein condensation. Thus Einstein, following Bose's derivation of Planck's law, had started a train of developments which were to be fully understood within the context of quantum mechanics. S.N. Bose took no part in these developments.
526
The Golden Age of Theoretical
Meeting
with Einstein
Physics
and stay in
Berlin
On taking up residence in Berlin Bose wrote to Einstein on 8 October 1925 requesting an appointment to see him. Einstein was on his annual visit to Leyden at that time, and Bose's long-cherished meeting with him took place only several weeks later when Einstein returned to Berlin. The meeting had lost the value of immediacy for Einstein, if it had any, after he had made his last decisive contribution to statistical thermodynamics in January of that year. He had now moved on to other things including the beginning of his search for a unified field theory, 72 a pursuit which would remain with him for the rest of his life. But for S.N. Bose 'the meeting was most interesting. He wanted to know how I had hit upon the idea of deriving Planck's law in this way. Then he challenged me. He wanted to find out whether my hypothesis, this particular kind of statistics, did really mean something novel about the interaction of the quanta, 73 and whether I could work out the details of this business.' Bose obtained a letter of introduction from Einstein. With its help he was granted certain privileges, such as the right to borrow books from the university library and to attend the physics colloquium. 'I made good use of these privileges.' In Berlin Bose had the opportunity of meeting numerous interesting people. 'Because of this letter of introduction from Einstein I could go everywhere. I met [Richard] von Mises, [Fritz] Haber, [Michael] Polanyi, [Otto] Hahn, and [Lise] Meitner. [Walter] Gordon was there [Gordon 74 was Laue's assistant at that time], and [Eugene] Wigner was then working with Polanyi. Gordon and I used to discuss a lot among ourselves. Sometimes I would meet Einstein. He was held in very great respect. But he was very accessible. Unlike Haber he was not an administrator. I met a lot of people in Berlin.' 'I tried to work out certain things along the lines Einstein had suggested. After I had been in Berlin for some time Heisenberg's 75 paper came out. Einstein was very excited about the new quantum mechanics. He wanted me to try to see what the statistics of light-quanta and the transition probabilities of radiation would like in the new theory. 76 I discussed with Gordon and tried to understand what constituted the new line. Einstein did not believe that it was by any means the final theory. 'I also discussed questions of relativity theory with Einstein. There was this Hilbert paper 77 which we discussed. And he told me about his new ideas on unified field theory.' 72 Early in the summer of 1926 Bose visited Gottingen and heard a lecture of Max Born on quantum mechanics. 'I was there for about one day. I met [Erich] Huckel. [Hiickel, who had earlier worked with Debye in Zurich, was now Bom's assistant in Gottingen.] Heisenberg was then in Copenhagen, and I couldn't see him.' S.N. Bose did not publish any scientific papers from Paris or Berlin, nor did he do so for a long time after his return to Dacca.
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
527
Return to Dacca In the late summer of 1926 Bose returned to Dacca. Before his return home his friends had advised him to apply for a professorship of physics at Dacca University, and since he did not possess a doctor's degree they suggested that he obtain a letter of recommendation from Einstein. Einstein was surprised at the request
because he thought that Bose's scientific work should entitle him to the position. Einstein's recommendation notwithstanding, the professorship at Dacca was offered to D.M. Bose 11 ' 17 who declined. S.N. Bose was then appointed Professor of Physics and head of the department in 1927. At Dacca University Bose became involved in the normal routine of teaching, guiding students in their studies, and administration. He had no well conceived plan or programme of research, and such scientific work as he did later on was desultory. His interest shifted from one probem to another. 'On my return to India I wrote some papers. I did something on statistics and then again on relativity theory, a sort of mixture, a medley. They were not so important. I was not really in science any more. I was like a comet, a comet which came once and never returned again.' Thus in 1936, twelve years after his celebrated paper on the derivation of Planck's law, he wrote two papers on 'The moment-coefficients of £) 2 -statistic' in which the population values of the invariances and co-variances were substituted for the corresponding sample estimates [8, 9]. In 1938 Bose investigated the conditions under which the electromagnetic waves get totally reflected in the ionosphere [12]. The following year, in a paper entitled 'Studies in Lorentz group,' he employed algebraic methods to arrive at the decomposition of the Lorentz group into two commutable factors [13]. In 1941 he published 'The complete solution of the equation: y2
^ - ^ 2
~
k2<
f> =
-4np(xyzty,
the inhomogeneous scalar Klein-Gordon equation, 'which seems to have acquired some importance in view of a recent work of H.J. Bhabha 78 on the mesotron' [14]. About three years later he wrote a note on the Dirac equation in the presence of an electromagnetic field, in which he sought to give 'a new treatment for solving Dirac's equations for hydrogenic atoms' [16], and, in 1945, a paper on 'An integral equation associated with the equation for hydrogen atom' [17]. Bose's early scientific interest had been aroused by reading Einstein's papers on relativity. When Bose arrived in Berlin Einstein had already launched his programme of research on a unified field theory. 72 In 1948 Einstein 79 published a new formulation of his generalized theory of gravitation, and in 1950 a paper on the Bianchi identities in the generalized theory.80 In 1953 Einstein 81 published a note in response to a criticism of the unified field theory. These were some of Einstein's last attempts in his search for a unified theory. Bose was inspired to consider certain aspects of this theory, and during 1953-55 he published five short papers on these [19-22, 24], but they broke no new ground.
528
The Golden Age of Theoretical
Physics
Bose sent one of these papers [19] to Einstein and asked for his opinion. Bose, in his paper, had claimed that 'our result is slightly more general [than Einstein's].' Einstein wrote him a detailed letter and, just before closing it, underlined the sentence, 'This is to show that the equations Ti = 0 do not involve any arbitrary assumption... has furthermore the advantage that it exhibits transposition invariance.' 82 At Dacca University Bose also kept up an interest in experimental physics, especially the studies of thermoluminescence amd crystal structure. In 1945 Bose accepted the offer of the Khaira Professorship of Physics in Calcutta University, and also served as Dean of the Faculty of Science for a few years. S.N. Bose retired from Calcutta University in 1956 and was appointed Professor Emeritus. On retirement from Calcutta he joined Viswa-Bharati University, Santiniketan, as its Vice-Chancellor. He left this office in 1959 on his honorific appointment as a National Professor. Other Activities During the late 1930s and early 1940s S.N. Bose became increasingly involved in the movement for Indian independence. He greatly loved his native Bengal and its division in 1947 caused him deep sorrow. With Tagore, Bose shared the conviction about the necessity of basic education through the mother tongue, and from his youth he had been interested in the propagation of science among the people in their own language. In 1935 he wrote an article on 'Einstein' in a Bengali magazine called Parichaya. Bijnan Parichaya, a science magazine in Bengali, was started under Bose's inspiration in 1941 at Dacca by a group of scientists there. At Calcutta University Bose even began to teach his physics courses in Bengali. In 1948 he founded Bangiya Bijnam Parishad (Science Association of Bengal) for popularizing science in Bengali, which has since been publishing Jnan O Bijnan (Knowledge and Science), a popular monthly magazine on science in Bengali. Bose himself contributed articles to it occasionally. Though Bose had a special love for the Bengali language and literature, he also appreciated European literature and had wide cultural interests. He liked poetry and read it in Bengali, Sanskrit, English, French, German and Italian. Travel After his return home in 1926 Bose went abroad again only in 1951 and visited France. During the next few years he travelled extensively. In July 1954 he visited Paris again to attend the Third General Assembly of the International Union of Crystallography, at which he presented a report on the study of thermoluminescence conducted in his department. On this occasion Bose also visited Copenhagen to see Niels Bohr and met G. de Hevesy as well. 'It was not long after the explosion of the hydrogen bomb by the Russians, and we had a discussion about the technological and political implications of the bomb. Some people thought that the Russians had
Satyendra
Nath Base, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
529
not done it at all, and had just contrived to obtain a picture of the explosion to scare. Hevesy made a suggestion that supposing they had opened a bottle of tritium and it exploded; it would look as if it was a hydrogen explosion. Bohr laughed outright. He said, "That's not possible at all. Well, if the Russians had exploded [the hydrogen bomb] before the Americans that would mean serious trouble for Oppenheimer."' During his visit to Europe in 1953 Bose also wanted to go to the United States and meet Einstein again. 'I never met him afterwards [after 1926]. I was never in America. Some friends urged me to go to see him. However, the McCarthy situation was active, and, since I had happened to go to Russia before, they thought of me as a communist and did not give me a visa.' In 1962 Bose visited Japan to attend a conference on science and philosophy. In Japan he obtained first-hand experience of what the mother tongue could do for the education of the people. On his return home he declared: 'I have returned home from Japan firmly convinced about the necessity of adopting the language of the province [the different Indian states] as the medium of instruction in the university.' 83 Honours S.N. Bose was elected president of the Physics Section of the Indian Science Congress in 1929. He became the General President of the 31st Session of the Indian Science Congress in Delhi in 1944. In 1949 Bose was elected President of the National Institute of Sciences of India, now called the National Academy of Sciences of India. S.N. Bose was elected Fellow of the Royal Soceity in 1958, and the Government of India named him a National Professor the following year. The President of India conferred on him the honour of Padma Vibhushan. The Inspiration of Bose's Life The great inspiration of Bose's life was the work and personality of Albert Einstein. To him Einstein's personality 'was beyond comparison,' and he was forever grateful to Einstein for the encouragement he had received from him at just the right time. Immediately after Einstein's death Bose wrote: 'During the upheavals between the two World Wars Einstein suffered much. In 1933 he was forced to leave Berlin and robbed of all his possessions His indomitable will never bowed down to tyranny, and his love of man often induced him to speak unpalatable truths which were sometimes misunderstood. His name would remain indissolubly linked up with all the daring achievements of physical science of this era, and the story of his life a dazzling example of what can be achieved by pure thought.' 84 S.N. Bose felt a deep affinity with the ideas of Pierre Teihard de Chardin. In an article on Teilhard de Chardin in Bengali, Bose paraphrased his words with approval: 'We have to overcome all obstacles and impediments heroically to build future civilization, irrespective of all religious and national differences. The whole of
530
The Golden Age of Theoretical
Physics
humanity will be included in it. This is the message of hope we learn from science. A scientific attitude to life, cooperation and love in place of envy and jealousy, this is what we need, and the history of evolution shows the way to this end.' 85 However, 'Of all human beings ever born, I revere Gautama Buddha most,' Bose had declared. 85 S.N. Bose had great faith in the survival and spiritual growth of humanity. He believed in the pursuit of objective truth through science and in selfless service to mankind. He was convinced that an irresistable social evolution will carry humanity to a higher level of moral and spiritual existence. S.N. Bose was a kind and unassuming man. His needs were simple. He loved cats and music. He enjoyed many warm and loyal friendships. His early work and its immediate recognition by Einstein had made his name familiar in modern physics, and he lived the legend of his fame modestly. He became a source of inspiration to a number of his students and friends. During his life Bose was loved, and on his death mourned, by many. I wish to thank the Council of the Royal Society for inviting me to write the biographical memoir of Professor S.N. Bose. I had the opportunity of meeting S.N. Bose twice. The first meeting took place on 21 July 1953 in Paris. It came about accidently. I was a graduate student at Gottingen and had gone to Paris to visit Louis de Broglie with an introduction from my teacher, Professor Werner Heisenberg. I learnt from the Indian musician Dilip Kumar Roy that Bose was in Paris, and I sought to meet him. I kept notes of my conversation with him. For the second meeting, on 30 August 1970,1 made a special trip from Brussels to Calcutta to see S.N. Bose. On this occassion I had a long list of properly organized questions and Professor Bose had agreed in advance to discuss them with me. I came prepared with a tape-recorder which worked on 110 or 220 volts a.c. In Professor Bose's home the electric installation was on d.c. The battery of my recording device had not been charged in advance, a grave error, and was almost dead. Fortunately, Professor Bose had a transformer which he used for his U.S.-made record-player, which saved the day. We had over five hours of discussion, much of it recorded on tape,* interrupted only by the miaowing of numerous cats who wished to enter the living room to rest comfortably close to their master but had been kept outside in the interests of the concentrated pursuit of an important chapter in the history of modern physics. The article 'Satyendranath Bose: Co-founder of quantum statistics' [W.A. Blanpied, Am. J. Phys. 40,1212 (1972)] gives a readable account of Bose's discovery and scientific career. Most of the articles on S.N. Bose, a few even by competent Indian physicists, published in Science today (a Times of India publication), January 1974, and in the Bose memorial issue of the Physics News Bulletin of the Indian Physics Association, vol. 5, no. 2, June 1974, are mawkish and seek to perpetuate legends. "The tape-recording of the conversation with Professor S.N. Bose will be deposited in the archives of the Royal Society.
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 531 T h e article by Virendra Singh on the discovery of Bose statistics, in t h e former, and by P.K. Roy on Bose, t h e scientist a n d m a n , in the latter, are exceptions. It was given t o S.N. Bose to m a k e one important discovery a n d t o write a paper of four pages a b o u t it. It h a d far-reaching consequences in modern physics. T h a t has often been more t h a n enough for one m a n in the history of science. T h e p h o t o g r a p h is taken from Satyendra
Nath Bose:
70th birthday
commemo-
ration volume, C a l c u t t a (1965), p a r t I. References 1. Sir C.V. Raman, F.R.S.; see 'Chandrasekhara Venkata Raman (1888-1970),' Biogr. Mem. Fellows R. Soc. 17, 565 (1971). 2. Meghnad Sana, F.R.S. (1893-1956); see Biogr. Mem. Fellows R. Soc. 5, 217 (1959). 3. Sir K.S. Krishnan, F.R.S. (1891-1961); see Biogr. Mem. Fellows R. Soc. 13, 245 (1967). 4. L. de Broglie, 'Recherches sur la theorie des quanta,' These, Masson & Cie, Paris, 1924; Annls. Phys. 3, 22-128 (1925). 5. A. Einstein, ' Quantentheorie des einatomigen idealen Gases,' S.B. preuss. Akad. Wiss., Phys.-math. Kl. (Berlin), [10 July 1924], 261-267 (1924); 'Quantentheorie des einatomigen idealen Gases. II,' ibid. [8 January 1925], 3-14 (1925); 'Zur Quantentheorie des idealen Gases,' ibid. [29 January 1925], 18-25 (1925). 6. E. Schrodinger, ' Quantisierung als Eigenwertproblem,' Annln. Phys. 79, 361 (1926); see p. 373. 7. Rabindranath Tagore (1861-1941), Indian poet; Nobel Prize in Literature, 1913, for Gitanjali, a collection of his lyric poems originally in Bengali, translated into English by Tagore himself and into French by Andre Gide. 8. Sir Jagadish Chandra Bose, F.R.S. (1893-1972); see Obit. Not. Fell. R. Soc. Lond. 3, 3 (1940). 9. P.C. Mahalanobis, F.R.S. (1893-1972); see Biogr. Mem. Fellows R. Soc. 19, 455 (1973). 10. 'Among other things he [Asutosh Mookerji] had given a very clever explanation of the Monge differential equation of the conies.' 11. 'The preliminary negotiations with Sir Asutosh were principally carried on by S.N. Ghosh, another young man who had taken a degree in physics at that time. He was a very enthusiastic person, but was also involved in revolutionary activities because of which he had to drop his project of doing research work. He convinced Sir Asutosh that the young people, such as myself and Saha, would be equal to the task if only he would give them suitable opportunities. Theoretically, it was on the University's programme to open post-graduate classes in physics. They had already offered two professorships in physics, one to C.V. Raman and the other to D.M. Bose. Raman had not joined; he was still in the Finance Department. D.M. Bose planned to go abroad for doing further research before taking up the appointment. Raman did his scientific research during leisure hours. He was hesitating to quit his Government position because he was not sure whether the University would be able to give adequate facilities. So, before the professors arrived Sir Asutosh chose five young people to teach classes, among them Saha and me. S.N. Ghosh was wanted by the authorities because of his revolutionary activity and he escaped to America. He was there for about twenty years and then returned.' 12. J. Willard Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press, 1902. 13. A. Einstein, 'Zur Elektrodynamik bewegter Korper,' Annln. Phys. 17, 891-921 (1905).
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The Golden Age of Theoretical Physics
14. H. Minkowski, '•Die Grundgleichungen fur die elektromagnetischen Vorgdnge,' in Nachr. Ges. Wiss. Gbttingen. 53-111 (1908). 15. A. Einstein, ' Grundlage der allgemeinen Relativitatstheorie,'Annln. Phys. 49, 769-822 (1916). 16. H.A. Lorentz, A. Einstein, H. Minkowski and H. Weyl, The Principle of Relativity, A collection of original memoirs on the special general theory of relativity (with notes by A. Sommerfeld), translated by W. Perrett & G.B. Jeffery, London: Methuen & Co. (1923). 16a. The Saha-Bose translation [26] of The Principle of Relativity was reviewed in Nature, Lond. 110, 26 August 1922. The reviewer said: 'The translation cannot be called a good one. In a work of this kind we expect a fairly literal translation, but in the present book there are numerous errors in translation, and the choice of English equivalents for German words is frequently unfortunate. In many instances the mathematics is faultily reproduced. The numbering of the pages is not continuous, but recommences at the beginning of Section 4, and the omission of the footnotes form the originals is regrettable. Provided it is studied with care, the translation will nevertheless be of service to those who are unfamiliar with German, and wish to grapple with the pioneer works on this subject, some of which are rather inaccessible.' 17. D.M. Bose was appointed to the second endowed chair of physics in Calcutta University, the Rash Behari Ghosh Professorship. He and M.N. Saha attended the International Congress of Physics held at Como, Italy, 11-20 September 1927, on the occasion of the hundredth anniversary of Alessandro Volta's death. D.M. Bose presented a paper on 'The Magnetic Moments of Ions of the Transitional Group of Elements.' Sana's talk at Como was 'On the explanation of Complicated Spectra of Elements.' D.M. Bose succeeded C.V. Raman in the Palit Professorship and held it from 1932 to 1937. M.N. Saha returned to Calcutta from Allahabad in 1938 as Palit Professor. 18. N. Bohr, 'On the Constitution of Atoms and Molecules. I, Phil. Mag. 26, 1-25 (1913); 'On the Constitution of Atoms amd Molecules. II,' ibid., 476-502; 'On the Constitution of Atoms and Molecules. Ill, ibid., 857-875; 'On the Effect of Electric and Magnetic Fields on Spectral Lines,' Phil.Mag. 27, 506-524 (1914); 'On the Series Spectrum of Hydrogen and the Structure of the Atom,' Phil. Mag. 29, 332-335 (1915); 'On the Quantum Theory of Radiation and the Structure of the Atom, Phil. Mag. 30, 394-415 (1915). 19. A. Sommerfeld, 'Zur Quantentheorie der Spektrallinien,' Annln. Phys. 5 1 , 1-94, 125167 (1916); (With W. Kossel) 'Auswahlprinzip und verschiebungssatz bei serienspektren,' Verh. d. phys. Ges. 2 1 , 240-259 (1919). 20. M.N. Saha, 'On Maxwell Stresses,' Phil. Mag. 3 3 , 256 (1917); 'On the Dynamics of the Electron,' Phil. Mag. 36, 76 (1918). 21. M.N. Saha, 'Ionization in the Solar Chromosphere,' Phil. Mag. 40, 472 (1920). This work came about as a result of Saha's teaching activity. 'It was while pondering over the problems of astrophysics, and teaching thermodynamics and spectroscopy to the M.Sc. classes that the theory of thermal ionization took a definite shape in my mind in 1919.' (M.N. Saha to H.H. Plaskett, University Observatory, Oxford, in a letter dated 18 December 1946. Full letter quoted in (2), pp. 221-222.) 'I was a regular reader of German journals, which had just started coming after four years of the First World War, and in the course of these studies, I came across a paper by J. Eggert [a pupil of Nernst] in the Phys. Z. (p. 573), Dec. 1919, "Uber den Dissoziationzustand der Fixsterngase," in which he applied Nernst's heat theorem to explain the high ionization in stars due to high temperatures, postulated by Eddington in the course of his studies on stellar structures I saw at once the importance of introducing
Satyendra Nath Bose, Bose—Einstein Statistics, and the Quantum Theory of an Ideal Gas 533
21a. 22. 23. 24. 25. 26. 27. 28. 29. 30.
31.
32.
33.
34. 35. 36.
the value of "ionization potential" in the formula of Eggert, for calculataing accurately the ionization, single or multiple, of any particular element under any combination of temperature and pressure. I thus arrived at the formula which now goes by my name.' Quoted by N.D. Sengupta in Science Today (a Times of India publication), January 1974, pp. 42-43. M.N. Saha was born on 6 October 1893 in the village of Seoratali, Dacca District. W. Pauli,' Uber das thermische gleichgewicht zwischen strahlung undfreien elektronen,' Z. Phys. 18, 273 (1923). A. Einstein & P. Ehrenfest, 'Zur Quantentheorie des Strahlungs gleichgewichts,' Z. Phys. 19, 301 (1923). A. Einstein, 'Quantentheorie der Strahlung,' Phys. Z. 18, 121 (1917). A.H. Compton, 'Secondary Radiations Produced by X-Rays and Some of Their Applications to Physical Problems,' Bull. Nat. Res. Coun. 4 (Part 2), October 1922. A.H. Compton, 'A Quantum Theory of the Scattering of X-rays by Light Elements,' Phys. Rev. 2 1 , 483 (1923). P. Debye, 'Zerstreung von Rontgenstrahlen und Quantentheorie,' Phys. Z. 24, 161 (1923). The name 'photon' was given to a light-quantum by G.N. Lewis in 1926: 'The Conservation of Photons,' Nature, Lond. 118, 874 (1926). P. Debye, 'Der Wahrscheinlichkeitsbegriff in der Theorie der Strahlung,' Annln. Phys. 33,1427 (1910). Debye considered the radiation cube (as Rayleigh and Jeans had done) and enumerated the totality of the proper modes of the electromagnetic vibrations. Debye ascribed the energy value hv to each of these proper modes. Accordingly, his energy quantum was not restricted to atomic or molecular features as with Planck but was associated with proper modes which are distributed over the whole cube. He found the probability for the quanta! distribution as well as the most probable state, just as Planck had done, by the variation method. The temperature thus entered via the entropy. The distribution function so obtained is Planck's radiation law. In the fourth edition of his monograph [Die Theorie der Warmestrahlung] Planck himself adopted Debye's method which he characterized 'as an extremely simple deduction' of his radiation law. M. Planck, 'Zur Theorie des Gesetzes der Energieverteilung im Normalspektrum,' Verh. d. Deutsch. Phys. Ges. 2, 237 (1900); 'Uber das Gesetz der Energieverteilung im Normalspektrum,' Annln. Phys. 4, 553 (1901). See Ref. 25 and the earlier publications: A. Einstein, 'Zur Quantentheorie der Strahlung,' Mitt. Phys. Ges., Zurich 16, 47 (1916); 'Strahlungs-emission und-absorption nach der Quantentheorie,' Verh. d. Deutsch. Phys. Ges. 18, 318 (1916). Einstein did not use Bohr's correspondence principle in deriving equation (3). He used Bohr's concept of the stationary states. Bohr wrote explicitly on what he later called 'the correspondence principle' first in 1918, and referred specifically to Einstein's work: 25 N. Bohr, 'On the Quantum Theory of Line Spectra Part I: On the General Theory,' Skriffer, nat. mat. Dan. Vid. Selsk. 8, 4, No. 1 (1918). See [6], pp. 178-179. See [6], p. 179. Planck's theory of the elementary region of phase space was first given in his Vorlesungen uber die Theorie der Warmestrahlung, Leipzig; J.A. Barth (1906), in Section 150, for an oscillator having one degree of freedom. O. Sackur ['Die Anwendungen der kinetischen Theorie der Gase auf chemische Probleme,' Annln. Phys. 36, 958 (1911)] and H. Tetrode [Die chemische Konsante der Gase, Annln. Phys. 38, 434 (1912)] extended it to motions having three degrees of freedom.
534
The Golden Age of Theoretical Physics
37. See M. Planck, 'Die Quantenhypothese fur Molekeln rait mehreren Freiheitsgraden,' Verh. d. Deutsch. Phys. Ges. 17, 418, 438 (1915). 38. See E.C.G. Sudarshan and J. Mehra, 'Classical Statistical Mechanics of Identical Particles and Quantum Effects,' Int. J. Theoret. Phys. 3, 245 (1970). 39. See Refs. 36 and 37, i.e., the ideas involved in Planck's derivation of the radiation law and in phase space quantization. 40. In 1905 59 Einstein introduced the concept of the light-quantum on the basis of thermodynamic considerations which he had developed earlier. 42 During 1906-9 43 he classified and strengthened the thermodynamical arguments underlying Planck's theory and drew definite conclusions about the microscopic phenomena themselves. Einstein pointed out clearly that the concept of the energy quantum, in the sense of Planck, was in contradiction to the continuum basis of classical electrodynamics. Einstein's report on the constitution of radiation in Salzburg (1909) 43 presented the important conclusion that the elementary processes must be directed not only for absorption but also for emission of radiation. In his subsequent work he supported his postulate of a directed emission process by strong thermodynamical arguments. Bohr's successful application of quantum theory to the explanation of line spectra removed the dependence on such particular systems as Planck's oscillators. By using assumptions which hold for all atomic systems, such as Bohr's conception of the staionary states, Einstein derived 25 Planck's radiation law. In this derivation he used general statistical laws for spontaneous and induced emission processes, and for absorption processes which are the inverse of the former; he assumed the validity of two general relations between the three coefficients which determine the frequency of these processes and which, if one of these coefficients is give, permits the computation of the other two. Besides this derivation of Planck's law Einstein also discussed, in the same paper, 2 5 the exchange of momentum between the atomic system and the radiation in a general way, using his theory of Brownian motion. However, the appropriate counting method for the light-quanta, in the sense of Boltzmann and Planck, had escaped Einstein during all these years. S.N. Bose was aware of it. It was, therefore, logical that he should send his paper to Einstein. (Also see the following reference.) 41. The only man, other than Einstein, who had reflected deeply about the role of the quantum of action, energy quantum, and the fundamental contradiction between classical electrodynamics and the blackbody radiation law, was Planck himself. One wonders what the consequence would have been if Bose had sent his paper to Planck instead of Einstein. It is quite possible that Planck would have arranged its publication, but it is very unlikely that he would have given Bose's derivation his endorsement, and still more unlikely that he would have proceeded to develop the quantum theory of ideal gases on its basis, the daring thing which Einstein set out to do. 42. A. Einstein, Kinetische Theorie des Warmegleichgewichtes und des zweiten Hauptsatzes der thermodynamik,' Annln. Phys. 9, 417 (1902); 'Theorie der Grundlagen der Thermodynamik,' Annln. Phys. 1 1 , 170 (1903); 'Allgemeine molekulare Theorie der Warme,' Annln.Phys 14, 354 (1904). See also J. Mehra, 'Einstein and the Foundation of Satistical Mechanics,' Physica 79A, 447-477 (1975). 43. A. Einstein, 'Theorie der Lichterzuegung und Lichtabsorption,' Annln. Phys. 20, 199 (1906); 'Zum gegenwartigen Stande des Strahlungsproblems,' Phys. Z. 10, 185 (1909); 'Entwicklung unserer Anschauungen iiber das Wesen und die Konstitution der Strahlung,' Phys. Z. 10, 817 (1909). Also Refs. 25 and 32. 44. 250 Indian rupees per month. 45. In fact the manner of deriving Planck's radiation law from equation (21) is not very different from Debye's, 30 which Bose criticized ([7], p. 384): 'Debye has shown that
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Gas 535 Planck's law can be derived from statistical mechanics. His derivation, however, is not completely indpendent of classical electrodynamics in so far as he uses the concept of eigen-oscillations of the ether and assumes that, in so far as the energy is considered, the spectral region between v and u + dv can be replaced by 8iri/2/c~3Vdv resonators whose energy can take on only those values which are integral multiples of hv. One can, however, show that the derivation might be altered in such a way that one does not have to borrow from the classical theory, %-KV2 / c~3V dv can be interpreted as the number of elementary regions in the six-dimensional phase space for the quanta. The rest of the calculation remains essentially the same.' 46. See [7], p. 388. 47. See [7], p. 391. 48. Einstein is not referring here to Bohr's correspondence principle. That a quantum result should pass into the classical result in the appropriate limit had been used as heuristic principle by Planck and Einstein from the beginning. 49. A copy, typed in 1960 from Einstein's original draft of the letter to Bose, was made available to me by Miss. Helen Dukas, Einsteins's former secretary and custodian of the Einstein archives at Princeton, N.J., in May 1970. Miss Dukas also provided copies of the letters of S.N. Bose to Einstein in this memoir. I wish to express my gratitude for this courtesy. 50. See Paul Langevin, F.R.S. (1872-1946), by F. Joliot in the Obit. Not. Fell. R. Soc. Lond. 7, 405 (1951). 51. When Madame Curie first arrived in Paris from Poland as a young woman she had enormous difficulties with the French language, even though she thought that she had learnt 'sufficient French' at home. It became one of her goals to master French, both written and spoken, as soon as possible — and she did. She believed ever afterwards that it was essential to know the language of a country really well in order to participate in its scientific and cultured life. The faith of converts is deep and sometimes unreasonable. See R. Reid, Marie Curie (London: Collins, 1974). 52. Maurice de Broglie (1875-1960) was a distinguished experimental physicist who worked primarily in the field of X-ray spectroscopy. In 1911 he had been one of the scientific secretaries (the other was R. Goldschmidt of the Universite fibre de Bruxelles) of the first Solvay Conference on La theorie du rayonnement et les quanta (ed. P. Langevin and M. de Broglie), Paris: Gauthier-Villars (1912). For a biographical notice of M. de Broglie see Dictionary of Scientific Biography, Vol. II, New York: Charles Scribner's Sons (1970), pp. 487-488. 53. I learnt from Louis de Broglie that Paul Langevin was at first a bit hesitant in accepting the ides contained in his thesis. He did not think that they were right. Since Langevin was close to Einstein and thought highly of his opinions, he informed Einstein about de Brogue's ideas and also sent him a copy of the thesis during summer 1924. Einstein's response, communicated in September or October 1924, was favourable: he found the thesis 'very interesting.' As a result Langevin modified his opinion in favour of de Broglie's work. At the occasion of the defence of de Broglie's thesis [examination] the committee consisted of J. Perrin (chairman), P. Langevin (who made the report), C -V. Maugin and E. Cartan. 'Mr. Cartan was not very interested.' 53a. In a recent meeting (18 June 1975) in Paris, Louis de Broglie told me that he did not meet S.N. Bose during his stay (1924-25) in Paris nor at any other time later on. 54. N. Bohr, H.A. Kramers and J.C. Slater, 'The Quantum Theory of Radiation,' Phil. Mag. 47, 785 (1924). 55. In this paper [Ref. 54] Bohr had proposed that energy is conserved only statistically in radiative processes. Einstein was totally against such ideas. Later on he would strongly
536
56. 57.
58. 59. 60.
61. 62.
63. 64.
65. 66. 67. 68.
69. 70.
The Golden Age of Theoretical
Physics
resist the claim that statistical interpretation, such as one has in quantum mechanics, could belong to a 'complete' or 'final' theory of natural phenomena. Einstein's concern about the 'statistical' as against 'complete' causality had arisen soon after his own work 25,32 on the absorption and emission of light. Thus, on 27 January 1920, he had written to Max Born: 'That business about causality causes me a lot of trouble, too. Can the quantum absorption and emission of light ever be understood in the sense of complete causality requirement, or would a statistical residue remain? I must admit that there I lack the courage of my convictions. But I would be very unhappy to renounce complete [Einstein's emphasis] causality' [The Born-Einstein Letters, London: Macmillan (1971)]. This was before Einstein's first meeting with Niels Bohr in Berlin later in April 1920. 'I don't think that this work came up for discussion when I met Einstein. It was, of course, not published.' W. Nernst, ' Uber die Berechnung chemischer Gleichgewichte aus thermischen Messungen,' Nachr. Ges. Wiss. Gottingen, 1906, 1-39; 'Uber die Bezeiehungen zwischen Warmeentwicklung und maximaler Arbeit bei kondensierten Systemen,' S.B. preuss. Akad. Wiss. 933-940 (1906). O. Sackur, Ber. dt. chem. Ges. 47, 1318 (1914). A. Einstein, 'Uber einen die Erzeugung und Verwandlung des Lichte betreffenden heuristischen Gesichtspunkt,' Annln. Phys. 17, 132-148 (1905). P. Ehrenfest had treated this question in the context of clarifying the distinction between Planck's energy quanta and Einstein's conception of the light-quantum [' Welche Zuge der Lichtquantenhypothese spielen in der theorie der Warmestrahlung eine wesentliche RolleV Annln. Phys. 36, 91-118 (1911)]. Ehrenfest and H. KammerlinghOnnes took up this question again in: Simplified deduction of the formula from the theory of combinations which Planck uses as the basis of his radiation theory, Proc. Amsterdam Acad. 870-873 (1914). These papers are included in Paul Ehrenfest's Collected Scientific Papers, Amsterdam: North-Holland Publishing Company (1959). P. Gunther, S.B. preuss. Akad. Wiss. 36, 720 (1920). W. Pauli's work on the Exclusion Principle [' Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren,' Z. Phys. 3 1 , 765-783 (1925)] had not yet appeared. Moreover, as I have already mentioned, Einstein did not make any rigid distinction between 'the elementary quanta of matter and radiation.' Indeed, when the Bose-Einstein statistics appeared many people thought that it was the only quantum statistics. For Einstein's earlier contributions to statistical thermodynamics, see (42). E. Fermi, 'Sulla quantizzazione del gas perfetto monoatomico,' Rend. Lincei 3,145-149 (1926); 'Zur Quantelung des idealen einatomigen Gases,' Z. Phys. 36, 902-912 (1926). P.A.M. Dirac, 'On the Theory of Quantum Mechanics,' Proc. R. Soc. Lond. A 112, 661-77 (1926) P.A.M. Dirac, 'The Quantum Theory of Emission and Absorption of Radiation,' Proc. R. Soc. Lond. A 114, 243-265 (1927). P. Jordan and E. Wigner, Z. Phys. 47, 631 (1928). W. Pauli, ' Theorie quantique relativiste des particules obeissant a la statistique de Eintein-Bose,' Annls. Inst. Henri Poincare 6, 137-152 (1936); 'The Connection Between Spin and Statistics,' Phys. Rev. 58, 716-722 (1940). W. Pauli, 'Uber Gasentartung und Paramagnetismus,' Z. Phys. 4 1 , 81-102 (1927). A. Sommerfeld, 'Zur Elecktronentheorie der Metalle und des Volta-Effektes nach der Fermi'schen Statistik' Atti del Congresso Internatzionale dei Fisici, 11-20 Septembre
Satyendra
71.
72. 73.
74. 75.
76. 77. 78. 79. 80. 81. 82.
83. 84.
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
537
1927, Bologna: Nicola Zanichelli (1928), vol. 2, pp. 449-473; Z. Phys. 47, 43-60 (1928); Ber. dt. chem. Ges. 6 1 , 1171-1180 (1928). F. London, 'The A-Phenomenon of Liquid Helium and the Bose-Einstein Degeneracy,' Nature, Lond. 141, 643-644 (1938); 'On the Bose-Einstein Condensation,' Phys. Rev. 54, 947-954 (1938). A. Einstein, 'Einheitliche Feldtheorie von Gravitation und Elektrizitat,' S.B. preuss. Akad. Wiss. 414-419 (1925). In order to understand this question properly one has to remind oneself that by 'quanta' Einstein meant in particular his light-quanta of 1905, 59 i.e. similar to independent (classical) particles and having the property of Boltzmann's molecules. The 'Bose light-quanta' or photons, on the other hand, were endowed with (classical) particle and (classical) wave properties at the same time. Hence, from the point of view of Einstein's original quanta, it was quite appropriate to ask whether the new statistics was equivalent to introducing a specific interaction between Boltzmann's independent molecules. Einstein's question was the more relevant as, at least since 1918, he had been thinking of deriving the quantum behaviour of microscopic objects and systems from solutions of a nonlinear (classical) field theory. During 1925, Einstein had again been occupied with this problem. 72 Within the framework of quantum theory, i.e. starting from a quantum field theory involving photons and electrons, the problem of the interaction of photons with each other was first attacked by M. Delbriick [Z. Phys. 84, 144 (1933)] who treated the scattering of light by an electrostatic field in second order, and then by O. Halpern [Phys. Rev., 44, 855 (1934) who considered the interaction of two photons via the virtual exchange of four electrons (the virtual pair production in the vacuum giving rise to nonlinear interactions between electromagnetic fields). Thus the bosons of the electromagnetic field (but not Einstein's 1905 light-quanta) can exert forces on each other leading to effects which, however, are small under normal conditions. Walter Gordon of the Klein-Gordon equation: W. Gordon, Z. Phys. 40, 117 (1926); O. Klein, Z. Phys., 41, 407 (1927). W. Heisenberg, ' Uber quantentheoretische Umdeutung kinematicher und mechanischer Beziehungen,' Z. Phys. 3 3 , 879-893 (1925); M. Born, W. Heisenberg and P. Jordon, 'Zur Quantenmechanik II, Z. Phys. 35, 557-615 (1926). This was done by Dirac early in 1927. 66 D. Hilbert, 'Die Grundlagen der Physik,' Math. Annln. 92, 1-32 (1924). H.J. Bhabha, Proc. R. Soc. Lond. A 172, 384 (1939). A. Einstein, 'A Generalized Theory of Gravitation,' Rev. Mod. Phys. 20, 35-39 (1948). A. Einstein, 'The Bianchi Identities in the Generalized Theory of Gravitation,' Can. J. Math. 2, 120-128 (1950). A. Einstein, 'A comment on a Criticism of Unified Field Theory,' Phys. Rev. (2) 89, 321 (1953). Letter to S.N. Bose from A. Einstein, 22 October 1953. [Professor Bose kindly provided me with a copy of this letter.] Bose remarked to me: 'In my paper I had said something to the effect that it went a little farther than his theory. He didn't like it very much. He stayed away from this [my] thing and wanted to give me some idea as to what had led him to do so.' [Conversation with Professor S.N. Bose, 30 August 1970.] Quoted by P. Ghosh in Sci. Today, p. 45. 2 1 a See [25], p. 516.
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85. Quoted by N.N. Ray, Professor S.N. Bose: 'An Impression of His Personality,' Satyendranath Bose: 70th Birthday Commemoration Volume, Calcutta (1965), part I, p. 11. Ray's article contains useful information about S.N. Bose, intermingled with a touching personal tribute.
Bibliography of S.N. Bose [1] 1918: (With M.N. Saha) 'On the Influence of the Finite Volume of Molecules on the Equation of State,' Phil. Mag. 36, 199. 1919: 'The Stress-Equations of Equilibrium,' Bull. Calcutta Math. Soc. 10, 117 'On the Horpolhode,' Bull. Calcutta Math. Soc. 1 1 , 21. 1920: (With Megh Nad Saha) 'On the Equation of State,' Phil. Mag. Ser. 6, 39, 456. 'On the Deduction of Rydberg's Law from the Quantum Theory of Spectral Emission,' Phil. Mag. 40, 619. 1924: 'Plancks Gesetz und Lichtquantenhypothese,' Z. Phys. 26, 178. 1924: ' Warmegleichgewicht im Strahlungsfeld bei Anwesenheit von Materie,' Z. Phys. 27, 384. 1936: 'On the Complete Moment-Coefficients of the Z? 2 -Statistic,' Sankhya: Indian J. Statist. 2, 385. 1937: 'On the Moment Coefficients of the Z) 2 -Statistic and Central Integral and Differential Equations Connected with the Multivariate Normal Population, Sankhya: Indian J. Statist. 3, 105. 'Recent Progress in Nuclear Physics,' Sci. Cult. 2, 473. (With S.R. Khastgir) 'Anomalous Dielectric Constant of Artificial Ionosphere,' Sci. Cult. 3, 335. 1938: 'On the Total Reflection of Electromagnetic Waves in the Ionosphere,' Indian J. Phys. 12, 121. 1939: 'Studies in Lorentz Group,' Bull. Calcutta Math. Soc. 3 1 , 137. 1941: (With S.C. Kar) The Complete Solution of the Equation: V V - (62ip/c2dt2) k2
Satyendra Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
539
[25] 1955: Albert Einstein. Sci. Cult. 20, No. 11. Book [26] 1920: The Principle of Relativity, Original papers by A. Einstein and H. Minkowski, translated into English by M.N. Saha and S.N. Bose, with an historical introduction by P.C. Mahalanobis, University of Calcutta.
Appendix* Just one week after submitting the German translation of Bose's paper to Zeitschrift fur Physik, Einstein presented a communication to the Prussian Academy on 10 July 1924, bearing the title ' Quantentheorie des einatomigen idealen Gases' ('Quantum Theory of the Monatomic Ideal Gas,' Einstein, 1924c). In a short abstract, published also in the Sitzungsberichte of the Academy, he summarized the content of the paper as: 'the method, upon which Mr. Bose based his derivation of Planck's radiation formula, may be applied also to ideal gases. Thus one arrives at a deviation from the classical equation of state for ideal gases at low temperatures' (Einstein, 1924c', p. 241). Several reasons might have motivated Einstein to deal with the quantum theory of ideal gases on the basis of Bose's method. Among them stood out the possibility of treating light-quanta and material particles, such as atoms and molecules, on the same footing, a possibility which Einstein had always kept in mind in his work on kinetic theory and relativity.* The other important reason was the fact that 'a quantum theory of monatomic gases, free of arbitrary assumptions, does not exist so far' (Einstein, 1924c, p. 261). The necessity for developing such a theory had arisen from the following observation. In classical theory monatomic, ideal gases were described by the equation of state, PV
= crRT,
(1)
relating the pressure p, volume V and temperature T of a gram-atoms, with R the universal gas constant. Equation (1) implied the equation for the energy of the gas, Ev = ^aRT,
(2)
from which the value cv followed for the specific heat (of a gram-atom) at constant volume. In his talk at the 84th Naturforscherversammlung in Miinster on 18 September 1912, Walther Nernst had emphasized that for very low and very high temperatures deviations must be expected from Eq. (2): he had claimed, for instance, that for low temperatures this equation should be replaced by an equation •Added in 1982. * Einstein emphasized this motivation in a later paper (his third on the theory of an ideal gas), where he stated: 'This theory appears to be justified in case one starts from the conviction that a light-quantum is distinguished (apart from its polarization property) from a monatomic molecule basically by the fact t h a t its rest mass is arbitrarily small' (Einstein, 1925b, p. 18).
540
The Golden Age of Theoretical Physics
similar t o one valid for m o n a t o m i c solids (Nernst, 1912a, p . 1066), t h a t is,* Ev
= 3*R
J"/**
exp(hv/kT)
(3)
. - 1
*• '
In the following years many physicists h a d attacked t h e problem of applying quant u m theory t o m o n a t o m i c gases, among t h e m O t t o Sackur (1911), H. Tetrode (1912), Willem Hendrik Keesom (1914b), Wilhelm Lenz a n d Arnold Sommerfeld (Sommerfeld, 1914b), P a u l Scherrer (1916a) and M a x Planck (1916b). Scherrer and, especially, Planck had applied the q u a n t u m theory of several degrees of freed o m to the m o t i o n of atoms; t h a t is, they h a d t r e a t e d t h e m o n a t o m i c molecules like system of t h r e e degrees of fredom a n d quantized their phase spaced For a monatomic ideal gas, consisting of n molecules, Planck h a d obtained t h e expression of entropy as S = nk In
r
/c\
2 nkT
M
^{T>wn<- " ^
i
(4)
where m denotes t h e mass of the molecules a n d k t h e B o l t z m a n n constant. (Evidently, one could replace t h e factor nk by aR, where a is the number of g r a m - a t o m s a n d R t h e universal gas constant.) Planck h a d later o n drawn certain consequences from his theory of monatomic gases, which seemed t o agree with t h e known behaviour of such gases (Planck, 1922a,b). Still, Planck h a d m a d e some specific assumptions and t h e s e had been criticized, for example, by P a u l Ehrenfest a n d Viktor Trkal (1920, 1921). Not only did there exist the problem of counting t h e distributions of molecules appropriately — about which Ehrenfest a n d Trkal a n d Planck disagreed — b u t t h e question of whether t h e atoms possessed zero-point energy or not also played a role. Hence, Einstein, who h a d been interested in t h e problem of zero-point energy previously, felt t e m p t e d t o clean u p the entire situation in t h e q u a n t u m theory of monatomic ideal gases. And for doing so, he welcomed Bose's method. In his communication to the Prussian Academy on 10 July 1924, Einstein described Bose's m e t h o d a n d its extension as consisting of four steps: [First:] The plane space of an elementary object (here of a monatomic molecule [and in Bose's case, of a light-quantum]), associated with a given (three-dimensional) volume, is divided into 'cells' of extension h3. [Second:] If many elementary objects are present then their (microscopic) distribution, which plays a role in thermodynamics, is determined by the particular manner how the elementary objects are distributed among these cells. [Third:] The 'probability' of a macroscopically defined state (in the sense of Planck) is equal to the number of different microscopic states by means of which the macroscopic state can be thought to be realize. [Fourth:] The entropy of the macroscopic state and, 'Nernst derived this conclusion by assuming that in the collision processes between atoms radiation might be created, which had to obey Planck's law. For high temperatures, deviations must occur because then the rotational degree of freedom of the atom will be excited. tThe previous authors had expanded the thermal motion of the gas into Fourier series involving the characteristic vibrations of various frequencies v\ they had then quantized the vibrations, i.e., assigned to each of them the average, temperature-dependent energy hv[exp(hu/kT) — l ] - 1 .
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
541
therefore, the statistical and thermodynamic behaviour of the system is determined by Boltzmann's theorem [relating entropy and probability]. (Einstein, 1924c, p. 26) Einstein then carried out these steps in the example of a monatomic ideal gas. In order to determine the phase-space volume he considered molecules of mass m with position variables, x, y, z, and momentum variables, px,Py,Pz, hi a volume V. The phase-space volume of molecules having energy between e and e + Ae was given by the expression 27rV(2m) 3 / 2 e 1 / 2 Ae; hence the number of cells was As, where As = 27r^(2me)V2^,
(5)
a large number, provided the volume V was taken to be large enough. If there were n molecules in volume V, of which An possessed energies between e and e+Ae, then the microscopic distribution of the An molecules was determined by the occupation of As cells, or by the probabilities p* (r = 0 , 1 , 2 , . . . , with ]>2r Pr = 1)> which stated that PQAS cells contained no molecules, p\As cells contained one molecule, p | A s cells contained two molecules, etc. The macroscopic probability of this particular state, then, could be obtained by computing the number of all possibilities to establish the microscopic distribution. For it, Einstein found the result As! w=
(6)
n~o(p?A s ) r
in agreement with Bose's result. From w followed the entropy 5 of the macroscopic state of n molecules in the volume V, of which An possessed an energy between es and ea + dea, according to Boltzmann's relation, or
S=-kJ2(PrhPr)-
(7)
r,s
The equilibrium state was defined as the state having the maximum entropy. Eintein computed this maximum by varying the right-hand side of Eq. (7) under the constraints of a constant number of molecules n (= ]T}r rp*) and a constant total energy E ( = YLr s e s^Pr)- This procedure yielded the result S
= ~k j E W l - exp(-[A + Bs2'*})} - An - ^E\
,
(8)
where, A, B and C were denned by the equations n = X > x p ( A + -Bs2/3) - I ] " 1 ,
(8a)
8 S2/3
•r-^
E = C
\eMA
+ BsW)-V
and h2 / 4
\~2/3
(8b)
542
The Golden Age of Theoretical Physics
T h e relation between entropy and t e m p e r a t u r e , dS/dE ratio B/C as
— 1/T, further fixed the
<M>
§-£•
E q u a t i o n s (8) a n d (8a-d) determined completely the behaviour of t h e m o n a t o m i c ideal gas. For example, they provided immediately t h e equation of s t a t e
P
~
d(E-TS) dV
2E ~3V
W
E q u a t i o n (9) established the same relation between t h e pressure p and t h e average kinetic energy E as t h e corresponding relation of t h e classical theory. Einstein noticed t h a t if the quantity exp(A + Bs2^3) were assumed to be large compared with unity, other results of classical theory could also be obtained. For instance, with this assumption, t h e entropy S became equal t o Planck's expression given by Eq. (4); a n d this expression agreed with the one derived from the classical theory u p to a constant. However, t h e new theory went far beyond the classical approximations. As a n especially important result Einstein emphasized the following: According to the theory presented here, Nernst's theorem is satisfied in the case of ideal gases. To be sure, our formulae cannot immediately be applied to extremely low temperatures, for we have assumed in their derivation that the psT change only relatively infinitely little if s is altered by 1. Still one recognizes at once that the entropy must vanish at the absolute zero of temperature. The reason is that then all molecules are in the first cell; and for this state there exists only one distribution of molecules according to our counting method. Hence our assertion is immediately proved to be correct. (Einstein, 1924c, p. 265)* T h e classical limit of t h e new q u a n t u m theory of ideal gases followed by taking large values of exp(A + Bs2?3) compared to unity. This implied t h a t the quantity exp(—A), which Einstein called 'a measure of t h e "degeneracy"' a n d denoted by A (Einstein, 1924c, p. 266), h a d to be a small quantity. For hydrogen under normal conditions (i.e., at room t e m p e r a t u r e a n d one a t m o s p h e r e pressure) A assumed t h e value of a b o u t 6 x 1 0 - 4 , hence the classical theory did describe t h e behaviour correctly; however, for helium close to t h e critical state, t h e situation was different, hence large deviations from t h e classical behaviour h a d t o be expected due t o a comparatively large value of A.t Q u a n t u m theory also changed t h e Maxwellian *At that time the validity of Nernst's theorem was regarded as a crucial test. On 25 July 1924 Nernst became sixty years old. Two days later Max von Laue celebrated this event by an address to the Physikalische Gesellschaft zu Berlin, in which he emphasized the importance of Nernst's heat theorem and remarked: 'In thermodynamics, this new theorem has proved its fruitfulness. What its statistical significance might be, will not be recognized in full clarity until the quantum mystery has been solved' (Laue, 1924, p. 43). tThe factor multiplying ffcT was (J27=i r~5/2^T)/(£,^=1 T-3/2XT), or (1 - 0.0318A) for small A (such that A2 could be neglected compared to A).
Satyendra
Nath Bose, Bose-Einstein
Statistics,
and the Quantum
Theory of an Ideal Gas
543
velocity distribution of the molecules: in a quantum gas slower molecules occurred more frequently.* At the end of his paper Einstein discussed a special problem: the mixture of two types of monatomic ideal gases. He claimed that the new theory implied that the gas possessed — as in the classical theory — it own, separate phase space; hence the entropies of the two gases, each given by the expression (8), added up to give the entropy of the mixture. This situation, however, implied a paradox, which Einstein described as follows: A mixture containing n\ and ri2 molecules of the first and the second kind, which differ arbitrarily little (especially with respect to their molecular masses m i and m.2) from each other, thus yields — for a given temperature — a different pressure and distribution of states than a uniform gas having a number of molecules ni -+- n.2 of practically the same molecular mass and the same volume. This appears to be a really impossible conclusion. (Einstein, 1924c, p. 267) In spite of the above-mentioned paradox, Einstein continued to investigate his new theory further. He wrote to Ehrenfest: 'The quantum thing has turned out to be very interesting. It seems to me more and more that much that is true and deep lies behind it' (Einstein to Ehrenfest, 2 December 1924). He presented new results in a second note, communicated to the Prussian Academy on 8 January 1925 (Einstein, 1925a). In this note he discussed the consequences derived from Eqs. (8) and (8a-d). He remarked at the outset that in quantum theory, in contrast to the classical theory, the volume of a given ideal gas at a given temperature could not be decreased below a certain finite value. This restriction followed from the fact that the degeneracy parameter A assumed values only between 0 and 1. Since A and the number of molecules n were related by the equation
Einstein concluded that if one tried to increase the density of the gas 'then a number of molecules, which increases monotically with the density, goes over to the first quantum state (state without kinetic energy), while the other molecules distribute themselves in accordance with the parameter value A = 1' (Einstein, 1925a, p. 4). Hence he claimed that a situation occurred similar to the one in which vapour was concentrated in a volume smaller than the saturation volume. That is, 'a separation [of the gas molecules] occurs; one part 'condenses,' the rest remains a 'saturated ideal gas' (A = 0, A = 1)' (Einstein, 1925a, p. 4). He then showed that the condensed part and the saturated gas were in thermal equilibrium, which implied that no 'supersaturated gas' existed.t *For small A, Einstein obtained the distribution, n 3 = (const.) exp(-€ s /fcT)[l + Aexp(-e x /fcT) + • • • ] . T
To show the existence of thermal equilibrium, Einstein evaluated the quantity S — (E + pV)/T; it was zero for the condensate's molecules because then S, E and pV were all separately zero. For the saturated gas the entropy turned out to be equal to (B + pV)/T.
544 The Golden Age of Theoretical Physics Einstein also derived the equation of state for the quantum gas, obtaining the result pV = oRTF{\), (11) where F denoted a function of the degeneracy parameter; it assumed the value 1 for A = 0 and decreased nearly linearly with growing A until it reached a value slightly above 0.5 for A = 1. Einstein found that for the gases existing in nature, saturation was never reached, but in the case of helium the critical density reached a value of 0.2 times the saturation density. Hence he concluded that the degeneracy might indeed become observable when investigating the equation of state of helium (and other ideal gases) at low temperatures. However, there was still another case, in which the influence of quantum theory might become even more important: the case of the electron gas. Due to the small mass of electrons their saturation density (i.e., the density obtained by substituting A = 1 on the right-hand side of Eq. (10) was rather low. Consequently, Einstein argued, only a small fraction of the electrons in metals should participate in thermal motion. These electrons would contribute only little to the specific heat, in agreement with observation. Still, a difficulty arose from the assumption that electrons obeyed Einstein's law of ideal, monatomic gases: in order to explain the observed electrical and thermal conductivity, the mean free paths of the thermal electrons had to be enormously large (of the order of 10~ 3 cm).* In his second contribution to the quantum theory of an ideal gas, Einstein discussed another important point in some detail: the statistical independence of the molecules. Since the independence was not guaranteed as in classical theory, a serious debate started concerning the consistency of the new gas theory, which went on in 1925 and even later. In early 1925 the problem of gas theory was not considered to be solved at all by Einstein's new proposals. Einstein was therefore happy to realize that his ideas gained theoretical support from another line of arguments, supplied by Louis de Broglie. De Broglie's wave theory of matter, which had just recently been expounded in his doctoral thesis at Paris (de Broglie, 1924e), claimed that all particles, whether they possessed mass (like gas molecules) or not (like light-quanta), exhibited wave properties; and these wave properties could be used to explain the peculiar statistical dependence of quantum-theoretical gas atoms. Thus, de Broglie's matter waves also helped in resolving the mixing paradox, which Einstein had mentioned at the end of his first paper on the quantum theory of monatomic ideal gases (Einstein, 1924c, p. 267). References t o the Appendix Einstein, A. (1924c), Sitz. her. Preuss. Akad. Wiss. (Berlin), pp. 261-267 (1924c': Attached Note.) Einstein, A. (1925a), Sitz. her. Preuss. Akad. Wiss. (Berlin), pp. 3-14. "According to Einstein, condensated electrons would not contribute to normal conductivity but might do so to superconductivity.
Satyendra Nath Bose, Bose-Einstein Statistics, and the Quantum Theory of an Ideal Oas 545 Ehrenfest, P., and V. Trkal (1921), Ann. d. Phys. (4) 65, 609-628. Ehrenfest, P., and V. Trkal (1920), Proc. Kon. Akad. Wetenseh. (Amsterdam) 162-183. Keesom, W.H. (1914b), Phys. Zs 15, 695-697. Laue M. Von (1924), Verh. d. Deutsch. Phy. Ges. (3) 5, 42-43. Nernst, W. (1912a), Phys. Zs. 13, 1064-1069. Planck, M. (1916b), Sitz. her. Preuss. Akad. Wiss. (Berlin), pp. 653-667. Planck, M. (1922a), Ann. d. Phys. (4) 66, 365-372. Planck, M. (1922b), Sitz. ber. Preuss. Akad. Wiss. (Berlin), pp. 63-70. Sackur, O. (1911), Ann. d. Phys. (4) 36, 958-980. Tetrode, H. (1912), Ann. d. Phys. (4) 38, 434-442.
23,
15 Louis de Broglie and t h e P h a s e Waves Associated with Matter* Relatively few French scientists contributed to the development of quantum theory. The reason for this paucity must be sought in the peculiarities of the French scientific establishment. 1 First, one should remind oneself that quantum theory grew up as an abstract theoretical scheme and the number of theoretical physicists in France was very limited at that time. There existed, for example, only about three chairs of theoretical physics in France, two of them in Paris. Leon Brillouin — son of Marcel Brillouin, professor at the College de France — who studied theoretical physics in the years between 1908 and 1920, reported later: 'There was really no career open for a theoretical physicist in the French organization. People who had curiosity for theory, would right away go into pure mathematics. There were dozens of chairs in mathematics. Even the smallest universities would have one or two mathematics chairs Many of my colleagues told me: "Are you crazy? To go into theoretical physics, there is no future" ' (Brillouin, AHQP Interview, 4 April 1962, p. 8). 2 Nor were any applied mathematicians available, who might be interested in Lecture delivered at the Fondation Louis de Broglie, Sorbonne, Paris, 18 May 1976; revised and enlarged version published (with Helmut Rechenberg) in The Historical Development of Quantum Theory (Springer-Verlag New York, 1982). In discussing the scientific situation in France, especially with respect to theoretical physics and the quantum theory during the first quarter of the twentieth century, we have made use of the interviews (of the Archives for the History of Quantum Physics) with Leon Brillouin and, to a lesser extent, the interviews with Jean Langevin (son of Paul Langevin) and Francis Perrin (son of Jean Perrin). 2 Leon Nicolas Brillouin was born on 7 August 1889 in Sevres, near Paris, as the son of Louis Marcel Brillouin. He studied physics at the Ecole Normale Superieure, Paris, from 1908 to 1912 (1911-1912 with Jean Perrin), at the University of Munich (1912-1913, with Arnold Sommerfeld), and at the University of Paris (1913-1914, 1919-1920; 1914-1919, military service), obtaining the degree of Docteur es sciences in 1920 under Paul Langevin. After t h a t he worked as scientific secretary of the reoorganized Journal de Physique et le Radium. In 1923 he was appointed an associate director of the physical laboratories at the College de France. In 1928, after the Institut Henri Poincare was established, Leon Brillouin was appointed professor there; in 1932, after the retirement of his father Macrel Brillouin in 1931, he was appointed professor at College de France. Prom 1939 t o 1941 he worked as Director General of the Radio Diffusion Prancaise. Then he went to the United States of America: visiting professor, University of Wisconsin, Madison, 19411942; professor, Brown University, Providence, Rhode Island, 1942-1943; research scientist at the National Defense Research Committee, Columbia University, New York, 1943-1945; professor
546
Louis de Broglie and the Phase Waves Associated
with Matter
547
quantum theory. 3 Because of this situation one may understand the lack of young French students who would have been eager to turn to the modern physical theories. Then the experimentalists also compounded the problem: while many physicists and chemists worked on the spectroscopy of atoms and molecules, very few among them cared for theoretical explanations. 4 Another fact, which distinguished the French situation from the one in other European countries and America, was the following: during World War I there occurred in France an almost complete standstill with respect to scientific investigations. As Brillouin recalled: 'Everybody was working — either mobilised or not — for the war.' In the military establishment, physics was not rated highly, and 'the only one who had some appreciation of scientific work was General Ferrie who [was in charge of] research in radio propagation' (Brillouin, AHQP Interview, 29 March 1962, p. 11). Hence, while in England and Germany many important contributions to experimental and theoretical physics were published between 1914 and 1918, in France physicists remained silent; even scientific journals, such as the Journal de Physique, ceased to appear during these years. After the war there occurred another circumstance, which helped to prevent French scientists from participating in the rapidly growing development of atomic physics and quantum theory. In France the scientific organizations stuck particularly rigidly to the policy, declared at the end of World War I, of interrupting all exchanges with scientists from Germany and Austria. Thus, in early 1920s, when scientific developments occurred fast and were frequently communicated orally or by letters, the French physicists had to learn the news from the more or less delayed publications. Of course, some scientists still obtained the necessary information on the research in Germany and Austria, either indirectly — via Swiss, Dutch, British of applied mathematics, Harvard University, Cambridge, Massachusetts, 1947-1949; worked for I.B.M. Corporation, Poughkeepsie, New York, and staff member at the I.B.M. Watson Laboratory at Columbia University, 1952-1954; adjunct professor at Columbia University, 1954-1979. He died in October 1979. Leon Brillouin did pioneering work on the propagation of electromagnetic waves, in which he had become interested during his stay with Sommerfeld in Munich in 1912-1913, and on which he continued to work in World War I. His doctoral thesis dealt with the quantum theory of an ideal solid. He continued to contribute to quantum theory in the following years, becoming one of the founders of modern solid state theory {Brillouin zones). He also investigated problems of classical and quantum statistical mechanics, and applied information theory to physics and to the design of computing machines. In the early 1920s, Leon and Marcel Brillouin worked on problems of atomic and quantum theory. With Paul Langevin as the scientific director and Leon Brillouin as secretary of Journal de Physique after World War I (starting with volume 1 of the sixth series in 1920), the favourable reception of papers on quantum theory in the journal was assured. In French universities applied mathematics was not a favourite subject. In Paris before World War I it was taught only at the jZcole Poly technique. After the war a new chair was created for applied mathematics at the University of Paris, but within a few years it was changed to pure mathematics. For example, the excellent experimentalist Charles Fabry (1867—1945), who had improved spectroscopic resolution enormously by applying a special interference method — the so-called FabryPerot interferometer — was not interested at all in theoretical problems. He rather preferred to use spectroscopy for investigating astronomical problems.
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The Golden Age of Theoretical Physics
or Scandinavian colleagues — or through personal acquaintances. For example, Paul Langevin and Madame Curie continued to maintain their personal contact with Albert Einstein in Berlin. Also, a number of French physicists — Marcel and Leon Brillouin, Maurice de Broglie, Marie Curie, Paul Langevin, Jean Perrin and Pierre Weiss — participated in the third Solvay Conference, held in April 1921 at Brussels, which was devoted to the recent progress in problems of atomic physics and electron theory. In spite of their reservation, one cannot say that French physicists opposed atomic and quantum theory. Leon Brillouin claimed: 'In France either people were not interested in quanta, or accepted it immediately. Langevin, Perrin and Madame Curie accepted the photon of Einstein literally right away without any discussion. They found it wonderful, they liked it and they used it' (Brillouin, AHQP Interview, 29 March 1962, p. 8). Langevin even included the light quantum in his lectures; for example, he discussed the problem of the Brownian motion of the particle emitting and absorbing light-quanta, similar to what Einstein had done in 1916. And the students had no problem in accepting Einstein's reasoning in favour of light-quanta. They argued: 'That's certainly a good idea. How far it will go nobody knows, for this is so new. It's hard to say exactly how far you can use it, but it is certainly right in principle' (Brillouin, loc. cit., p. 8). Some spectroscopists, including especially Maurice de Broglie and his collaborators, made use of Bohr's theory of atomic structure in order to explain the discrete X-ray spectra of elements. They not only referred to Bohr's original theory, but also to its extensions that were made by Arnold Sommerfeld and others and published in German journals. In this connection one should note an important feature, which distinguished the French discussions of atomic and quantum-theoretical problems from those in Berlin, Copenhagen, Gottingen or Munich. In France these problems did not stimulate heated debates, but one took things more or less for granted. The authors of articles appearing, say in Comptes Rendus or Journal de Physique, rarely seemed to worry about fundamental problems, which bothered Bohr or Einstein so much. Moreover, no French journal existed — similar to the Zeitschrift fur Physik — a large part of which was devoted to problems of quantum theory. Paul Langevin, the French physicist who was perhaps most concerned with the deeper aspects of quantum theory, worked on some problems but did not publish his results. Brillouin recalled: 'Langevin, after 1920, never wrote one paper. Everything he did he gave in his lectures at College de France to a group of students. He had a very good class of from 30 to 40 students, regularly year after year, and that was the place where he communicated his new work. Then, after he had given it to the students, he was tired of the whole thing. He wouldn't take a minute to write it up' (Brillouin, AHQP Interview, 29 March 1962, pp. 11-12). Still, among Langevin's students there were only few who picked quantum problems to work on: besides Leon Brillouin, another senior student became engaged upon them; he was Louis de Broglie.
Louis de Broglie and the Phase Waves Associated
with Matter
549
Louis Victor Pierre Raymond de Broglie was born on 15 August 1892 at Dieppe, the youngest son of Prince Victor de Broglie.5 In 1901 the family moved to Paris, where Victor de Broglie was a member of L'Assemblee Nationale. Louis finished high school in 1910 and began to study history and law at the University of Paris in order to prepare himself for entering the civil service. However, under the influence of the books of Henri Poincare he soon became interested in science, the field in which his elder brother Maurice had graduated in 1908 with the degree of Docteur es sciences.6 In fall 1911 Maurice de Broglie served as one of the scientific secretaries at the first Solvay Conference in Brussels. During the sessions of the Conference he prepared, together with Paul Langevin, the notes of the discussions following the reports of the speakers, which were included in the proceedings of the Conference (Langevin and de Broglie, 1912). Louis got to read these notes, and he remembered his reaction to them as follows: 'With the ardour of my age I became enthusiastically interested in the problems that had been treated, and I promised myself to devote all my efforts to achieve an understanding of the mysterious quanta, which Max Planck had introduced ten years earlier into theoretical physics, but whose deep significance had not yet been grasped' (L. de Broglie, 1953, p. 458). Thus he changed, still in 1911, to the faculty of sciences and began to improve his knowledge of physics mainly by reading the literature on recent results in theoretical physics. Especially, he studied the works of Henri Poincare, Hendrik Antoon Lorentz, Paul Langevin, Ludwig Boltzmann and Josiah Willard Gibbs, as well as the papers of Planck and Einstein on quantum theory and relativity. He concerned himself with the role of classical dynamics in the quantum theory and with the modification of the concepts of space and time in relativity theory. In 1913 he received his Licence es sciences. During the same year he entered the engineering corps of the French army to perform his obligatory military service. On the recommendation of his brother Maurice, he soon became associated with wireless telegraphy. Before he could leave the army, World War I broke out, and Louis de Broglie had to remain in service until August 1919. During the war wireless telegraphy experienced a rapid advance, and both de Broglie brothers and Leon Brillouin became involved in developing its techniques. Brillouin and the de Broglies even did some experiments together. 7 In spite of 5 Victor de Broglie and his wife, Pauline d'Armaille, had five children: Albertine, Maurice, Philippe, Pauline and Louis. The de Broglies were descended from a noble Piedmontese family. In the middle of the seventeeth century Francois Marie Broglia, Comte de Revel, joined the French army. During the eighteenth century several members of the family became marshals of France; one of them, Francois Marie, received the title of 'Due' and changed his name to 'de Broglie.' In 1816 Due Victor de Broglie married Albertine de Stael, daughter of Germaine de Stael-Holstein (the Madame de Stael of literary fame). Their grandson was Due Louis Amedee Victor, the father of Maurice and Louis de Broglie. (L. de Broglie, Conversations with Mehra; also see Tonnelat, 1966.) Victor de Broglie died in 1906, upon which Maurice — who was seventeen years older — became responsible for Louis' education. At first he did not try to influence Louis towards physics, for he seemed to be suited for a diplomatic or political career. Louis de Broglie and Leon Brillouin worked in Paris in nearby offices. As Brillouin recalled: 'I was at the laboratory of the Signal Corps and Louis de Broglie was at the transmitting station
550
The Golden Age of Theoretical
Physics
exciting and successful work during the war, Louis de Broglie did not feel quite happy. As Maurice de Broglie recalled: 'My brother deplored the interruption of his thinking [on the fundamental problems of quantum theory and relativity]; he complained later that his impetus had been broken, and it took him several years to regain it' (M. de Broglie, in Louis de Broglie, Physicien et Penseur, 1953, p. 427). As soon as he returned to civilian life, Louis de Broglie picked up his studies where he had left them in 1913.8 His main goal was to formulate his earlier ideas on quanta in a doctoral thesis. Leon Brillouin reported about this work as follows: 'His brother Maurice encouraged him very much At the same time Louis de Broglie was coming regularly to the Ecole de Physique et Chimie to meet Langevin and myself. I think we were the only three people to whom he talked during the time he wrote his thesis' (Brillouin, AHQP Interview, 29 March 1962, p. 7). De Broglie's research was not only characterized by the fact that he interacted with just a few persons in scientific matters, he also did not especially depend on information about the most recent status of problems in which he was interested. Hence the particular disadvantage of French science, the isolation from the main centres of quantum theory in Germany and Copehangen, did not affect Louis de Broglie's progress seriously. Indeed, his brother later claimed that the situation rather favoured Louis de Broglie's programme: in 1920 Niels Bohr's ideas on atomic structure had been settled and confirmed enough to attract Louis' attention and they became fruitful in his thinking. Besides reading the papers of Bohr and Sommerfeld, Louis de Broglie learned about the theory of atomic structure by discussing the results of the spectroscopic work performed in his brother's laboratory. Having adapted himself to the atmosphere of a scientific laboratory during the war, he now participated at times in the investigations of brother Maurice and his closest collaborator Alexandre Dauvillier. Thus he earned his initial scientific reputation by publishing short papers on problems of X-ray spectroscopy. His first two papers dealt with the absorption of X-rays on the basis of Bohr's theory (L. de Broglie, 1920a,b). Then he analyzed with Maurice de Broglie the measurements of the spectra arising from the photoelectric effect (M. and L. de Broglie, 1921a,b), and with Dauvillier he derived certain consequences at the Eiffel Tower which was just a few blocks away' (L. Brillouin, A H Q P Interview, 29 March 1962, p. 7). While Brillouin performed only minor experiments with Louis de Broglie, he carried out an important one with Maurice de Broglie, which he described as follows: Maurice de Broglie, who was much older and was a retired Navy officer, thought of a new method for receiving radio signals in submarines. His idea was t h a t even when a submarine is under water, it could receive radio signals and radio messages with a large coil outside the submarine and a powerful amplifier inside. Maurice de Broglie built a large coil, and I at that time happened to have built the most powerful amplifier in Paris. So we put the two things together. (L. Brillouin, AHQP Interview, 29 march 1962, p. 6) The two of them then went to Toulon and experimented in a submarine. They found that the idea worked perfectly, and that one indeed received with the apparatus long radio waves even from America. Within a few months the submarines of the Allies had these receiver sets. Louis de Broglie augmented his knowledge of physics by attending several lecture courses of Paul Langevin at the College de France: for example, the course on quantum and atomic theory in May and June 1919; the course on relativity theory from December 1921 to May 1922; and the course on electron theory from December 1924 to January 1925. He also thought about his own research work.
Louis de Broglie and the Phase Waves Associated with Matter
551
from a systematic investigation of the X-ray spectra with respect to the electronic structure of atoms of heavy elements (Dauvillier and L. de Broglie, 1921a,b). In the following years he continued to contribute to the theoretical analysis of spectroscopic data with his brother and other collaborators. This work provided him a detailed knowledge of certain aspects of the theory of atomic structure; for example, in a new paper on the absorption of X-rays he explored a specific application of the correspondence principle (L. de Broglie, 1921b).9 At the same time, the intense discussions with Maurice on the properties of X-rays contributed importantly to de Broglie's ideas about light-quanta because, as he said later: 'These long conversations with my brother... were very useful to me, for they made me reflect deeply about the necessity of always connecting together the points of view of waves and corpuscles' (L. de Broglie, 1953, p. 459). The study of the processes of X-ray absorption and the photoelectric effect directed de Broglie's attention to the problem of blackbody radiation. 10 On 26 January 1922 the Journal de Physique received the manuscript of a paper entitled 'Rayonnement noir et quanta de 1^X111^ ('Blackbody Radiation and Light Quanta,' L. de Broglie, 1922b). De Broglie stated its goal in the introduction as being 'to establish a certain number of results, which are known in the theory of radiation by means of reasoning based entirely on thermodynamics, kinetic theory and quantum theory, without any reference to electrodynamics' (L. de Broglie, 1922b, p. 422). That is, he adopted the light-quantum hypothesis completely and, in his calculations, regarded the blackbody radiation of a given temperature 'as a gas consisting of atoms of light of energy W = his' (L. de Broglie, 1922b, p. 422). In his treatment he neglected 'the molecules of light [consisting] of 2 , 3 , . . . , n atoms hi/,' being aware of the fact that this would be consistent only with the validity of Wien's law for blackbody radiation. Thus, with respect to the radiation law, it did not go beyond what Einstein had found already in 1905. However, he proceeded by applying a different idea: he introduced a massive light-quantum by associating with it an energy quantum hi/ and rest mass mo, defined by the equation W = hv=
,"***
,
(1)
where v is the velocity of the light-quantum, which he thought to be infinitely close to c, the velocity of light in vacuo. Then he treated an assembly of such massive 'light-atoms' according to the rules of kinetic gas theory and derived the following 9 A t this point we may mention that the collaboration with Dauvillier on X-ray spectra had a negative influence on Louis de Broglie's reputation in Copenhagen because of Dauvillier's involvement in the celtium-hafnium controversy. 10 I n a paper, entitled 'Rayons X et Equilibre Thermodynamique' ('X-rays and Thermodynamic Equilibrium') and submitted in June 1921, de Broglie considered the absorption of X-rays in connection with blackbody radiation (L. de Broglie, 1922a, received by Journal de Physique et le Radium on 1 June 1921 and published in the issue of February 1922). Similarly he treated later the problem of the spectrum of X-rays produced from atoms by the impact of electrons or of primary X-rays (L. de Broglie, 1921a, a note presented to the Academy of Sciences, Paris, on 5 December 1921, and published in the same month).
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The Golden Age of Theoretical
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results. First, the energy distribution of the light-quanta was given by the equation
where n is the number of atoms per unit volume, k Boltzmann's constant and T the absolute temperature. Second, each 'atom of light' possessed an average energy 3kT, which was twice as much as gas molecules having nonrelativistic energies. Third, he derived the Stefan-Boltzmann law of radiation (which claimed that the total radiation energy depended on the fourth power of the temperature). And, finally, he derived Wien's radiation law. 11 He also noted in passing that by a mixing of the monatomic, diatomic, triatomic, etc., 'molecules of light' one might arrive at Planck's law of blackbody radiation, but he declared that the mixing procedure itself represented an arbitrary assumption. De Broglie concluded his paper by emphasizing two points: first, he had been able to deduce Wien's law by treating the light-quanta as relativistic particles of extremely small mass according to the rules of ordinary statistical mechanics; second, in his approach the constant in Stefan-Boltzmann's law corresponded to the so-called chemical constant of monatomic gases. 12 Louis de Broglie took the next step in the theory of massive light-quanta in a note presented to the Academy of Sciences, Paris, at the session of 6 November 1922 (L. de Broglie, 1922c). In it he considered the problem of interference, claiming that one had to introduce an assumption concerning the behaviour of light-quanta in order to explain the observed interference phenomena. To discover the nature of this assumption he referred to Einstein's analysis of blackbody radiation obeying Planck's law in thermal equilibrium in terms of energy fluctuations (Einstein, 1909a). He rewrote Einstein's expression 1909 as a sum of terms, (E - £)2 = 5 > f a / E n ,
(3)
n=l
where En=
8irhi/3V ( nhv\ ^ exp(-—j
, „ „ (n = 1,2,3,...).
(3a)
(E denoted the actual radiation energy in volume V, and E = X^^Li En its average value in thermal equilibrium.) He interpreted the right-hand side of Eq. (3) as follows: 'From the point of view of light-quanta, the interference phenomena may seem to be connected with the agglomeration of light-atoms, whose motions are not independent but coherent. Therefore, it is natural to assume that if one day the theory of light-quanta will be able to explain interference phenomena, it will have to make use of such agglomeration of quanta' (L. de Broglie, 1922c, p. 813). 11 I n order to establish quantitative agreement, de Broglie had to multiply his calculated expression for the energy density of radiation by a factor 2, which he interpreted — in agreement with Peter Debye (1910) — as arising from the two possibilities of polarization. 12 T h e entropy of radiation was given by the expression ^AT3, with A the constant in StefanBoltzmann's law (Eq. (3)). This analogy had been mentioned already earlier by Frederick A. Lindemann (Lindemann, 1920).
Louis de Broglie and the Phase Waves Associated
with Matter
553
That is, only by means of the agglomeration effect could one expect to describe the wave-like behaviour of radiation including interference. 13 Looking superficially at the papers of Louis de Broglie up to late 1922 it might appear that he had not really progressed beyond what had been reported earlier in the literature. 14 All that he had added, one could argue, was the vague statement about the origin of the interference phenomena in light-quantum theory and the rather uncommon assumption that the light-quanta possessed a very small mass. Not the slightest sign of the idea of matter waves had occurred in the papers published so far. And yet, Leon Brillouin later recalled numerous discussions in 1921 or 1922, in which Louis de Broglie definitely mentioned to him the crucial steps leading to the hypothesis of matter waves. As Brillouin reported: He [L. de Broglie] looked at the experiment. You see particles emitted from a radioactive source. You put on a magnetic field and some [particles] are bent to the right and some are bent to the left, and some travel straight ahead. So his idea was: 'Well, all this must be very similar. Either they are all waves or they are all particles.' Since the idea of the photon being a particle had already been used by Einstein, he tried the opposite idea to see if he couldn't make everything waves. That's how he got the idea of wave mechanics. I remember seeing him sit down in front of me and point to a picture of particles, some going straight ahead. (Brillouin, AHQP Interview, 29 March 1962, p. 7) Brillouin explained the reason why de Broglie did not mention this idea in his publications before 1923: 'The very important thing was the moment when he discovered the relation between momentum and wavelength. Everything started from there. That was really the first mark. And the rest had to follow from it. Each step took a long time of thinking and wondering before he made up his mind about the way to adjust a new term' (Brillouin, AHQP Interview, 3 April 1962, p. 1, our italics). Now in the paper on the blackbody radiation, submitted in January 1922, the relation between the momentum and the frequency of the light-quanta was indeed expressed in the remark: 'Their momentum [i.e., the momentum of light-quanta with energy hv] is hisfc' (L. de Broglie, 1922b, p. 422). In a letter to Fritz Kubli, spring 1964, Louis de Broglie confirmed Brillouin's claim that the phase-wave hypothesis started from this innocent-looking remark. 15 13 T h e wave term, (c3/8ms2dv)(E2/V), in Einstein's equation (79) was represented in de Broglie's equation by the sum, 2 ^ 1 2 ( n — l)hvEn. 14 We recall, for example, that the idea of 'light-molecules' had been treated earlier in a paper of Mieczylaw Wolfke, published in Physikalische Zeitschrift (Wolfke, 1921). It is not that de Broglie did not read German scientific articles; for example, in his paper on blackbody radiation he referred to the article of Robert Emden, ' Uber Lichtquanten' ('On Light-Quanta,' Emden, 1921), which was published in Physikalische Zeitschrift several months after Wolfke's paper (see L. de Broglie, 1922b, p. 428). Emden's paper, submitted in August 1921, dealt with the thermodynamical properties of a quantum gas in detail; in addition, it contained a derivation of Planck's radiation formula, in which Emden also introduced 'light-molecules.' (See the last equation on p. 515 of Emden's paper in which a sum, J ^ nhvexp(—nhu/kT), appears in the numerator.) 15 F . Kubli has discussed the development of Louis de Broglie's thesis of 1924 in detail in an article, entitled 'Louis de Broglie und die Entwicklung der Materiewellen' ('Louis de Broglie and
554
The Golden Age of Theoretical Physics
When I started to obtain, in the period 1922-23, the basic ideas of undulatory mechanics, it was my intention to extend the coexistence of waves and particles — which Einstein had discovered existed in the case of light — to all particles. Hence I began with the formulae established by Einstein for the photons [i.e., light-quanta], W = hv and p = hv/c = h/X. Applying these formulae to particles other than the photons, I was led, of course, to write for them: W = mQc2 / y/l - (32 = hv and p = m0v/^l-f32 = h/X,[/3 = v/c]. However, it then seemed to me that the symmetry with the theory of Einstein would not be complete unless one attributed to the photon a proper mass /io, thus allowing one to write Einstein's equations in the form: W = MoC 2 /\/l — ft2 = hv and p = IM,v/y/T^~(P = h/\. T h e latter formulae could indeed be found already in de Broglie's p a p e r on blackb o d y radiation (L. de Broglie, 1922b, p . 422). It remained, however, t o extend t h e quantum-theoretical relation (valid for the light-quantum) t o particles with masses considerably larger t h a n t h e hyothetical light-quantum m a s s . T h i s step, which may be considered as a n a t u r a l consequence of introducing massive lightq u a n t a , was not expounded by de Broglie before September 1923: on t h e 10th of t h a t m o n t h J e a n Perrin communicated Louis de Broglie's notes on 'Ondes et Quanta' ('Waves and Q u a n t a ' ) to the Academy of Sciences, Paris (L. de Broglie, 1923a). In this n o t e de Broglie associated 'an internal periodic p h e n o m e n o n ' {'un phenomene periodique interne,' L. de Broglie, 1923a, p . 508) with any massive particle including t h e light-quantum. In a second note, which was communicated two weeks later t o t h e Academy of Sciences in Paris, he investigated t h e consequences of his assumption of m a t t e r waves in t h e case of light-quanta (L. de Broglie, 1923b), while in a t h i r d note — presented t o t h e Academy on 8 October 1923 — he dealt with the statistical properties of gas atoms having such wave characteristics (L. de Broglie, 1923c). T h e publication of the three notes of September a n d October 1923 indicated t h a t a breakthrough h a d occurred in Louis de Broglie's work in t h e course of summer 1923. He described it four decades later as follows: As in my conversations with my brother we always arrived at the conclusion that in the case of X-rays one hand [both] waves and corpuscles, thus suddenly — I cannot give the exact date when it happened, but it was certainly in the course of summer 1923 — I got the idea that one had to extend this duality to the material particles, especially to electrons. And I realized that, on one hand, the Hamilton-Jacobi theory pointed somewhat in that direction, for it can be applied to particles and, in addition, it represents a geometrical optics; on the other hand, in the quantum phenomena one obtains quantum numbers, which are rarely found in mechanics but occur very frequently in wave phenomena and in all problems dealing with wave motion. Then I said to myself that there must exist a wave property associated with the quantum phenomena, and I wrote three notes for Comptes Rendus in September and early October 1923, in which I tried to formulate these ideas exactly. These three notes together contain the essential things which later entered into my thesis, which I wrote afterwards. (L. de Broghe, Conversations; AHQP Interview, 7 January 1963) the Development of Matter Waves,' Kubli, 1971). The remarks quoted here are in the article on p. 62.
Louis de Broglie and the Phase Waves Associated with Matter
555
De Broglie opened the first of these notes, the one on ' Ondes et Quanta' ('Waves and Quanta'), with a general remark, namely, that according to the principle of inertia of energy a particle of proper mass mo possesses an internal energy TUQC . He continued: 'On the other hand, the quantum principle leads one to attribute this internal energy to a periodic phenomenon of frequency VQ such that hu0 = m0c2 ,
(4)
where c always denotes the limiting velocity of relativity theory and h is Planck's constant' (L. de Broglie, 1923a, p. 507). On first inspection, the hypothesis contained in Eq. (4) appears to be rather strange, for it implies that even with a particle of mass mo at rest a frequency had to be connected. However, in summer 1923 for de Broglie — who had just arrived at the conviction that a general duality existed between matter and waves — Eq. (4) followed immediately from a result, which he had obtained early in 1922 for the massive light-quanta, i.e., from Eq. (1) if the object under consideration moved with velocity zero. For a particle moving with a uniform velocity v, de Broglie now considered two different periodic motions with frequencies v and v\, respectively. On one hand, an observer at rest would conclude from the total energy of the particle, m0c2/y/l — v2/c2, a frequency v, where y2\-l/2
/ V
=
VQ 1
;
\ -*)
(5)
on the other, the same observer would also notice an internal periodic phenomenon, whose frequency was vi, where
/
v2\1/2
V\ = Vo I 1 - ^J J
(6)
and whose amplitude was proportional to sin(27ri/1t). The two frequencies were related as
1 = "(l-5)-
CD
The question now arose as to what the two periodic motions had to do with each other. De Broglie suggested the following interpretation. He first considered the wave motion with frequency u, propagating in space with the velocity c2/v and coinciding with the particle in space at time zero. Due to the fact that the velocity of the wave surpasses the velocity c, the wave would not transport any energy; hence de Broglie looked at it 'like a Active wave associated with the movement of the particle' (L. de Broglie, 1923a, p. 508). At time t then the particle could be found at the space point x = vt, with an amplitude of its internal periodic motion proportional to sm(2ni/ix/v). At the same space-point, however, the particle would meet the 'Active' wave, having an amplitude proportional to sin[27rz/(< — xv/c2)]. As a consequence of the relation between v and v\ then, the two wave motions were always in phase. Conversely, the requirment that the Active wave motion of
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The Golden Age of Theoretical Physics
frequency u associated with the particle must always be in phase with the internal periodic motion of frequency v led to the condition that the fictive wave must move with velocity c2/v. De Broglie immediately applied the idea of a twofold periodic motion to the propagation of light-quanta and the electrons moving in orbits in atoms. While he derived no further consequence in the case of massive light-quanta, he found an important result for orbiting electrons. Due to their different velocities the fictive wave and the electron would meet on the orbit after a time r, given by the equation v2lc2 ' jT , 1 - v2/c2 r
" -
(8)
where Tr is the period of revolution of the electron. 16 By requiring the coincidence of phases of the fictive wave and the internal wave, de Broglie obtained a condition for the period of revolution Tr, that is, mov2 y/1 -
V2/c
fTr
= nh,
(9)
n being an integer. 17 He then showed that Eq. (9) corresponded to the quantum condition for the relativistic motion of the electron, described by the position variables, x, y and z, and the momentum variables, px,Py and pz, that is, / Jo
(pxdx + pydy + pzdz) = / Jo
° v 1-
dt = nh,
(9)
v2/c2
which Einstein had formulated earlier (Einstein, 1917b). This quantum condition passed over, for small electron velocity v, into
I
2-n
p^d(j) — nh,
(9')
that is, Bohr's quantum condition for an electron moving on a circular orbit (described by the momentum variable p<j, = mowa2 and the angular variable <j>).ls The connection of the idea of waves associated with material particles with Bohr's quantum condition might be regarded as a somewhat surprising result. It had, however, been indicated earlier in a paper by Marcel Brillouin, to which de Broglie referred in the introduction of his note 10 September (L. de Broglie, 1923a, p. 507, footnote l ) . 1 9 Marcel Brillouin's paper, entitled 'Action mecanique a heredite discontinue par propagation; essai de theorie dynamique de I'atome a 16
De Broglie obtained Eq. (8) by considering the fact that the paths travelled by the fictive wave motion and the electron must be the same, i.e., 17 De Broglie derived Eq. (9) by assuming the phase of the internal wave, 27ri/iT. to be an integral multiple of 2n. 18 Bohr's quantum condition was mouia? = nh/2ir, with a the radius of the orbit. 19 Louis Marcel Brillouin was born on 19 December 1854 at Martin-les-Melles, Deux-Severs. He entered the Ecole Normale Superieure, Paris, in 1874 and became maitre de conferences at the
Louis de Broglie and the Phase Waves Associated
with Matter
557
quanta.' dealt with the periodic motion of a particle in an elastic medium having a much larger velocity than the velocity of sound in that medium (M. Brillouin, 1919). Brillouin had shown that the elastic waves, emitted by the moving particle, hit the particle again creating an 'inherited field,' i.e., an action reminding the electron of a finite number of its previous positions. He had further noticed that a periodic or quasi-periodic motion of the particle in the elastic medium could be characterized by the integral number n, 'It thus appears,' he noted, 'that one can formulate a dynamical hypothesis having the capacity to represent the essential properties of the Bohr atom, provided one is able to decide upon the appropriate emission law' (M. Brillouin 1919, p. 1319). Brillouin had assumed, then, that the ether possessed a characteristic velocity far below c (say, of the order of 10 km/s), and that the quantum phenomena emerged from the fact that the electrons in atomic orbits moved much faster than this characteristic velocity; thus an electron was hit by the 'inherited field,' emitted by the electron from a number of previous points on its orbit. The discrete positions of these points could be connected with an action integral, to which Brillouin had assigned the value of Planck's constant (M. Brillouin, 1919, p. 1320). Although Marcel Brillouin's attempted derivation of Bohr's quantum condition in 1919 and Louis de Broglie's derivation four years later rested on completely different ideas, they shared the assumption of an extra periodic motion in addition to the periodic orbital motion of the electron. In de Broglie's theory, the extra periodic motion belonged to the electron and had nothing to do with the ether; moreover, it had a velocity greater than the velocity of the electron. 20 In his note of 10 September 1923, de Broglie presented, without any justification, an estimate for the proper mass of the light-quantum, namely, mo ^ 1 0 - 5 0 g. The origin of such a small number came, as he explained later, from the following consideration: 'And I asked myself, why shouldn't one take a very small proper mass? There is the objection that the velocity of light would depend on the frequency. However, if the proper mass is extremely small, the dependence will be extremely feeble' (L. de Broglie, Conversation with F. Kubli, quoted in Kubli, 1971, p. 63). Especially, one had to think of possible effects of the mass of the light-quantum on astronomical observations. These observations seemed to admit a mass below 10 _ 4 5 g, but, because of the role of the velocity c in special relativity, de Broglie put the limit still lower. He explored further consequences of the light-quantum mass in his second communication to the Academy of Sciences in Paris, entitled ' Quanta de same institution three years later. In 1900 he was appointed professor of physics at the College de France, a position which he occupied until 1931. He died on 16 June 1948 at Melle, Deux-Sevres. Marcel Brillouin, who started out as a mathematician, soon began to concern himself with mechanical problems and finally with problems of other fields of physics. He contributed primarily to hydrodynamics, thermodynamics and statistical mechanics (e.g., to the theory of diffusion and viscosity of gases). But he was also interested in problems of quantum theory and special and general relativity. 20 I t should be pointed out that a year before de Broglie, Erwin Schrodinger had obtained a derivation of Bohr's quantum condition with the help of an argument based on general relativity. He had also applied a specific phase relation (Schrodinger, 1922c).
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The Golden Age of Theoretical
Physics
lumiere, diffraction et interferences' ('Light-Quanta, Diffraction and Interference,' L. de Broglie, 1923b). 21 In this note de Broglie introduced a new description of the wave phenomena, which he had associated with the material particles: he spoke of just one wave motion. But with each light-quantum, for instance, he associated a group of waves having the 'group velocity' vg, the same velocity as the massive light-quanta had; the phase velocity of the waves, however, was c2/v, the velocity of propagation of the Active wave motion, which he now called 'the phase wave' (L. de Broglie, 1923b, p. 549). The phenomenon of diffraction, de Broglie continued, implied that the lightquanta did not move in a straight line. Hence one had to replace the usual principles of particle dynamics by the following postulate: 'At each point of this trajectory, a free particle follows in uniform motion the ray of its phase wave, what is (in an isotropic medium), the normal to the surfaces of equal phase' (L. de Broglie, 1923b, p. 549). With the help of this postulate de Broglie formulated mathematically the conviction, at which he had arrived in summer, that light-quanta and electrons should be considered as particles and waves at the same time. According to the new postulate, he claimed that the motion of the quantum-theoretical particles agreed with the one derived from the usual dynamics of mass-particles (determined by the principle of least action) or of geometrical optics (determined by Fermat's principle). De Broglie asserted: 'But if the particle has to pass through a slit, whose dimensions are small in comparison with the wavelength of the phase wave, its trajectory will bend in general like the ray of a diffracted wave' (L. de Broglie, 1923b, p. 549). 22 Thus he predicted the diffraction of light atoms 'however small their number'; that is, any moving particle would be diffracted under suitable conditions. Especially, an electron beam going through a slit would do so, and de Broglie added: 'It is in this direction that one must look for the experimental confirmation of our ideas' (L. de Broglie, 1923b, p. 549). De Broglie interpreted the phase wave as 'guiding the propagation of the energy'; their very existence allowed him to perform 'the synthesis of the waves and the quanta' (Ha synthese des ondulations et des quanta' L. de Broglie, 1923b, p. 549). The new dynamical description included the previous point dynamics (and its relativistic generalization) as well as geometrical and wave optics. With its help de Broglie now explained the interference patterns in the following way. He assumed that every atom possessed a probability of absorbing a light-quantum, and this probability was determined by the vector describing the phase wave of the light-quantum. If a light-quantum was emitted by an atom, its phase wave tried to establish phase coherence with the phase waves of the light-quanta emitted by other, neighbouring atoms. Hence he concluded that the phase waves of all or many light-quanta 21 W h e n he submitted the note of 10 September 1923, de Broglie had already obtained the main result of his second note, for he announced: 'As of today, we are in a position to explain the phenomena of diffraction and interference by taking into account the light-quanta 1 (L. de Broglie, 1923a, p. 510). 22 De Broglie stated that in the process of diffraction the momentum of the light particles will not be conserved because they might transfer momentum to the atoms at the border of the slit.
Louis de Broglie and the Phase Waves Associated
xaith Matter
559
emitted by atoms had a tendency to coincide, thereby giving rise to a coupling of the light-quanta, which seemed to result in the formation of 'light-molecules' — as had been noticed in the analysis of blackbody radiation. Returning to the problem of diffraction, he assumed that the passage of 'light-atoms' through a slit was guided by their phase waves. The phase waves were than responsible for the bright and dark fringes emerging in the space behind the slit. Thus the same diffraction pattern should be established as in the wave theory of light, no matter how small the intensity of the incident radiation might be. In his third note of October 1923, entitled lLes quanta, la theorie cinetique des gaz et le principe de Fermat' ('Quanta, the Kinetic Theory of gases and Fermat's Principle'), de Broglie finally looked at the description of the properties of gases from the point of view of the phase-wave theory (L. de Broglie, 1923c). He declared that his main goal in this work was the justification of Max Planck's assumption concerning the elementary phase-space regions associated with each atom. For this purpose he considered that with each atom (of proper mass mo and velocity v) was associated a group of phase waves, having the phase velocity c2/v, frequency (moc2/h)(l — v2/c2)~ll2 and the group velocity v. Then he made the following assumption: 'The state of the gas will be stable only if the waves corresponding to all the atoms from a system of stationary waves' (L. de Broglie, 1923c, p. 630). That is, he took over the treatment, which Lord Rayleigh and especially James Jeans had applied to radiation in thermal equilibrium: they had considered the stationary modes in a given volume. 23 De Broglie also took over Jeans' method of counting the number of stationary waves of a given frequency interval, v and v + dv, contained in a unit volume; he found this number to be nvdv, where n
»
d u
=
( 2/ z
\i
v
d u
z
=
—
v
d
v
(10
•
4
v(c lv) c By substituting the relation between v and w, the kinetic energy of the gas atoms (i.e., hu = w + mnc 2 ), he wrote Eq. (324) as 47T
nvdv = -r?mlc{l + a) y/a(a + 2)dw, (10') ha where a = W/TUQC2. NOW each wave could transport either zero, or one, or several atoms, their number being determined by the canonical (temperature) distribution. Hence the number of atoms, connected with the phase wave of energy hu and having kinetic energy in the interval w and w + dw, in a volume element of the six-dimensional phase space became dnw = (const.)^m 2 Q c(l + a)y/a(a
+ 2)- ^
^{-nhu/kT)
dwdxd dz
y
'
For small values of a, i.e., in the nonrelativistic approximation, all terms except the first in the sum in the numerator on the right-hand side might be neglected, and 23
J a m e s Jeans had applied such a treatment in several papers around 1910 (e.g., Jeans, 1910) as well as in his report at the first Solvacy Conference in 1911, which Louis de Broglie knew in detail.
560
The Golden Age of Theoretical Physics
de Broglie rediscovered Planck's result for gas atoms. 24 In the case of light-quanta, however, a was always large; hence one had to sum over all terms on the right-hand side of Eq. (11). At the end of his short paper, de Broglie reformulated his ideas in a still more accurate manner. He assumed that a phase wave as associated with any particle, being in phase with the particle at any space-point and having its frequency and velocity determined by the energy and velocity of the particle. These phase waves should especially have the property that: 'The rays of the phase waves coincide with the trajectories [which are] dynamically possible' (L. de Broglie, 1923c, p. 632). In a situation with variable dispersion — for example, in the case where light-quanta were diffracted by a dispersive medium — de Broglie claimed that the rays of the phase wave must be described by the (optical) principle of Pierre de Fermat, which he formulated as the condition for the path integral extending between two space points, that is, 8[jg-=6[ 2
moV
dl = 0. (12) J (C /V) J hy/1 - V2/C2 ^ ^ Fermat's principle coincided with Pierre de Maupertuis' principle of least action. 'The fundamental link which unites the two great principles of geometrical optics and of dynamics is thus also brought clearly to light,' de Broglie concluded proudly. He further pointed out that certain among the dynamically possible trajectories of quantum-theoretical trajectories are distinguished as being 'in resonance with the phase wave' (L. de Broglie, 1923c, p. 632). Thus, for instance the Bohr orbits of electrons were such resonance trajectories; they were determined by the fact that the integral over the closed path, §vdl/{
He found, in particular, dnw = const.—m Q ' \/2iuexp ( — — I • dxdydz.
(See, for instance, Planck, 1921b, Section 132.) 25 De Broglie's paper was communicated to the Philosophical Magazine by Ralph Fowler. C. D. Ellis, a guest from the Cambridge, who had worked with Maurice de Broglie in Paris in 1923 on the photoelectric effect, had encouraged de Broglie to write the paper and had corrected the translation. We should mention here that Louis de Broglie had already presented an outline of his ideas earlier in Nature, in a letter entitled 'Waves and Quanta,' dated 12 September and published in the issue of 13 October 1923 (L. de Broglie, 1923d).
Louis de Broglie and the Phase Waves Associated
with Matter
561
associate motion of body and propagation of wave, and that this idea gives a physical interpretation of Bohr's analytical stability conditions. Diffraction seems to be consistent with an extension of Newtonian Dynamics. It is then possible to save both the corpuscular and the undulatory characters of light, and, by means of hypotheses suggested by the electromagnetic theory and the correspondence principle, to give a plausible explanation of coherence and interference fringes. Finally, it is shown why quanta must take a part in the dynamical theory of gases and how Planck's law is the limiting form of Maxwell's law for the light-quanta gas. (L. de Broglie, 1924a, p. 457) He claimed that though the ideas may be criticized and undergo further changes in the future, the 'real existence of light-quanta' should not be doubted. 'Moreover,' he stated finally, 'if our opinions are received, as they are grounded on the relativity of time, all the enormous evidence of the "quantum" will turn in favour of Einstein's conceptions' (L. de Broglie, 1924a, p. 457). One of the changes, which de Broglie found appropriate to introduce later on, was the systematic relativistic formulation of the relation between the energy of the particle and the frequency of the wave. He suggested the first steps already in a postscript added in proof to his paper in the Philosophical Magazine (L. de Broglie, 1924a, pp. 457-458). There he defined a four-dimensional vector 0M, whose time component was equal to the frequency v of the phase wave divided by c, and whose space components were the components of a vector of magnitude v over the phase velocity (or A - 1 ) drawn along the ray of the phase waves. He connected this fourdimensional phase vector to the four-momentum vector JM ( = (W/c,px,py,pz)) of the particle by the equation Jti = hQtl
(^ = 0,1,2,3).
(13)
He presented this relation again in his note lSur la definition generate de la correspondance entre onde et mouvemen? ('On the General Definition of the Correspondance between Wave and [Particle] Motion'), submitted in the begining of July 1924 to the Academy of Sciences, Paris (L. de Broglie, 1924b). Especially, he considered in it how the phase wave propagated (i.e., in the direction normal to the plane of equal phases) and how the motion of the phase wave was related to the motion of the particle. Thus he arrived at the equations E = hv
and
p= — = — ,
(13')
where p denoted the momentum of the particle at a given point in the direction of the tangent to its trajectory, and vph the phase-wave velocity in the same direction (which was also the direction of the ray of the phase wave). With the help of the Hamiltonian equation of motion, he also proved that v, the velocity of the particle, was equal to vg, the group velocity of the phase wave, because _ dx _ dW ~li--dp-
v
=
du _ d(v/vph)-Vg-
(14)
At that time, in summer 1924, Louis de Broglie received some support from his Paris colleague Leon Brillouin in dealing with the light-quantum theory. Brillouin
562
The Golden Age of Theoretical Physics
studied the reflection and refraction of radiation in a note, presented by his father to the Academy of Sciences in Paris on 19 May 1924, and showed that the usual behaviour of radiation (i.e., the behaviour due to classical electrodynamics) could be obtained by applying Maupertuis' principle of minimum action to light-quanta (L. Brillouin, 1924). In the course of this investigation he also found the second Eq. (13'). 26 De Broglie himself returned to the problem of interference in fall 1924. In a note, communicated to the Academy of Sciences, Paris, on 17 November 1924, he showed that the interference patterns of light-quanta might arise without invoking specific properties — such as the absorption probabilities (see L. de Broglie, 1923b) — of the atoms forming the apparatus (e.g., the border of the slit), just because of the propagation of the phase waves (L. de Broglie, 1924d). This note, and the slightly earlier one on a problem of atomic theory (L. de Broglie, 1924c) — in which de Broglie showed that the correspondence between the transition frequencies of atoms and the classical radiation frequencies could be deduced from the dynamics of the phase waves connected with the electrons in the neighbouring orbits of higher quantum numbers — were the last ones which de Broglie published in 1924. During the same year, however, he also completed his doctoral thesis, an important piece of work bearing the title 'Recherche sur la theorie des quanta' ('Investigations on the Theory of Quanta') (L. de Broglie, 1924e, 1925). As de Broglie remarked many years afterwards, he began to write the thesis immediately after publishing the three notes in Comptes Rendus in September and October 1923 (L. de Broglie, Conversations with Mehra). 27 In summer 1924 he had completed the work and had it printed (L. de Broglie, 1924e), so that it was ready for his doctoral examination, which was held on 25 November 1924 at the Sorbonne. The thesis was also published in the January-February 1925 issue of Annates de Physique (L. de Broglie, 1925). In his thesis Louis de Broglie attempted to present a systematic and logical report on the results, which he had obtained during the previous years. He organized the material in an historical introduction, seven chapters and one appendix. By the time de Broglie began to write the thesis, he was already thirty-one years old and had pondered about the problems for over ten years. The historical introduction described the perspective in which he wished to place his own studies: namely, the development of the main physical theories since the Renaissance, in a line with the principal ideas created by such men as Newton, Maupertuis, Hamilton, Maxwell, Boltzmann and Gibbs in mechanics, as Fermat, Huygens and Fresenel in optics, as Volta, Ampere, Faraday, Maxwell and Lorentz in electrodynamics, as Lorentz, FitzGerald and Einstein in relativity theory, and as Planck and Einstein in quantum theory. After describing the results of quantum theory, he pointed to the main problems and difficulties which had remained in this theory: 26
During his stay with Sommerfeld in Munich in 1912-1913, Leon Brillouin had concerned himself with the problem of diffraction of electromagnetic radiation in dispersive media and written a detailed paper on it (L. Brillouin, 1914). Leon Brillouin, in his interviews for the Archives for the History of Quantum physics, recalled that de Broglie had begun to compose his thesis already in 1920 and that the printing took a long time. He probably had in mind the earlier versions of the thesis; after all, de Broglie had started to think about his thesis before World War I.
Louis de Broglie and the Phase Waves Associated with Matter
563
namely, the fundamental nature of the quantum conditions of Bohr and Sommerfeld and the simultaneous corpuscular and undulatory nature of radiation, which seemed to have renewed the historic debate between Newton and Huygens. He concluded: In short, the moment seems to have arrived to try to make an effort with the goal of uniting the corpuscular and the undulatory points of view and thus penetrating a bit into the real nature of quanta. This is what we have done recently, and the main goal of the present thesis is to present a more complete treatment of the new ideas which we have proposed, of the successes to which they have led, and also of the many gaps they still contain. (L. de Broglie, 1925, p. 30) In Chapter One de Broglie discussed the phase wave and its properties. He introduced the energy-frequency relation (1) and derived from it the existence of the wave motions with the two frequencies u and v\, given by Eqs. (5) and (6); by requiring agreement of the phases of the two waves at all times, he arrived at the phase wave, as he had shown in his notes of 1923 (L. de Broglie, 1923a,b,c). He showed, in particular, that one could connect two velocities of propagation with the phase wave, the so-called phase velocity and group velocity (denoting the velocity of groups of phase waves), and proved the equality of the group velocity and the velocity of the particle, with which the phase waves were associated. (See the content of the paper, L. de Broglie, 1924a, discussed above). The hypothesis of the phase wave connected with all material particles enabled de Broglie, in Chapter Two, to enter into an analysis of two fundamental dynamical principles: the principle of Maupertius and the principle of Fermat. Although they had been used earlier to describe different phenomena of mechanics and optics, respectively, de Broglie demonstrated that both could be applied alternatively to describe the dynamics of just one object: the quantum-theoretical particle. For this purpose he first formulated Maupertuis' principle of particle dynamics as rB
' /
f
Y\pidli = 0,
(15)
where pi and qi(i = 1 , . . . , / ) denoted the momentum and position variables of a given system (of / degrees of freedom) and the integral extended from the initial state (position) A to the final state B. For a relativistic particle the principle read
'l
]rJM
(/i = 0,1,2,3),
(15')
where JM is the four-momentum of the particle and P and Q are two points in the four-dimensional space. Second, he wrote Fermat's principle, Eq. (12), similarly in the relativistic form, rQ
3
(12') Mdx* = 0 , P Z=S where the phase-wave four-vector 0M = (y/c, —(v/vph) cos(x, 1), — (v/vph). cos(y, I), — (y/v-ph) cos(z, 1)), where 1 denoted a position vector in the three-dimensional space J
564
The Golden Age of Theoretical
Physics
(hence |1| cos(x, 1) = lx, |1| cos(y, 1) = ly, |1| cos(z, 1) = lz) describing the direction of the ray of phase waves. Now he assumed the validity of the quantum-theoretical equation (13) connecting the particle's four-momentum vector J^ with the phasewave vector Ofj,. Comparing then, the content of the two dynamical principles, he concluded: 'Fermat's principle applied to the phase wave is identical to Maupertuis' principle applied to the [moving] particle; the dynamically possible trajectories of the particle are identical with the possible rays of the wave' (L. de Broglie, 1925, p. 56). In the following chapters de Broglie drew certain consequences from his dynamical theory of quanta. Thus, in Chapter Three, he studied the stability of electron orbits in atoms, and presented in some detail the ideas contained in his note of 10 September 1923 to the Academy of Sciences, Paris (L. de Broglie, 1923a). Especially, he showed how the quantum condition for multiply periodic systems followed from the assumption that the electrons in atoms were accompanied by phase waves. When the electron moves in a closed orbit, de Broglie argued, its phase wave should behave analogously to a wave of a fluid in a closed channel of finite length I. A stationary situation will exist in the channel filled with the fluid only in the case of 'resonance,' i.e., when an integral multiple of the wavelength of the fluid motion, n\, is equal to I. De Broglie now imposed the same resonance condition on the phase wave of the electron in the atom, writing it in a form involving the integral of Fermat's principle ( / dl/X with A = vph/v) as
/
— dl=T=n, Uph
n = 1,2,3,...,
(16)
A
where the line integral on the left-hand side extended over the closed path of the electron in the atom. By applying the relation between Fermat's and Maupertuis' principles—i.e., by replacing the integral, f(v/vph)dl, up to a constant factor by the integral, mo § vgdl, de Broglie arrived at Bohr's quantum condition for the circular orbits in the hydrogen atom. Finally, he proved that, in the case of a multiply periodic system the resonance condition for the phase wave led to several quantum conditions: for that purpose he first showed that the sum of the phase integrals had to be an integral multiple of Planck's constant h (see, e.g., Eq. (9); the phase-wave resonance condition then implied that each individual phase integral became an integral multiple of h. In Chapter Four, de Broglie treated the problem of the relativistic hydrogen atom. There occurred a special difficulty in that problem, for one did not know how to distribute the electrostatic energy between the hydrogen nucleus and the electron. De Broglie, therefore, proposed to assign to the electron an effective mass, m e - aP/c2, and to the nucleus an effective mass, M - (1 — a)P/c2, with -P denoting the potential energy, m e , and M the rest masses of the electron and the hydrogen nucleus, respectively, and a a constant between 0 and 1. He estimated the maximal effect caused by the change of the electron's rest mass on R, the Rydberg constant of the hydrogen atom, as calculated from the method due to Bohr; he found SR/R ~ 1 0 - 5 , a correction which was several orders of magnitude smaller
Louis de Broglie and the Phase Waves Associated
with Matter
565
than the usual corrections (due to the motion of the nucleus) considered so far. Neglecting the mass correction, de Broglie then applied his resonance condition, Eq. (1), to the relativistic hydrogen atom and obtained the equation
2nmMR + r?
f ^ ^ )
v/l-(M/(me+M))(v2/c2) V
= n h j
(17)
v
where w is the angular velocity and v (= (R + r)u>) the velocity of the centre of mass of the system, which he assumed to move on a circle of radius R + r (r and R being the radii of the motions of the electron and the nucleus, respectively). In the nonrelativistic limit (v -C c), the condition (331) passed over into the one used by Bohr in 1913 (i.e., 2nme(M/(M + me))uj{R + r)2 = nh, see Bohr, 1913d). In chapter Five, de Broglie presented his theory of light-quanta, which he assumed to be — as in his previous publications (L. de Broglie, 1922b, 1924a) — particles having an extremely small mass mo and, in contrast to electrons (which he pictured to be spherically symmetric objects), possessing 'an axis of symmetry corresponding to the polarization' (L. de Broglie, 1925, p. 77). 28 They should propagate with a frequency-dependent velocity v, where
^V1-^-
(18)
From the known measurements of the velocity of propagation of radio waves having wavelengths of several kilometers, de Broglie concluded that the mass of the light-quanta had to be less than 1 0 - 4 4 g. Then he discussed several results of the wave theory of radiation and compared them with the ones found from the new theory, especially the Doppler effect and the reflection from a mirror; in all cases no difference showed up. Further, the same relation between the pressure of blackbody radiation and its energy density (i.e., p = |/o) could be obtained as in classical theory. Finally he treated qualitatively the phenomena of interference and coherence of radiation and demonstrated that Bohr's frequency condition of atoms was consistent with his theory. De Broglie devoted the following Chapter Six to problems of the interaction of X-ray and 7-rays with matter. After discussing the classical theories of Joseph John Thomson and Peter Debye describing the passage of radiation through matter, and the light-quantum theories of Arthur Compton and Debye describing the Compton effect, he proposed a generalization of the Compton-Debye theory by treating the collision of light-quanta with electrons having a finite initial energy. To the individual collision processes, involving a light-quantum of initial frequency v\ and an electron of initial velocity i>i, he applied the laws of energy and momentum conservation and found for v^, the frequency of the scattered light-quantum, the equation „
= v 2
l-(t;i/c)co8fli *1 - (vi/c) cos
- v'f/c2 '
28 D e Broglie proposed to consider the light-quantum like a 'doublet of the electromagnetic theory', without specifying this concept further; he just noted that he expected a profound modification of the electromagnetic theory in the future (L. de Broglie, 1925, p. 77).
566
The Golden Age of Theoretical
Physics
where 9\ and
A = - % = JL,
(20)
where mo is the rest mass of the gas atoms. If an atom of the gas was allowed to move a linear distance I, the resonance condition implied that an integral multiple of A be equal to I; as a consequence, the thermal velocity of the gas atom would also assume a value that is an integral multiple of vo = h/mol. By applying this consideration to a gas contained in a three-dimensional volume, de Broglie reproduced the results of his earlier paper on the quantum theory of gases (L. de Broglie, 1923c). He also discussed again the theory of blackbody radiation and concluded that the phasewave hypothesis implied a general statistical dependence of quantum-theoretical particles. He gave the following reason: 'If two or several atoms possess phase waves which superpose exactly, and thus one can say that they are transported by the same wave, then their motions cannot be considered as being entirely independent of each other; hence these atoms should not be treated as independent objects in the calculus of probability' (L. de Broglie, 1925, p. 116). That is, the phase wave, if associated with several atoms, gave rise to a certain coherence of their motions. Since the elementary object in statistical mechanics thus appeared to be the stationary phase wave, de Broglie had to define the concept of an elementary stationary
Louis de Broglie and the Phase Waves Associated
with Matter
567
phase wave. He did so by constructing a stationary wave from the superposition of two waves of the same frequency v, the same wavelength A, and the same phase constant
568
The Golden Age of Theoretical Physics
method, through which Einstein had obtained the light-quantum; that is, he had associated with all material particles a wave motion. This phase-wave hypothesis allowed him indeed to formulate the principles of a new, quantum-theoretical dynamics describing particles and light-quanta. While the applications of the new dynamics seemed to reproduce all the good results derived so far in quantum theory, he was aware that he had not answered an important question concerning the physical interpretation of his theory. 'I have intentionally left quite vague the definitions of the phase wave and the periodic phenomenon, as well as of the lightquantum,' de Broglie noted in concluding his thesis, and added: 'The present theory should therefore be considered rather as a scheme whose physical content is not fully defined, than as a consistent doctrine which is definitely established' (L. de Broglie, 1925, pp. 127-128). The principal results of de Broglie's doctoral thesis had become known in Prance (especially through the publication of his earlier notes in the Comptes Rendus) even before he formally presented it. After all, members of the Academy of Sciences of Paris, such as the experimentalists Jean Perrin and Henri Deslandres had communicated these notes. Perrin was also the chairman of the committee which examined Louis de Broglie at the Sorbonne on 25 November 1924; the other three members were the mathematician Elie Cartan and the physicists Charles Victor Mauguin and Paul Langevin. They all found that the candidate gave an elegant presentation of his ideas and defended his thesis quite well. Langevin, who had formally supervised and accepted the thesis, emphasized in his report on it — besides the originality of the thoughts — the fact that de Broglie had been capable 'of having pursued with remarkable mastery an effort aimed at overcoming the difficulties which encumbered the physicists' (Langevin, quoted in Tonnelat, 1966, p. 28). In spite of the good impression which de Broglie left on the members of the committee, and in spite of the fact that his scientific work was well known, he did not entirely convince his examiners of his ideas. 31 As Mauguin recalled nearly thirty years later: At the time of the defence of the thesis, I did not believe in the physical reality of the waves associated with the particles of matter. Rather I regarded them [the waves] as very interesting objects of imagination, which allowed one — for the first time — to avoid the completely empirical character of the quantization rules, providing the latter with a simple, almost familiar, interpretation analogous to the laws of vibrating strings. (Mauguin, 1953, p. 434) Nevertheless they let de Broglie pass the examination. Francis Perrin described many years later the main difficulty, which had stood in the way of the acceptance of the phase-wave hypothesis, as follows: At that time Louis de Broglie had spoken of the fact that when an electron hits a target it gives X-rays and so we are more or less used to this transformation of particles into 31 Francis Perrin, son of Jean Perrin, recalled later: 'I think Mauguin and my father had looked through the paper, before the thesis was expounded. Langevin had thoroughly studied it and probably my father had spoken with Langevin, maybe, already before the day of the exposition of the paper' (F. Perrin, AHQP Interview, 12 January 1963, p. 2).
Louis de Broglie and the Phase Waves Associated with Matter
569
waves. But it was obvious that the wavelength of the X-rays connected with an electron [in the production of X-rays by electron impact] was quite different from the wavelength associated by de Broglie to the particle itself. So this argument of this easy production of waves by collision and transformation of momentum into production of X-rays was not at all a good connection between waves and matter particles. (F. Perrin, AHQP Interview, 12 January 1963, p. 2) Another question concerned the experimental consequences following from the assumption of the matter waves. As Francis Perrin recalled: 'At the soutenance de these [defence of the thesis] my father asked de Broglie if there coould be some real experimental verification of his ideas. And Louis de Broglie stated at that time that maybe diffraction of particles through very narrow slits could be observed and show the existence of waves associated with the particles' (F. Perrin, loc. cit, p. 1). After the doctoral examination de Broglie tried to persuade the experimentalists to investigate the problem. He recalled later one of these efforts: 'I suggested to Mr. Dauvillier — who was a very skillful experimenter, well versed in electronics at that time — to attempt the experiment, but being absorbed in other researches he did not do it' (L. de Broglie, Conversations with Mehra; AHQP Interview, 7 January 1963, p. 6). Paul Langevin was not so much bothered about the direct experimental consequences. He rather admired the theoretical consistency of de Broglie's idea about matter waves. As Francis Perrin recalled later, Langevin remarked at that time: 'It is really this intrinsic invariance relation of the formulation of relativity which seems to me very coherent and true in showing that it is something [fundamental]' (F. Perrin, loc. cit, p. 1). Langevin's opinion was shared by Albert Einstein, whom he considered to be a very high authority. Louis de Broglie recalled Einstein's involvement with pleasure: I had my thesis typed and had naturally given one copy to Mr. Langevin, so that he could prepare his oral report of it. One day Mr. Langevin wrote to me — or telephoned — and said: 'I spoke to Einstein about your thesis and it interested him; he would like to have a copy, do you have a typewritten one?' I told him: 'Yes, luckily I had three copies made, and I do have one.' Well, I sent him a copy, which he forwarded to Einstein. (L. de Broglie, Conversations with Mehra; AHQP Interview, 7 January 1963, p. 7) Einstein's reaction was very favourable, and he wrote to Langevin: 'He [de Broglie] has lifted a corner of the great veil' (lEr hat eine Ecke des groflen Schleiers gelufet,' Einstein to Langevin, 1924; quoted by Kubli, 1971, p. 28). Einstein wrote in greater detail to Hendrik Lorentz about de Broglie's work: 'A younger brother of the de Broglie known to us [i.e., Maurice] has made a very interesting attempt to interpret the Bohr-Sommerfeld quantization rules (Paris Dissertation, 1924). I believe that it is the first feeble ray of light to illuminate this, the worst of our physical riddles. I have also discovered something that supports his construction' (Einstein to Lorentz, 16 December 1924). Einstein's very positive and friendly interest in de Broglie's work convinced Langevin of its value. Einstein, however, did more: he advocated de Broglie's ideas in his second communication to the Prussian Academy on the quantum theory of an ideal gas; he showed that the phase-wave hypothesis supported
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The Golden Age of Theoretical Physics
the q u a n t u m statistics of S. N. Bose and himself, thus providing another evidence of the correctness of Louis de Broglie's ideas (Einstein, 1925a). Einstein's paper t h e n served to acquaint m a n y physicists outside France with de Broglie's work. References Bohr, N. (1913d), Nature 92, 231-232. Brillouin, M. (1919), Comptes rendus (Paris) 178, 1696-1698. Dauvillier, A., and L. De Broglie (1922a,b), Comptes rendus (Paris) 175, 685-688; 755-757. De Broglie, L. (1921a,b), Comptes rendus (Paris) 173, 1160-1162; 1456-1458. De Broglie, L. (1922a,b), J. Phys. et rad. (6) 3, 33-45; 422-428. De Broglie, L. (1922c), Comptes rendus (Paris) 175, 811-813. De Broglie, L. (1923a,b,c), Comptes rendus (Paris) 177, 507-510; 548-550; 630-632. De Broglie, L. (1924a,b,c,d), Phil. Mag. (6) 47, 446-458; Comptes rendus (Paris) 179, 39-40; 676-67; 179, 1039-1041. De Broglie, L. (1924e), Recherche Sur la theorie des quanta, 25 November 1924, Paris: Masson et cie. De Broglie, L. (1925), Annales de physique (10) 3, 22-128. De Broglie, L. (1953), in Louis de Broglie: Physicien et Penseur, Paris: Albin Michel, pp. 457-486. De Broglie, M. (1953), in Louis de Broglie: Physicien et Penseur, Paris: Albin Michel, p. 427. De Broglie, M., and L. de Broglie (1921a,b), Comptes rendus (Paris) 172, 746-748; 173, 527-529. Einstein, A. (1909a), Phys. Zs. 10, 185-193. Einstein, A. (1917a), Phys. Zs. 18, 121-128. Einstein, A. (1925a), Sitz. her. Preuss. Akad. Wiss. (Berlin), pp. 18-25. Emden, R. (1921), Phys. Zs. 22, 513-517. Jeans, J.H. (1910), Phil. Mag. (6) 20, 943-954. Kubli, F. (1971), Archive for History of Exact Sciences 7, 26-68. Langevin, P., and M. De Broglie (1912), La Theorie du Rayonnement et les QuantaRapports et Discussions de la Reunion tenue a Bruxelles, du 30 Octobre au 3 Novembre 1911, Paris: Ganthier-Villars. Lindemann, F.A. (1920), Phil. Mag. (6) 39, 21-25. Maugin, C. (1953), in Louis de Broglie, Physicien et Penseur, Paris: Albin Michel, pp. 430-436. Planck, M. (1921b), Vorlesungen iiber die Theorie der Warmestrahlung, fourth edition, Leipzig: J. A. Barth. Poincare, H. (1912a,b), J. Phys. (Paris) (5) 2, 5-34; Revue Scientifique (5) 50, 225-232. Tonnelat, M.A. (1966), Louis de Broglie et la mecanique ondulatoire, Paris: Seghers. Wolfke, M. (1921), Phys. Zs. 22, 375-379.
16 Wolfgang Pauli and t h e Exclusion Principle* Wolfgang Pauli had since long considered the correct distribution of electrons in atoms as an important problem of atomic theory, and he had been aware of the fact that Bohr's theory of the periodic system did not solve the problem. 'The weakness of the theory is,' he had written to Sommerfeld already on 6 June 1923, 'that it does not provide any explanation for the values 2, 8, 18, 3 2 , . . . of the length of periods, because one cannot derive definite conclusions as to where exactly the periods are closed.' In Pauli's opinion this weakness could not be removed on the basis of the fundamental principles of Bohr's theory; the failure of classical mechanics — i.e., of an essential part of the fundamental principles of atomic theory — which was reflected by the unsuccessful calculation of the stationary states of the helium atom and by the inability (of the Bohr-Somerfeld theory) to explain the phenomena of the anomalous Zeeman effects, made it obvious that the known difficulties of the existing theory could not be resolved by a slight modification of the assumption or equations that had been used so far. New ideas, more or less radically different from the ones involved in the Bohr-Sommerfeld atomic theory, were needed, and Pauli thought strenuously about how to proceed further after he had left Copenhagen — where he had spent most of the academic year 1922-1923 — and resumed his position as assistant to Wilhelm Lenz at the University of Hamburg. He was soon promoted to Privatdozent, and as the topic of his inaugural public lecture he chose ' Quantentheorie und periodisches System der Elemente' ('Quantum Theory and Periodic System of Elements'). 1 Later, in his Nobel lecture, Pauli recalled: 'Lecture delivered at the University of Geneva, 1977, ETH, Zurich, 1978, Cornell University, 1978, and Universite libre de Bruxelles, 1978. This revised version was published (with Helmut Rechenberg) in The Historical Development of Quantum Theory (Springer-Verlag New York, 1982). 1 Pauli submitted the paper on the thermal equilibrium between radiation and free electrons (Pauli, 1923b) as his Habilitation thesis. The members of the Hamburg science faculty, especially the physicists Wilhelm Lenz and Otto Stern, knew about the great scientific qualities of the candidate and were eager to have him as a colleague. Thus the mathematician Erich Hecke, then dean of the faculty, wrote to Pauli on 7 February 1924: 'In its meeting of 30 January 1924, the faculty has decided to accept your application for Habilitation and to exempt you — in view of your scientific achievements — from the trial lecture and the colloquium. Hence I confer upon you the venia legendi [the right to teach] for theoretical physics. The faculty would welcome it if you would give in due course a scientific lecture in its intimate circle.' Pauli gave his inaugural lecture on 23 February 1924. 571
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The Golden Age of Theoretical Physics
Very soon after my return to the University of Hamburg, in 1923,1 gave there my inaugural lecture as Privatdozent on the periodic system of elements. The contents of this lecture appeared very unsatisfactory to me, since the problem of the closing of the electron shells had been clarified no further. The only thing that was clear was that a closer relation of this problem to the theory of multiplet structure must exist. I therefore tried to examine again critically the simplest case, the doublet structure of the alkali spectra. According to the point of view then orthodox, which was taken over by Bohr in his... lectures in Gottingen, a non-vanishing angular momentum of the atomic core was supposed to be the cause of this doublet structure. (Pauli, 1948, p. 133) Towards the end of the year 1923 Pauli concerned himself with a different problem: the heat conduction in solid bodies (see Pauli to Kramers, 19 December 1923). He continued to work on this problem in early 1924, and in a letter to Bohr, dated 11 February 1924, he reported: 'About my own work I cannot report much at the moment. With the problem of heat conduction in solids I have still not succeeded completely, but I hope to finish it before long.' 2 He added: 'In any case, it has been very good for me that I took leave of atomic theory for some time. Later, perhaps very soon, I will return to it with fresh forces' (Pauli to Bohr, 11 February 1924). Pauli, who spoke in the same letter to Bohr about the problems of atomic theroy, such as the anomalous Zeeman effects — 'with which I have tormented myself in vain and which are much too difficult for me' — indeed hoped for a fresh start. But he soon realized that the problems of atomic theory were more difficult than he had anticipated. Thus he wrote again to Bohr: 'In contrast to the point of view, which you still had last fall, I now believe that also with respect to the quantum number k (not only in the case of j) the theory of [multiply] periodic systems does not represent the principal features of the real situation' (Pauli to Bohr, 21 February 1924). Pauli mentioned further that he would like to adopt a very radical point of view with respect to atomic theroy, abandoning the very concept of 'definite orbits of electrons in the stationary states.' However, he did not make any progress in this direction. We are aware that Pauli did not consider the atomic core-series electron model of many-electron atoms as a good candidate for the correct description of the multiplet spectra and the anomalous Zeeman effects. Nevertheless he greeted Lande's paper on the nature of the relativistic X-ray doublets, in which Lande had tried to explain all doublets by using the atomic core-series electron model (Lande, 1924a,b), as an essential step forward. 'I am very enthusiastic about your new paper [Lande, 1924b], and I congratulate you and admire your courage for stating these things so boldly, although you know exactly what a madness it is,' he wrote to Lande on a postcard dated 30 June 1924. Pauli added that he was 'very satisfied about the fact that the discrepancies between observations and the presently accepted principles of quantum theory, which have already showed up in the case of the anomalous Though Pauli continued to work on the problem of heat conduction in solids, he did not publish anything on it in 1924. On 8 February 1925, at the Gottingen meeting of the German Physical Society, he discussed another problem of solid state physics: the absorption of residual rays of crystals (Pauli, 1925c).
Wolfgang Pauli and the Exclusion
Principle
573
Zeeman effect, now become still sharper' (Pauli to Lande, 30 June 1924).3 He still did not believe that the final solution would come from the atomic core-series electron model, and preferred to pursue other ideas. 4 In this context he noticed with great interest the intensity measurements of the line multiplets by Leonard Ornstein and his collaborators and their explanation, especially by Sommerfeld. Sommerfeld drew attention to the occurrence of ratios of integers in the description of the data. In a talk on 21 June 1924 at the meeting of the German Physical Society at Cologne he noted: 'The integer numbers, which determine the position of the lines in the series and multiplets, also determine the intensity ratios of the line components' (Sommerfeld, 1925b, p. 8). That is, he claimed that the intensity of spectral lines was determined by simple functions of the quantum numbers and the likewise integral statistical weights rather than by correspondence arguments. 5 Though Pauli pointed out that these intensity rules were still consistent with the correspondence principle (Pauli to Sommerfeld, 29 September 1924), he shared in fall 1924 Sommerfeld's opinions about quantum problems: he agreed with Sommerfeld's criticism of mechanical models and emphasis on the role of integral numbers; and he supported the negative attitude towards the Bohr-Krammers-Slater radiation theory (see Pauli to Sommerfeld, November 1924, 6 December 1924). And there was still a further point, about which Sommerfeld and Pauli were equally convinced, namely, the validity of relativity theory in atomic physics.6 Pauli's renewed intensive interest in the theory of atomic structure in summer 1924 coincided with his work on an article, entitled ' Allgemeine Grundlagen der Quantentheorie des Atombaues' ('General Foundations of the Qunatum Theory of Atomic Structure'), for Miiller-Pouillet's Lehrbuch der Physik, which Pauli completed in November 1924, but was published five years later (Pauli, 1929b). 7 In the course of this work Pauli had the opportunity to think about various problems 3 I n Pauli's published Scientific Correspondence one finds the word 'nich[t]' instead of 'noch' (see Pauli, 1979, p. 156). We believe that our transcription is the correct one. (See also Forman, 1968, p. 171). Only in a letter to Naturwissenschaften, dealing with the hyperfine structure of certain spectral lines did Pauli use the conventional core-series electron model. (See Section VI.4.) "Sommerfeld repeated the same statements, which he had made in his talk on 23 September 1924 at the Innsbruck Naturforscherversammlung (Sommerfeld, 1924c, p. 1048). 6 At no time did Sommerfeld and Pauli have any doubt about the belief that relativistic considerations played an important role in atomic theory. In contrast to them some physicists claimed that the new situation in the doublet spectra contradicted relativity theory. So, for intance, Ernst Lau, in his review of the situation with respect to fine structure (Lau, 1924, p. 65), referred to an older paper of Ernst Gehrcke, in which Gehrcke had proposed to explain the optical and the X-ray doublets without employing relativistic considerations (Gehrcke, 1920b): Gehrcke had assumed that the atomic nucleus was surrounded by an 'ether,' which formed shells with a discrete number of free spaces — ring-like zones — between them; the electrons would fall or j u m p from one ring-like zone to another and, since each ether-free ring had a finite radial extension, the energy difference could assume several values, which then led to the fine-structure doubles and X-ray doublets. In 1924 Pauli wrote two articles for the Muller-Pouillet Lehrbuch der Physik: one on radiation theory (Pauli, 1929a), and the other on atomic structure (Pauli, 1929b). Both were published as Chapters 27 and 29 of the 11th edition of volumes II.1 and II.2 in 1929, together with an addendum (Nachtrag) to Chapter 29, dealing with the recent developments in electron spin and quantum mechanics (Pauli, 1929c).
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connected with atomic structure, and he turned to them, right after finishing the article. 8 'When I described, in composing this article, the usual theory of the normal Zeeman effect and the Larmor theorem,' he reported to Niels Bohr six weeks later, 'the idea occurred to me that already according to the usual quantum theory [of multiply periodic systems] the relativistic dependence of the electron's mass on the velocity must result in deviations from this [i.e., Larmor's] theorem' (Pauli to Bohr, 12 December 1924). After discussing some spectroscopic consequences of his theoretical considerations in several letters to Alfred Lande in Tubingen (Pauli to Lande, 10 November, 14 November, 24 November 1924), Pauli completed a paper, 'Uber den Einflufl der Geschwindigkeitsabhangigkeit der Elektronenmasse auf den Zeemaneffekt' ('On the Influence of the Velocity Dependence of the Electron Mass on the Zeeman Effect'), which was received by Zeitschrift fur Physik on 2 December 1924 and appeared in February 1925 (Pauli, 1925a). In his new paper Pauli treated the problem of the influence of the relativisitic corrections on the Zeeman effects. Sommerfeld had found in 1916 that this influence should be small in the case of the hydrogen atom and the helium ion, which exhibited a normal Zeeman effect in his calculation (Sommerfeld, 1916d). However, if one considered alkali atoms, a different conclusion followed, at least when one described them by a model in which the two if-electrons formed the core with a nonvanishing angular momentum and this momentum — by assuming discrete positions with respect to the orbit of the series electron — caused the doublet structure and the anomalous Zeeman effect. Pauli argued: 'Since, for higher atomic numbers, the velocity of the if-electron already approaches appreciably the velocity of light and their mass therefore deviates considerably from the rest mass..., one must consequently expect that for elements with higher atomic numbers the influence of the relativistic change of mass in the ivT-shell on the Zeeman effect of optical spectra cannot be neglected at all' (Pauli, 1925a, pp. 373-374). To prove this expectation, he considered the ratio of the absolute value of the magnetic moment |M| (with M = (e/2c)(r x v), where e is the charge of the electron, and (r x v) the vector product — averaged over a full orbit — of the radial vector r, joining the nucleus and the electron, and v the velocity of the electron; c is the velocity of light in vacuo) and | J | the absolute value of the angular momentum of a bound electron (with J = m e ( r x v ) / ( l - v2/c2)ll2, me being the mass of the electron); this ratio was given by the expression |M| _ |J|
|e| 2m e c
(1)
As a result all the energy terms of the atom and their separation will be multiplied by the factor 7, the average over an orbit of the expression for the relativistic square root, i.e., °Pauli mentioned his renewed involvement in atomic theory first in a letter to Alfred Lande, dated 10 November 1924. (See also Pauli to Sommerfeld, November 1924.)
Wolfgang Pauli and the Exclusion Principle
575
1/2
7 = ( l - 3 )
•
da)
In particular the perturbation energy of a one-electron atom in a magnetic field of strength |H| will be (2)
^magnetic = M 7 ^ L ,
where vi(— -e|H|/47rm e c) is the normal Larmor frequency, \± the magnetic quantum number and h Planck's constant. For the factor 7, Pauli found the result
,
w mec2
u [
a2 2
z
r1/2
[n - k + Vk2 - a2Z2} J
^
where n and k denoted the principal and azimuthal quantum numbers of the electron's orbit, Z the atomic number and a(= 2-rce2/he) Sommerfeld's fine-structure constant. For the many-electron atoms, whose spectral lines showed complex structure and anomalous Zeeman effects, the above consideration had some interesting consequences, provided one explained it on the basis of the usual atomic core-series electron model. The relativistic correction to the series electron could then be neglected because one had to replace the quantity Z in Eq. (3) by a small number of the order of unity. However, in case the .ftT-electrons contriuted significantly to the core's momentum — as was usually assumed — important relativistic corrections had to be taken into account, which could be computed by substituting n = k = 1 and (Z-l) instead of Z in Eq. (3). Thus the relativity factor 7 ( = y^l - a2(Z - l) 2 ) became 0.924 for barium (with Z- 1 = 56), or 0.812 for thallium (with Z-l = 80). That is, 'due to the relativity corrections the ratio of the magnetic moment to angular momentum of the K-shell — already when calculated completely classically — must deviate considerably from its normal value in the case of elements with higher atomic number (i.e., it must turn out to be smaller)' (Pauli, 1925a, p. 379). The deviation of the ratio | M | / | J | from its classical, normal value (|e|/2m e c) should be reflected directly in the formulae for the complex structure; especially, the so-called 3-factor should be described by the equation g = l+jS,
(4)
instead of being given by 1+6, with S depending on the angular-momentum quantum numbers of the core (r), the series electron (k) and the total angular momentum (j). (If one assumed — again as usual — the magneto-mechanical anomaly (i.e., an anomalous gyromagnetic ration) of the -K"-shell, then one had to multiply the ratio | M | / | J | for the if-shell by a factor 2, and the y-factor would be given not by Eq. (4), but by the equation ff = l + ( 2 7 - l ) * .
(4')
The experimental g-values did not show any dependence on the atomic number, as implied by Eq. (4) or (4'): for Ba + -lines the deviation from the value with 7 = 1
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The Golden Age of Theoretical Physics
would be 7%, for the mercury doublet lines 18%; rather, all observations confirmed Lande's ^-factor formula up to 1%.9 Hence Pauli concluded: 'If one sticks to the assumptions that even closed electron groups, in particular the K-shell, give rise to the magnetic anomaly, then one must admit not only a doubling of the ratio of magnetic moment to angular momentum in these groups, as compared to its classical value, but also a compensation of the relativity corrections' (Pauli, 1925a, p. 383). Although such compensations appeared to be logically possible, Pauli preferred 'to doubt the correctness of the assumption that the noble gas configuration in atoms contributes essentially to the complex structure and the anomalous Zeeman effect by exhibiting core—momentum with anomalous magnetic behaviour' (Pauli, 1925a, p. 383). He justified his opinion by referring to several difficulties of the usual view: first, only the complete K-shell should exhibit a residual angular momentum, not the other complete shells; second, associating a different gyromagnetic ratio with the atomic core and the series electron did lead to problems if the series electron and another electron in the core were equivalent electrons (i.e., both having the same energy); third, the analysis of Millikan and Bowen, or Lande, showed that the atomic core-series electron model did not represent the data quantitatively correctly (wrong dependence on the effective atomic number). To avoid these difficulties Pauli proposed an alternative intepretation of the data, which he formulated in the following statements: The closed electron configurations are supposed to contribute nothing to the magnetic moment and the angular momentum of the atom. In particular, in the case of alkali elements, the angular momentum values of the atoms and their energy shifts in the presence of an external magnetic field will be viewed essentially as being entirely due to the action of the series electron, which is also considered to be origin of the magneto-mechanical anomaly. The doublet structure of the alkali spectra, as well as the violation of Larmor's theroem, is caused, according to this view, by a peculiar, classically not describable kind of duplicity of the quantum-theoretical properties of the series electron. (Pauli, 1925a, p. 385) Pauli knew perfectly well — and he expressed it clearly at the end of his paper — that the new alternative interpretation he had proposed also encountered serious difficulties. In particular it could not easily be connected to the correspondence approach to spectra. However, he hoped that it would lead to a better understanding of the hitherto unexplained observations, such as the breakdown of Larmor's theorem. In order to make progress he pursued further the idea that the series electron and not the atomic core was responsible for the complex structure and anomalous Zeeman effects exhibited by many-elecron atoms. Already in a letter to Alfred Lande, dated 24 November 1924, he drew attention to the possibility of associating four quantum numbers — the principal quantum number n, the azimuthal quantum number k, and two magnetic quantum numbers, mi and mi — with any electron bound in an atom and to establish a theory of atomic structure on this basis. He emphasized the fact that 'the Aufbauprinzip [the building-up principle] 9 In letters to Lande, Pauli asked about the accuracy of the data. (See also Pauli, 1925a, p. 383, footnote 1.)
Wolfgang Pauli and the Exclusion Principle
577
is strictly valid in my view. This fact seems to me to endow the proposal made here — as against erstwhile considerations — with such superiority that I regard the former [i.e., his (Pauli's own) association of quantum numbers with electrons in atoms] to be more correct physically in spite of all the difficulties against it.' And he pointed out a second reason favouring his interpretation, namely, that 'in it the equivalent orbits find a most natural place' (Pauli to Lande, 24 November 1924). Pauli mentioned further that he could derive important support from a paper of Stoner in the Philosophical Magazine and that he was able to obtain Stoner's results from a single prescription, according to which 'it should be forbidden, that more than one (equivalent) electron with the same n has the same quantum numbers k, mi and m2' (Pauli to Lande, 24 November 1924). The prescription would not only explain the closure of electron groups in atoms and thus the lengths of the periods in the periodic system of elements, but also the other hitherto not understood phenomena, such as the nonexistence of a ground state in the triplet system of alkaline-earth atoms. Pauli considered the scheme of Edmund C. Stoner (Stoner, 1924) — and J.D. Main Smith (Main Smith, 1925) — as a confirmation of his ideas about the role of the series electron in the complex structure. He wrote to Sommerfeld: 'Earlier when I had this issue of Philosophical Magazine in my hands, I had completely skipped over this [Stoner's] paper. However, as soon as I read your preface [to the fourth edition of Atombau und Spektrallinien], I ran to the library and read Stoner's paper' (Pauli to Sommerfeld, 6 December 1924). Pauli agreed with Sommerfeld about the importance of Stoner's work, and he believed that he could extend it to apply to atoms having nonclosed electron shells. As he wrote Sommerfeld: 'If in future my generalization of Stoner's ideas is substantiated by experience in complicated cases as well, then this would mean that in the problem of closing the electron groups in atoms you were completely right in 'putting greater hope in the magic power of quanta than in correspondence and stability considerations' [Sommerfeld, 1924d, p. 192]. Indeed, I do not believe that the correspondence principle has anything to do with this problem' (Pauli to Sommerfeld, 6 December 1924). Pauli expressed the same opinion in a letter to Niels Bohr, dated 12 December 1924. Together with this letter he sent the manuscript of a new paper, entitled ' Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren' ('On the Connection between the Closing of Electron Groups in Atoms and the Complex Structure of Spectra'), to Copenhagen. In his letter to Bohr, Pauli emphasized that he considered his conceptions as an extreme alternative to the usual conceptions, especially the atomic core-series electron model, as 'nonsense conjugate to the usual nonsense up to now' (' Unsinn zu dem bisher ublichen Unsinn konjugiert,' Pauli to Bohr, 12 December 1924). Instead of criticizing Pauli's ideas too much, the Copenhagen physicists encouraged him to publish them. Pauli, after confirming certain consequences of his ideas in discussions with Lande and Ernst Back — during a personal visit to Tubingen on 9 January 1925 — submitted his new paper to Zeitschrift fiir Physik, where it
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was received on 16 January and published in the issue of 21 March 1925 (Pauli, 1925b). In the paper Pauli organized his results into two sections: he devoted the first — ' 1 . The Permanence of Quantum Numbers (Building-Up Principle) in Complex Structure and Zeeman Effect' — to the formulation of a revised 'Aufbauprinzij?; in the second — '2. On a General Quantum-Theoretical Rule for the Possibility of the Occurrence of Equivalent Electrons in an Atom' — he developed his new prescription about electrons in atoms. Pauli based his treatment on the result of his previous paper, especially on the conclusion that the electron groups in atoms and the complex structure of spectral lines originated from a special property of the series electrons, 'a classically not-describable duplicity of the quantum-theoretical properties' (Pauli, 1925b, p. 765). This conclusion, he remarked, led to a characterization of, say, the series electron in alkali atoms with the help of four quantum numbers: the principal quantum number n; the azimuthal quantum number k, or hi, as Pauli denoted it; a second azimuthal quantum number &2 (which was connected with the inner quantum number j in the case of alkali atoms as fo = j + ^); and the magnetic quantum number mi (which could be identified with the component of j — the total angular momentum quantum number of the atom — in the direction of an external magnetic field H). In the presence of the field H , for a given set of three quantum numbers n, k\ and fo, 2Ai2(= 2j + 1) term components arose for a doublet atom; if only n and k were given, the number of states was 2(2fci — 1). In a strong magnetic field, Pauli replaced the quantum number &2 by a second magnetic quantum number 7712, which assumed the two values m^ = 7711 + 5 a n d T7i2 = m i — 5 . 1 0
With the classification of the spectral terms by four quantum numbers, Pauli began to establish a theory of the periodic system of elements. This theory possessed — as he had written to Lande — the advantage that it satisfied fully the principle of the 'permanence of quantum numbers,' or the 'Aufbauprinzip' (the 'building-up principle'). He demonstrated this fact first in the example of alkaline-earth atoms, whose spectral terms could be separated in singlet and triplet terms, such that for a given azimuthal quantum number k\ of the series electron there existed {2k\ — 1) singlet and 3(2fci — 1) triplet components in a magnetic field. According to Pauli these terms arose from assuming that in the magnetic field the atomic core obtained two states — corresponding to the doublet state of the alkali atom preceding the alkaline-earth element under consideration in the periodic system — and that the series electron assumed 2{2k\ — 1) states. In contrast to this picture, the previous atomic core-series electron model violated the principle of (a strict) permanence of quantum numbers because one had imagined (in the core-series electron model) 10 P a u l i emphasized that the occurrence of the half-integral values of the quantum numbers (£2, etc.) had to be related to the anomalous values of the gyromagnetic factor and the 'anomaly of the relativistic correction' (Pauli, 1925b, p. 766). By the latter anomaly he had in mind the fact that the separation of the doublet terms (of atoms in the absence of external magnetic field) was described by a relativistic formula involving twice the usual quantum number of the core.
Wolfgang Pauli and the Exclusion
Principle
579
that any state of the atomic core would turn, under the action of a magnetic field, either into one (for the singlet terms) or three terms (for the triplet terms), hence it did not behave at all like a doublet atom. Moreover, the so-called 'branching rule' that applied to the term structure of many-electron atoms could also be explained. Lande and Heisenberg had found that if one added to the atomic core — whose energy may develop in an external magnetic field into N separate terms — an electron with azimuthal quantum number fci, then the new system — consisting of the old core plus the fci-electron — developed two types of terms: the terms of the first type split in an external magnetic field into (N + l)(2ki — 1) components, and the terms of the second type into (N — l)(2ki — 1) components (Lande and Heisenberg, 1924). Now, in Pauli's scheme, also 2iV(2A;1 - 1) — i.e., (JV + l)(2/fei - 1) + (N - l)(2*i - 1) — states did arise, but in a different way: one had just to assume that the core could possess N energy terms, and the additional electron (2ki - 1) energy terms; this gave altogether 2N(2ki — 1) components and was in complete agreement with the principle of the permanence of quantum numbers. Finally, Pauli showed how to obtain the magnetic perturbation energy for many-electron atoms in strong magnetic fields as a sum of the magnetic energy of the atomic core (which could be taken from the previous element in the periodic system of elements) plus the magnetic energy of the series electron. Pauli emphasized that in the calculation of the magnetic perturbation energy only the contributions from electrons in the not-yet-completed shell had to be considered. For instance, in the case of alkaline-earth elements only the two valence electrons contribute. 11 In concluding the first section, Pauli mentioned two outstanding difficulties of his theory of the periodic system of elements. First, it did not provide the slightest explanation of why different term systems existed for many-electron atoms, e.g., why there were singlet and triplet states in alkaline-earth atoms; nor did it account for the position of these terms in the case of unperturbed atoms, i.e., without external magnetic field (i.e. Lande's interval rule). Second, and even more important, there seemed to be no connection with Bohr's correspondence principle; hence it was not possible to derive, e.g., the selection rules for atomic spectra. Pauli stated: 'Now the difficult problem arises as to how the occurrence of the type of motion, which is required by the correspondence principle for the series electron, can be interpreted physically, independently of its hitherto assumed, specifically dynamical interpretation, which can hardly be maintained any further' (Pauli, 1925b, p. 771). He declared immediately that the solution of this problem might give the key to the calculation of the energy terms of the atoms. In the second section of his paper, Pauli turned to another important aspect of the theory of atomic structure: the problem of equivalent electrons in an atom, i.e., of electrons having the same binding energy (or the same principal quantum The sums of the magnetic terms in the case of the alkaline-earth atoms were — 2i>i,h, 0, 0, and +2v^h, respectively, for s-terms.
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number n). He first drew attention to the fact that the observed term structure of a given atom allowed one to decide the question whether the atom contained equivalent electrons or not. For example, in the case of alkaline-earth atoms the ground state was a singlet s-state with two equivalent electrons, while in the lowest triplet state the two electrons had different principal quantum numbers. From this and other cases Pauli concluded that the theoretical rules describing equivalent electrons in atoms must be formulated in such a way that certain energy terms of the atoms do not occur at all, or certain multiplet terms conincide with others. An example of the latter situation seemed to be present in the neon atom, in which two triplet terms coincide. The task now arose to relate the empirical observations to the electron distribution in atoms due to Stoner. Stoner's separation of electrons into subgroups could be expressed in two rules: first, for each value of the azimuthal quantum number fci there existed 2(2&i — 1) electrons in a complete shell — i.e., 2 for ki = 1, 6 for fci = 2, 10 for ki = 3, etc.; second, for each pair of values of the quantum numbers fci and ki there existed 2ki electrons. Pauli generalized these rules into another rule, namely: ' There can never exist two or more equivalent electrons in the atom for which, in strong [external magnetic] fields, the values of all quantum numbers n,k\,k2,mi (or, what amounts to the same, n,ki,rni,m2) coincide. If the atom contains an electron for which these quantum numbers assume definite values (in an external field), then this state is "occupied"' (Pauli, 1925b, p. 776). Pauli claimed that the new rule applied not only to atoms in strong, but also to atoms in weak magnetic fields, if one assumed the invariance of statistical weights of the states under adiabatic transformations. Under this assumption one could immediately derive the lengths of the periods in the periodic system of elements: i.e., the period 2, 8, 18 and 32 resulted from taking the sum over the number of all electrons permitted by the above rule in the shells characterized by the principal quantum number n = 1, 2, 3, 4, etc. (thus for n = 1, one had k\ = k2 = 1, hence the number of electrons was 2&2 = 2, etc). Similarly, the nonexistence of the triplet sstate for alkaline-earth atoms could be explained as follows: for fci = Ai2 = 1, the two electrons in the outer shell must have different values of the quantum number m i , i.e., mi = + | and —A; these add to mi = 0, or j = 0, for the total atom, hence the lowest state must be singlet s-state, while the triplet s-state was forbidden. Pauli further showed how his rule yielded similar structures for the X-ray spectra and the optical doublet spectra of alkali atoms: the X-ray spectra emerged by removing an electron from a complete shell, leaving a gap; the removed electron possessed a magnetic quantum number mi, whose value was minus the sum of the magnetic quantum numbers of the electrons remaining in the shell, i.e., — 5 Z m i (because in a complete shell the sum of mi's was zero); hence the emitted X-ray spectrum must have the structure of alkali spectrum. Pauli generalized this result into what he called a 'general reciprocity law' stating: lFor any distribution of electrons there exists a conjugate distribution, in which the gap values of mi and the occupied values
Wolfgang Pauli and the Exclusion
Principle
581
of mi are interchanged1 (Pauli, 1925b, p. 778). 12 That is, the spectra of two atoms, whose electronic structures possesed conjugate organizations — i.e., their respective numbers of electrons in a given subgroup added up to the number in a complete subgroup — , should have the same structure (because in both atoms the internal quantum numbers will be the same). As a consequence, for example, the spectra of elements at the end of a period and at the beginning of the following period in the periodic system (such as chlorine and potassium) must be related. Of course, that similarity would not determine the energy terms of the atoms and their differrences (i.e., the spacing of the multiplets). As a particularly interesting application of his rules, Pauli took the case of neutral carbon. According to the Lande-Heisenberg branching rule, the spectrum of carbon should be similar to the neon spectrum. Pauli, however, arrived at a different conclusion: in his scheme, carbon should have five different p-terms, as compared to one for the neon atom. Pauli's result, as well as the g-values which he derived for atoms having four electrons in the outer shell (like carbon or lead), seemed to be substantiated by the experimental data. 1 3 Because of the reciprocity theorem the results from carbon could be transferred immediately to the case of oxygen, which contained four electrons in the k\ — 2 subshell. In this case, as well as in other examples (say of atoms having five and seven electrons in a given shell), Pauli just briefly sketched the situation; however, sufficient data were not available to check the conclusions. Pauli's rule for equivalent electrons, which later became known as 'Pauli's principle' (Born, 1926d, p. 65), or 'Pauli's ban' ('Pauli's Verbot,' Heisenberg. 1926b, p. 423), or 'Pauli's exclusion principle' (Dirac, 1926f, p. 670), was accepted without delay by the physicists working in spectroscopy and atomic theory. The available evidence, much of which had been accumulated by Pauli in his paper, proved its validity. As we have mentioned earlier, even Bohr and the other Copenhagen physicists, took it seriously right away. And Werner Heisenberg — who, with Alfred Lande, had invented the atomic core-series electron model (including the magnetic coupling, according to which the energy differences of the atomic multiplet terms were caused by the different angles between the angular momenta of the atomic core and the series electron) — applied Pauli's principle immediately in his next paper on multiplet structure and Zeeman effects; he also showed in this paper that the principle was compatible with an extended formulation of the correspondence principle (Heisenberg, 1925b). 14 By the end of 1925, the exclusion principle belonged to ^ P a u l i ' s 'reciprocity law' expressed in a rational way what Edmund Stoner had earlier intuitively explained by referring to 'a sort of Babinet principle' (Stoner, 1924, p. 733). Stoner had in mind the principle of the French physicist and astronomer Jacques Babinet (1794-1872), stating that similar optical phenomena arise from illuminating complementary objects. (An illuminated screen leads essentially to the same diffraction pattern as a slit having the same shape.) The experimental information on lead was first presented by Hertha Sponer at the Gottingen meeting of the German Physical Society on 8 February 1925 (Sponer, 1925). Pauli, however, knew about the results already earlier from the experimentalist Ernst Bach of Tubingen. 14 We shall discuss Heisenberg's work in Volume 2, Sec. III.6.
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the accepted laws of atomic physics (see, e.g., Franck and Jordan, 1926). Not only did the spectroscopists work with it; in a paper presented on 7 February 1926 at the Accademia dei Lincei in Rome, Enrico Fermi developed a quantum-theoretical statistical method for particles, which, like the electrons, obeyed the exclusion principle (Fermi, 1926a). 15 Fermi proceeded in a way similar to what Satyendra Nath Bose and, especially, Albert Einstein, had done in developing their quantum statistical method; he derived an equation of state for gases consisting of mass-particles, whose state was characterized by a set of quantum numbers such that any two of them were not in a state described by the same values of this set of quantum numbers. This equation of state was h2n5/3 _ /27rmfcT\
p=
^^ p Uwj'
(5)
where p denoted the pressure of n particles of mass m, T the absolute temperature, and k and h Boltzmann's and Planck's constants, respectively. The expression P(2irmkT/h2n2/3), represented a function which, for very large values of the argument, could be approximated by the argument (i.e., 2irmkT/h2n2/3) itself, and for very small values of the argument it (i.e., the function P) approached a constant (i.e., 32/3^1/3/5 . 2 1 / 3 ). That is, for high temperatures T and small particle numbers n (per given volume), Eq. (5) passed over into the classical equation of state of an ideal gas; for low T and high n, on the other hand, the pressure p satisfied an equation describing a degenerate gas, with zero-point pressure (po = ^(6/7r) 3 / / 2 (/i 2 n 5 / , 3 /m)) and a low-temperature dependence given by a term proportional to n 1 / 3 T 2 . 1 6 The physicists worked with the exclusion principle in spite of the fact that Pauli was unable to provide 'a more precise justification' for it (' eine ndhere Begriindung fur diese Kegel,'1 Pauli, 1925b, p. 776). Pauli, however, expressed the hope that 'the problem of a better foundation might be attacked successfully after a further deepening of the fundamental principles of quantum theory' (Pauli, 1925b, p. 783). But even after the 'deepening' of the fundamental principles of quantum theory 15 B . Fermi was born in Rome on 29 September 1901. He studied physics at the University of Pisa (1918-1922), obtaining his doctorate in 1922. After a year's work as a private tutor in Rome, he received a travelling scholarship from the Italian Government, with the help of which he spent periods of stay at the University of Gottingen (in Bom's Institute, winter 1923-1924) and at University of Leyden (with Ehrenfest, fall 1924). Then he became a lecturer in physics at the University of Florence and, in 1926, professor of theoretical physics at the University of Rome. He left Italy for the United States in 1938 (where he became a naturalized citizen in 1945). He first accepted a professorship at Columbia University (1939-1942); then he moved to the University of Chicago (1942-1945, work on the atomic bomb project; 1946-1954, professor of physics at the Institute of Nuclear Studies). He died on 28 November 1954 in Chicago. Fermi did outstanding theoretical and experimental research in many fields of physics: spectroscopy, statistical mechanics, quantum mechanics and quantum field theory (theory of /3-decay), neutron physics (nuclear transformations by neutron-nucleus collisions), nuclear fission (fission with slow neutrons; building of the first atomic pile in Chicago, 1942), high energy and elementary particle physics (cosmic rays, pion-nucleon interaction). He received the 1938 Nobel Prize in Physics for his work on nuclear reactions stimulated by neutron impact. At low temperatures, the specific heat of the Fermi (i.e., electron) gas was proportional to the absolute temperature T.
Wolfgang Pauli and the Exclusion Principle 583 had occurred, no better justification of the exclusion principle became available. As time went on, a wave-mechanical formalism was developed, including the Pauli exclusion principle in a natural way (by taking wave functions for systems of several electrons, which were odd against the permutation of two electrons: Dirac, 1926f). Dirac would also succeed in obtaining a relativistic equation for the electrons (Dirac, 1928); and from this equation would follow, for instance, all the details of the fine structure of the spectral lines of hydrogen, in the same way as from Sommerfeld's relativistic formula of 1915 (Sommerfeld, 1915c). Dirac's equation for the electron would become the prototype of an equation describing the dynamical behaviour of a large clas of elementary particles, the so-called 'fermions' — including the electron, the proton and the neutrino. The other class of elementary particles, the so-called 'bosons' — like the light-quanta (photons) or 7r-mesons — which did not obey the exclusion principle and the Fermi statistics, but the Bose statistics, would be described by similar, relativistic field equations. In the late 1930s Pauli, after investigating in detail these relativistic equations for fermions and bosons, would expound another important principle of atomic theory, the spin-statistics theorem (Pauli, 1940). This theorem would then allow a deeper explanation of the exclusion principle, the fundamental principle governing the behaviour of electrons (and many other elementary particles), for which Pauli would receive the Nobel Prize in Physics for the year 1945. By mentioning the word 'spin,' we have already gone a step beyond what Pauli had conceived at the turn of the year from 1924 to 1925. The spin, an extra mechanical property of the electron, was not in the spirit of the ideas which the author of the exclusion principle developed. On the contrary, Pauli refrained from invoking any mechanical — dynamical interpretation of his rules, and he even opposed the occurrence of the concept of spin in spring 1925, as we shall see in the following essay. References Born, M. (1926d), Problems of Atomic Dynamics, Cambridge, Massachusetts: The M.I.T. Press. Dirac, P.A.M. (1926f), Proc. Roy. Soc. (London) A112, 661-677. Dirac, P.A.M. (1928), Proc. Roy. Soc. (London) A117, 610-324. Fermi, E. (1926a), Rend. R. Accad. Lincei 3, 145-149; reprinted in Collected Papers I, pp. 186-195. Forman, P. (1968), Isis. 59, 156-174. Franck, J., and P. Jordan (1926), Anregung von Quantenspriingen durch Stofie, Berlin. J. Springer-Verlag. Gehrcke, E. (1920b), Phys. Zs. 25, 378-381. Heisenberg, W. (1925b), Z. Phys. 32, 841-860. Heisenberg, W. (1926b), Z. Phys. 38, 411-426. Lande, A. (1924a,b), Z. Phys. 24, 88-97; 25, 46-52. Lande, A., and W. Heisenberg (1924), Z. Phys. 25, 279-286. Lau, E. (1924), Phys. Zs. 25, 60-68. Main-Smith, J.D. (1925), Phil. Mag. (6) 50, 878-879. Pauli, W. (1925a,b), Z. Phys. 31, 373-385; 765-783.
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Pauli, W. (1929a,b,c), Muller-Pouillets Lehrbuch der Physik II: Lehre von der strahlende Energie (Optik), Part 2.1, pp. 1483-1553; Muller-Pouillets Lehrbuch der Physik II/2.2, pp. 1709-1792; Muller-Pouillets Lehrbuch der Physik II/2.2, pp. 1793-1842 (Nachtrag to Chapter 29). Pauli, W. (1940), Phys. Rev. (2) 58, 716-722. Pauli, W. (1948), Nobel lecture on 'Exclusion Principle and Quantum Mechanics,' in Les Prix Nobel en 1946, Stockholm, pp. 131-147; Collected Scientific Papers 2, pp. 1080-1096. Pauli, W. (1979), Wissenschaftler Briefwechsel mit Bohr, Einstein, Heisenberg, u.a., Volume 1: 1919-1929, New York-Heidelberg-Berlin: Springer-Verlag. Sommerfeld, A. (1915c), Sitz. ber. Bayer. Akad. Wiss. (Miinchen), pp. 459-500. Sommerfeld, A. (1924c), Naturwiss. 12, 1047-1049. Sommerfeld, A. (1924d), Atomban und Spektrallinien, fourth edition, Braunschweig: Fr. Vieweg. Sommerfeld, A. (1925b), Zeitschrift fur technische Physik 6, 2-11. Sponer, H. (1925), Z. Phys. 32, 19-26. Stoner, E.C. (1924), Phil. Mag. (6) 48, 719-736.
17 The Discovery of Electron Spin*
The progress in atomic theory rested on the use of relationships involving integral numbers, such as the number of electrons occupying a given energy level in the atom under investigation. It happened at times that the quantum laws of this kind did not reflect the dynamical situation in atoms properly, especially the fact that the electrons moved on orbits under the attractive force of nuclei and the repulsion of neighbouring electrons. One could hardly expect that such a dynamical situation would yield relationships describable in terms of ratios of small integral numbers, like the ones accounting for the intensity ratios of multiplet components. On the contrary, one could foresee that the electron-electron interaction in many electron atoms must destroy any simple ratios. If one further believed in the principles of the Bohr-Sommerfeld atomic theory, one had real difficulty in answering the question of why the dynamical perturbation due to the mutual interaction of electrons should not play the expected role. The problem was even more complicated. Niels Bohr himself had claimed repeatedly in the early 1920s that, in order to understand atomic structure and atomic spectra in detail, one had to take into account the interaction between the charged particles constituting atoms and their coupling to the radiation field. And the progress achieved with the help of the dispersiontheoretic approach in explaining various properties of spectral lines justified his claim. Now the similarly successful Main Smith-Stoner-Pauli theory of atomic structure did not refer to any interaction or coupling in atoms. Pauli stated clearly that the new approach had nothing to do with the previous dynamical picture of atoms, which he belived to be fundamentally wrong. The physicists around Bohr in Copenhagen, while acknowledging the progress brought about by the ideas of Stoner and Pauli, judged the old atomic theory to be better than what Pauli thought. They rather agreed with what Pauli had himself cautiously stated in his paper of December 1924, after having discussed the respective consequences of the old and new ideas in the description of the anomalous Zeeman effects: 'Perhaps the final solution of the problems considered here will lie in the direction of a Lectures delivered at the University of Texas at Austin, 1969, University of Leyden, 1976, University libre de Bruxelles, 1977, Cornell University, 1978, and the University of California at Irvine, 1980. Revised and enlarged version published (with Helmut Rechenberg) in The Historical Development of Quantum Theory (Springer-Verlag New York, 1982).
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compromise between both of these points of view' (Pauli, 1925a, p. 385). This encouraged many theoreticians to seek to combine Pauli's exclusion principle with the dispersion-theoretic approach when they treated problems of atomic theory. 1 In the standard atomic model of Bohr and Sommerfeld the energy states were determined by the electric interaction between the nucleus and the electrons plus the quantum conditions. In order to account for the multiplet structure of spectral lines and the terms involved in the corresponding transitions, Alfred Lande and Werner Heisenberg had further introduced the so-called magnetic coupling. That is, the splitting of the components of term multiplets originated from an interaction of the magnetic moment of the atomic core with the magnetic field — the 'internal' magnetic field — created by the orbiting series electron. This magnetic interaction was, of course, different from the one which Walther Ritz had earlier used in his theory of optical spectra. 2 However, in Ritz' theory as well as in the Lande-Heisenberg theory the atom possessed a central core, which acted like an elementary magnet. 3 The Lande-Heisenberg picture of a many-electron atom allowed one to describe the observed data on multiplet structure and anomalous effects rather satisfactorily (as we have discussed in Section IV.4). But it also exhibited several unpleasant features: first, the ratio of magnetic moment to the angular momentum of the atomic core had to be assumed to be twice as large as on the electrodynamical theory, i.e., the core's gyromagnetic factor was g = 2; second, for the interpretation of the data, one had to use half-integral quantum numbers, especially for the angular momentum of the core. When the physicists disregarded these features, which could really not be understood on the accepted principles of atomic theory, they were able to derive — with the help of correspondence arguments — formulae for the intensity ratios of the multiplet lines and their Zeeman components. These formulae fitted the data, provided especially by the Utrecht experimentalists (Ornstein and Burger, 1925; Ornstein, Burger and Geel, 1925; Geel, 1925), perfectly. (See, e.g., Heisenberg, 1925a; Kronig, 1925a; Sommerfeld and Hdnl, 1925.) An especially interesting problem was posed by a type of multiplets, whose gfactors did not obey the usual formula. Alfred Lande, when analyzing the neon spectrum in 1923, had already drawn attention to this problem (Lande, 1923d). Lande and Heisenberg had then developed a formal theory of such multiplets, which they The physicists dealing with multiplet structure and the anomalous Zeeman effects had so far used various versions of the atomic core-series electron model. Among the few exceptions we might mention the work of Arthur Bramley, a student of Edwin Plimpton Adams at Princeton University; he tried to derive the frequency of the multiplet components from a relativistic theory, involving four phase integrals rather than three (Bramley, 1925). Ritz had assumed that spectral lines came about when the charged constituents of atoms performed harmonic oscillations in the magnetic field of a small elementary magnet constituting the centre of the atom (Ritz, 1908a). The necessity of taking into account the magnetic forces in an atom had also been discussed by Herbert Stanley Allen (1873-1945) of King's College, London, in several notes and papers published in 1914 and 1915 (Allen, 1914a,b; 1915a,b). Allen had considered as model of the atom a central core, carrying electric charge and producing a magnetic field similar to t h a t of an elementary magnet, surrounded by electrons in orbital motion.
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called 'multiplets of higher order' or 'supermultiplets' ('Multiplett hoherer Stufe1), with the help of their 'branching rule' (' VerzweigungsregeV); it stated that the neon core did not simply possess the angular momentum j (in unit of h/2ir) of a singlyionized neon atom (Ne + ), but from each Ne + -state there developed two states of the neon core with angular momenta j + \ and j — \ (Lande and Heisenberg, 1924).4 According to the branching rule, the terms of the neon atom were multiplets of the 'second order' ('zweiter Stufe'); they had to be characterized by several indices instead of the one index (r) describing the angular momentum of the atomic core. Lande and Ernst Back investigated the neon multiplets further in their contributions to the issue of Annalen der Physik in honour of Friedrich Paschen's sixtieth birthday (Lande, 1925a; Back, 1925a). In a paper published several months later Back considered another example of multiplets (Back, 1925b): these multiplets showed up in the analysis of the calcium spectra and belonged to what Lande and Heisenberg had called multiplets of the 'first order' ('erster Stufe'), but violated Lande's ^-formula of 1923. In dealing with the Zeeman effects of these multiplets, Back therefore spoke about 'the irregular Zeeman effects of the multiplets of the first order.' At the same time he noted: 'The line groups of Ca investigated by the author belong to the sustem of 'new terms,' which has been identified and interpreted most recently by Russell and Saunders in the spectra of alkaline earths' (Back, 1925b. p. 580). The paper to which Back referred was entitled 'New Regularities in the Spectra of the Alkaline Earths' and published in the January 1925 issue of the Astrophysical Journal (Russell and Saunders, 1925). It had been submitted in October 1924 by Henry Norris Russell, then professor of astronomy at Princeton University, and Frederick Albert Saunders, then professor of physics at Harvard University. 5,6 Russell 4
We shall discuss further details of the Lande-Heisenberg paper later on. H.N. Russell was born on 25 October 1877 at Oyster Bay, New York. He studied astronomy at Princeton University, obtaining his A.B. in 1897 and his doctorate in 1899 (under Charles Augustus Young). Prom 1903 he spent two years at the Cambridge Observatory (with Arthur Robert Hinks) as a research assistant of the Carnegie Institution. Returning to Princeton University in 1905, he became instructor in astronomy (1905-1908), assistant professor (1908-1911), professor (1911-1927) and research professor (1927-1947). Since 1912 he was also director of the Princeton University Observatory, remaining in this position until he retired in 1947. He died in Princeton on 18 February 1957. Russell contributed both to experimental and theoretical astrophysics: he proposed, in particular, a theory of stellar evolution and he developed with Hinks a method for the photographic determination of star parallaxes. He received many honours, including the gold medal of t h e Royal Astronomical Society in 1921, the Henry Draper medal of the National Academy of Sciences of U.S.A., and the Lalande medal of the Academy of Sciences, Paris. 6 F . A . Saunders was born in London, Ontario, Canada, on 18 August 1875. He studied physics at the University of Toronto (B.A., 1895) and Johns Hopkins University (Ph.D., 1899). In 1899 he became instructor in physics at Haverford College (Pennsylvania); two years later he moved to Syracuse University, where he was appointed associate professor of physics in 1902 and professor in 1905. He spent the academic year 1913-1914 in Europe at the Universities of Cambridge and Tubingen, and then joined Vassar College as a professor. Finally, he went to Harvard University as associate professor of physics (1919-1923) and became full professor (1923-1941). He retired in 1941 and died on 9 June 1963 at South Hedley, Massachusetts. At first Saunders worked mainly on optical spectroscopy; later on he changed his field to acoustics, being interested in the mechanical properties of stringed instruments. He was a member of the National Academy of Sciences. 5
"v.
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and Saunders devoted their paper to an analysis and interpretation of 'many prominent lines in the spectra of calcium and the allied metals which do not belong to the regular series and yet are evidently closely related to them' (Russell and Saunders, 1925, p. 38). By these prominent lines they meant 'several very conspicuous groups of lines, of exactly the character now recognized as multiplets, which are evidently combinations between known triple terms and other terms for which no place can be found in the recognized series' (Russell and Saunders, 1925, p. 42). The earliest mention of such lines dated back to 1894 (Rydberg, 1894). Later on they had been investigated occasionally, e.g., by Arthur S. King of Mt. Wilson Solar Observatory (King, 1918), but around 1920 the interest in them had increased because these lines did not seem to fit into a Ritz-like formula and into Bohr's theory of atomic structure. Walter Grotrian had proposed to interpret some of the non-Ritz terms in Priedrich Paschen's analysis of the neon spectrum (Paschen, 1919) as connected with the excitation of one of the four electrons existing in the Z-orbit of lower energy, i.e. having the quantum numbers nk = 2i. 7 Thus he had concluded that, since the ionization potential for electrons on the 2i-orbit was higher for the 22-electrons, the series limit of the anomalous lines must be higher than the one for the normal lines; hence, if that fact were taken into account, the anomalous lines also satisfied a Ritz formula (Grotrian, 1921). In his Wolfskehl lectures at Gottingen (especially in Lecture Six, see Bohr, 1977, p. 401), Bohr had applied a similar argumentation in order to explain the anomalous terms of alkaline-earth atoms. 8 Gregor Wentzel had finally worked out in a note, submitted to Physikalische Zeitschrift in early 1923, a systematic theory to explain the unusual or 'primed' ('gestrichenen') terms of many-electron atoms (Wentzel, 1923a). His interpretation of the anomalous terms of calcium, which he had attributed to a series electron on the 33-orbit (while the series electron of the normal terms was on a 4i-orbit), had been criticized: Lande had claimed that the series electron of the anomalous terms should have the same azimuthal quantum number as the normal series electron; that is, it should occupy a 4i-orbit (Lande, 1924c). The Wentzel-Lande controversy still continued at the turn of the year from 1924 to 1925 (see, e.g., Laporte and Wentzel, 1925; Lande, 1925b). Russell and Saunders already were familar with the problem of the anomalous terms before they presented their long memoir in the Astrophysical Journal (see, e.g., Saunders and Russell, 1923). They knew that these terms could be described like the normal terms by quantum numbers. That is, there existed p-, d-, and / terms, whose energy was shifted with respect to the corresponding normal ones. More accurately, there existed two types of anomalous terms: those which combined with the usual terms having the same azimuthal quantum number and those which combined with the usual terms having different azimuthal quantum numbers; the former were denoted as p'-, d'-, and /'-terms, and the latter as p"-, d"-, and ^The usual ('Ritz') terms of t h e neon atom were associated with the excitation of a series electron in the higher Z/-orbit, i.e. the n*. = 22-orbit. ^These terms had been analyzed in accordance with the Bohr-Sommerfeld theory of atomic structure (especially the inner quantum number) by Raimund Gotze (1921).
y
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/"-terms. Anomalous lines occurred rather frequently; for example, in King's table of the calcium lines, 81 were normal and 39 were anomalous lines. The intensities of the lines could also be compared quite well: some of the new lines belonged to the strongest ones in the calcium spectrum. Further, it was possible to arrange the new lines in series. 9 So far the properties of the normal and the anomalous lines were
more or less the same. However, in several respects the anomalous terms — and the anomalous lines arising from them — showed a behaviour different from that of the normal terms. Especially, the primed terms appeared to have strange combination properties. For example, the d'-terms (i.e., the anomalous triplet terms) of alkaline-earth atoms combined with normal terms of types d (triplet), D (singlet terms with azimuthal quantum number 3), and S (singlet term with azimuthal quantum number 1), thus behaving like a normal p-term; the /"-terms gave strong combination lines with normal d- and Z?-terms, behaving sometimes like p-terms and sometimes like /-terms. Russell and Saunders thus summarized the situation as: ' The azimuthal quantum numbers of the new terms are in some cases the same as for ordinary terms having the same inner quantum number, and in other cases differ by one, or perhaps by two, units' (Russell and Saunders, 1925, pp. 56-57). A particularly surprising feature was that some of the anomalous terms, e.g., the 3p'-term in calcium, possessed a negative ionization potential, if referred to the zero level of normal terms (Saunders and Russell, 1923; Wentzel, 1924b). Neverthless, these anomalous terms belonged evidently to the neutral and not to the ionized atom. A final property of the calcium p'-term provided, in the opinion of Russell and Saunders, 'the clue to the whole problem of anomalous terms.' They argued as follows: 'If more than enough energy to pull one electron entirely out of the atom can be put into the atom, without doing so it must be divided between two (or more) electrons, each of which is shifted to a higher energy-level without removing any. It is probable that all 'anomalous' terms correspond to states of this sort' (Russell and Saunders, 1925, p. 58). In the case of the calcium anomalous states this agreed with Wentzel's interpretation, according to which the p'-terms correspond to states, in which one of the two electrons (as usual) occupied successive orbits with the same azimuthal quantum number but increasing principal quantum numbers, while the second remained in an orbit with fixed quantum numbers. Having this interpretation in hand, Russell and Saunders proposed to extend Lande's vector model for the new terms. With that goal in view, they assumed a model in which the alkaline-earth atom consisted of a core and two series electrons. In the case of the anomalous triplet states the core should have the angular momentum R = §, and the two electrons have angular momenta K\ and K2, while for the anomalous singlet states R should be | (in units of h/2n). The angular momenta of the two electrons then combined to give the resultant 'angular momentum' K of the atom, its values 9 Russell and Saunders noticed, for instance, that the (pp')- transition frequencies obeyed a formula of the Ritz type: the frequencies of the series first decreased a bit and then increased strongly, showing a behaviour similar to the one in other alkaline-earth series.
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being \, § , . . . , and the angular momentum of the core should be quantized in the direction of K. Evidently, the new vector model differed considerably from Lande's for the normal terms, in which only the combination of the angular momentum of the atomic core with that of one series electron played a role. On the other hand, the selection rules for transitions depended — as for normal terms — only on the value of the azimuthal quantum number k of the electron. Their new interpretation and model allowed Russell and Saunders to describe all observed data on anomalous lines. Ernst Back then investigated experimentally, in the paper mentioned earlier, the g-factors of the anomalous calcium terms. He suspected that 'the interpretation proposed by Russell and Saunders about the origin of the new terms renders it quite probable that these terms deviate at least partially also in their magnetic behaviour from the usual terms' (Back, 1925b, p. 581). 10 The data confirmed his suspicion: the separation of the components in the Zeeman effects of some anomalous lines were not given by Lande's ^-formula and the interval rules could be violated; evidently, other selection rules were applicable besides those in the anomalous Zeeman effects of normal lines. 11 While Back carried on the work of Russell and Saunders with his experimental investigations, Werner Heisenberg provided some theoretical support. In a paper, entitled 'Zur Quantentheorie der Multiplettstruktur und der anomalen Zeemaneffekte1 ('On the Quantum Theory of the Multiplet Structure and the Anomalous Zeeman Effects'), which was received by Zeitschrift fiir Physik on 10 April 1925 and published in July 1925, he discussed the various models of many-electron atoms used in the description of data: (i) the standard atomic core—series electron model, (ii) Pauli's model, in which the electron was assigned four quantum numbers, and (iii) the model proposed by Russell and Saunders (Heisenberg, 1925b). According to Heisenberg the Russell-Saunders model for the alkaline-earth atoms could be interpreted as an atomic core-series electron model, involving a core which reflected some of the combined properties of the two outer electrons. From this model he derived formulae describing the anomalous energy terms and their Zeeman effects, finding among other results a violation of Lande's g-formula.12 Russell and Saunders had claimed that the p-values (derived from the anomalous Zeeman effects of the anomalous lines) of the anomalous terms obeyed in most cases Lande's formula of 1923. Anomalous g-factors occurred, e.g., for / " - t e r m s and for the so-called a;-terms. Back noticed, however, that the products of the g-values of the related / " - and i-terms were equal to the g-values of the normal / - t e r m s . The selection rule for transitions involving anomalous states stated that certain m —> m' transitions, which were allowed for normal lines, were forbidden. Thus in the cases of the p " — d transitions of magnesium and calcium, certain transitions from p"-states with magnetic quantum number m — 0 were suppressed. 12 We shall discuss the details of Heisenberg's paper in Volume 2, Section III.6, especially with respect to its role in the process of extending the correspondence principle It was concerning this paper of Heisenberg's that Born wrote to Einstein: On the whole my young people, Heisenberg, Jordan and Hund, are brilliant. I find that merely to keep up with their thoughts demands at times a considerable efforts on my part. Their mastery of the so-called 'term zoology' is marvellous, Heisenberg's latest paper [Heisenberg, 1925b] soon to be published, appears rather mystifying but
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The progress achieved in the understanding of anomalous terms greatly stimulated the research of many physicists. For example, Ralph Kronig extended his considerations on the intensity of multiplet lines and their Zeeman components to anomalous lines (Kronig, 1925b), and Gregor Wentzel completed the ordering of alkaline-earth terms (Wentzel, 1925e). Friedrich Hund carried out a detailed analysis of the spectra of all elements: in three papers, submitted in June and July 1925 to Zeitschrift fiir Physik, he developed a more or less complete interpretation of the complex spectra (Hund, 1925c,d,e). Hund — like Pascual Jordan in the special case of neon (Jordan, 1925c) — tended to prefer Pauli's scheme based on the exclusion principle to the atomic core-series electron model, though both Hund and Jordan avoided any discussion about which model rested on a more satisfactory theoretical foundation. This question was approached by other people, who concerned themselves with the physical interpretation of Pauli's fourth quantum number of the electron. At the same time as the analysis of the data proceeded, they introduced a new property of the electron: the intrinsic angular momentum or 'spin' of the electron. The first to do so was Kronig. Ralph de Laer Kronig learned about Pauli's scheme of four quantum numbers for the electron during his stay in Tubingen in early 1925. 13 He had been interested in atomic theory even as a student at Columbia University. As he recalled later: 'I had spent a good deal of time before I left America in 1924 studying the theory of the anomalous Zeeman effect and the whole spectroscopy. I was one of the people who had the rules of spectroscopy, which were mostly empirical at that time, more or less at [my fingertips]. I knew about the Burger-Dorgelo sum-rules before I came to Holland' (Kronig, AHQP Interview, 11 December 1962, p. 5). In Leyden he began to work on the intensity sum-rules and wrote a note on their theoretical derivation together with Samuel Goudsmit, which was submitted in December
is certainly true and profound; it enabled Hund to bring into order the whole of the periodic system with all its complicated multiplets. (Born to Einstein, 15 July 1925). 1-3 R. de L. Kronig was born on 10 March 1904 in Dresden, the son of an American citizen (who was an artist). He went to the Gymnasium in Dresden until 1918, when his family moved to the United States. From 1919 to 1924 he studied at Columbia University (B.A. in 1922; Ph.D. in experimental physics with A.P. Wills in 1924). With a Bayard-Cutting Traveling Fellowship he returned to Europe in 1924, visiting the Universities of Cambridge, Leyden, Tubingen, Copenhagen and Rome. Early in 1926 he returned to New York, taking up a lectureship at Columbia University. He spent the academic year 1927-1928 on the International Education Board Fellowship with Niels Bohr in Copenhagen. During the year 1928-1929 he was assistant to Pauli at the E.T.H. in Zurich, then he went (from 1929 to 1930) to England as a lecturer (University of Cambridge, Imperial College). He finally settled in Holland: University of Groningen (1930-1931, conservator; 19311939, lecturer in mechanics and quantum mechanics); Technical University of Delft (1939-1969, professor of theoretical physics). Kronig worked on many problems of theoretical physics, especially on atomic and quantum physics, solid state and low-temperature physics and high energy physics. His name is connected with the so-called dispersion relations, which he and Kramers found for X-ray scattering and which later were extended to describe collisions between elementary particles.
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1924 to Naturwissenschaften (Goudsmit and Kronig, 1925).14 Kronig expected to continue this work on the sum-rules at Tubingen, supported by the well-known experimental and theoretical spectroscopists Back and Lande. But he got into an even more exciting situation, which he later recalled: 'When I arrived in Tubingen, Lande said to me, "Oh, you are coming at a good moment, tomorrow Pauli will be here"' (Kronig, loc. cit, p. 8). 1 5 Kronig did not yet know Pauli personally; but, being interested in the anomalous Zeeman effects, he was acquainted with Pauli's papers on this subject, and he looked forward to meeting him. Further, Lande showed Kronig Pauli's letter, which contained the new ideas about atomic structure (Pauli to Lande, 24 November 1924). He reported thirty-five years later: Pauli's letter made a great impression on me, and naturally my curiosity was aroused as to the meaning of the fact that each individual electron of the atom was to be described in terms of quantum numbers familiar from the spectra of the alkali atoms, in particular the two angular momenta I [i.e., the earlier k] and s [i.e., the earlier r] encountered there. Evidently s could now no longer be attributed to a core, and it occurred to me immediately that it might be considered as an intrinsic angular momentum of the electron. (Kronig, 1960, pp. 19-20) The idea that electrons might possess an intrinsic angular momentum had been enunciated before. Especially, Arthur Compton had assumed in a paper, presented on 27 December 1920 at the American Association for the Advancement of Science, that the electron should be considered as an extended object spinning rapidly around an axis through its centre (Compton, 1921d). Such an electron then exhibited not only an angular momentum, but also a magnetic moment, which accounted for the ferromagnetic properties of metals. However, from the magnitude of the intrinsic angular momentum of the electron — which Compton had assumed to be h/2ix — an important consequence followed. As Compton remarked: 'If an electron with such an angular momentum is to have a peripheral velocity which does not approach that of light, it is necessary that the radius of gyration of the electron shall be greater than 10 11 cm' (Compton, 1921d, p. 150). And he argued that such a large electron radius seemed to be in agreement with experiments on the scattering of X-rays and hard 7-rays by matter. Kronig did not think of Compton's, or anybody else's, intrinsic rotation of the electron. 16 He was just interested in interpreting the fourth quantum number, which Pauli had associated with the electron. 'In the language of the models which before 14 Kronig first visited the Cavendish Laboratory at Cambridge. There he met Paul Ehrenfest, who invited him to Leyden. In December 1924 Lande came to Leyden to give a lecture. As a result, Kronig visited him in Tubingen before going on to Copenhagen. 15 Kronig arrived in Tubingen on 7 January 1925, and talked to Lande the next day. Pauli came on the afternoon of 9 January. 16A year after Compton, Earle Hesse Kennard had proposed, apparently independently, the idea of an electron rotating around its axis: he assumed the electron to be a uniformly charged sphere and calculated the gyromagnetic factor of 2 for this object, in agreement with the observations of the Einstein-de Haas effect (Kennard, 1922).
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the advent of quantum mechanics were the only basis for discussion one had, this could only be pictured as due to rotation of the electron about its axis,' he later recalled about his procedure (Kronig, 1960, p. 20). He knew, of course, that this picture brought about several serious difficulties, but he found it 'a facinating idea' and proceeded the same afternoon, as he had read Pauli's letter to Lande, to derive consequences from it. It was clear to Kronig that an electron, which possessed an intrinsic angular momentum of \ (in units of /i/27r) and moved around a nucleus, could assume two states, one parallel to the direction of the internal field and the other antiparallel to it. He carried out the calculation of the energy states. In order to obtain the inner magnetic field he chose a coordinate system in which the electron was at rest and the field was then created by the moving nucleus. He found that the difference between the doublet states obeyed the Z4-law, as borne out by the data and predicted by the relativistic theory. Thus the assumption of an intrinsic angular momentum of the electron enabled Kronig to obtain the essentially relativistic formula without any relativistic consideration. He immediately informed Lande, who had previously tried in vain to derive the Z 4 -law from the magnetic coupling theory. Lande was rather happy about the result, and both he and Kronig were anxious to hear Pauli's opinion. Pauli came the next day, and as soon as an occasion arose, Kronig approached him with his idea. To his surprise Pauli replied: 'That is surely quite a clever idea, but nature is not like that.' ('Das istja ein ganz witziges Apergu, aber so ist die Natur schon meat,' reported by Kronig, AHQP Interview, 11 December 1962, p. 16.) Pauli expressed his negative view with such vehemence that Lande, when alone afterwards with Kronig, remarked: 'Yes, if Pauli says so, then it is [surely] not like that.' (lJa, wenn der Pauli das sagt, dann wird es schon nicht stimmen'; reported by Kronig, loc. cit., p. 16.) Kronig was indeed aware that his proposal implied certain difficulties. For example, a closer examination showed that the formula for doublet separation A^, Av-2nH
-
167r4
™ee8
Z e 4 ffect ,
(1)
(where /z is the absolute value of the intrinsic magnetic moment of the electron, which Kronig assumed to be equal to e/i/47rmec; Hi the strength of the average internal magnetic field: Hi = (e/mec)(kh/2i:)(Z /r3), where e and m e are the charge and mass of the electron, c the velocity of light in vacuo, hh/2ir the angular momentum of the orbiting series electron, and (Z/r3) the average value over an orbit of the effective charge divided by the cube of the distance from the electron to the nucleus) reporduced the data only up to a factor 2. That is, Eq. (1) predicted the doublet splitting to be twice as large as given by experiment. A second difficulty was that the anomalous gyromagnetic factors could not be explained on the basis of the new assumption. Nevertheless, Kronig was quite surprised about Pauli's strong reaction. Why did Pauli not believe in the possibility that an electron possessed an intrinsic angular momentum, in spite of the fact that it led to the correct dependence
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of the doublet separation on the effective charge of the nucleus? Surely, he could not completely reject the idea that intrinsic angular momenta of charged particles played a role in atomic theory. After all, he had himself used, in a note dated 17 August 1924 in Naturwissenschaften, the assumption of an intrinsic angular momentum for atomic nuclei in order to explain the hyperfme structure of spectral lines (Pauli, 1924c). 17 Now, if the nucleus could possess a finite angular momentum, why not the electron? The answer must be sought in the relativistic consideration, which Pauli had applied to the anomalous Zeeman effect a few weeks earlier (Pauli, 1925a). There Pauli had shown that the magnetic moment of a charged massparticle, rotating with a velocity comparable to the velocity of light in vacuo, was not a constant of motion, but depended strongly on the relativistic increase of the mass. Hence an electron rotating around its own axis with relativistic speed could possess neither a finite magnetic moment nor a fixed angular momentum |(/i/27r), as demanded by Kronig. 18 This argument against an intrinsic angular momentum did not apply in the case of the nucleus: since its mass was several thousand times larger than the electron mass, the rotational velocities derived from an angular momentum of the order of h/2n were far below the relativistic range; hence the nuclear magnetic moment could assume a fixed, quantized value, in agreement with Pauli's interpretation of the hyperfme structure (Pauli, 1924c). Thus his previous analysis of the relativistic electron and the Zeeman effect had made Pauli allergic to the assumption of a fixed finite magnetic moment of the electron. Such a property, apart from the fact that it had not been observed by experiment, also contradicted his fundamental conclusion that the phenomena of complex structure and of anomalous Zeeman effects were rooted in the 'classically non-describable duplicity of the quantum-theoretical properties' of the electron. No, the simple assumption that the electron performed an intrinsic rotation with a given angular momentum and therefore also possessed a finite magnetic moment, which — when coupled to the inner field — caused the doublet structure of X-ray and alkali lines, would not be able to solve the deeply rooted problems of atomic theory. In Pauli's opinion, the solution did not have anything to do with a more detailed mechanical model of the electron or the atom; rather, the solution had to be found by completely abandoning all mechanical models in favour of genuine quantum-theoretical concepts, of which the 'non-describable duplicity' — or the 'non-mechanical stress,' as Bohr and his associates called it (see Heisenberg, 1925b, 17
T h e hyperfine structure had been noticed by Hantaro Nagaoka and his associates: they observed in particular certain 'satellites' of mercury lines. (See, e.g., Nagaoka, Suguira and Mishima, 1924.) 18 T h e above argument was presented in a late letter by Ralph Kronig to Nature (Kronig, 1926a). Kronig stated there that the magnetic moment of the electron, following from the intrinsic angular momentum hypothesis, implied that the internal velocity (i.e., the velocity of rotation) of the electron was 'close to that of light' (Kronig, 1926a, p. 550). In fact, one could show that this velocity even exceeded the velocity of light c. However, the argument — and we can be sure that Kronig in early 1926 presented Pauli's old argument of January 1925 — that faster-than-light velocity entered into the picture was not decisive, for even relativistic motions of the electron with velocities below c would be incompatible with the assumption of a constant magnetic moment for the electron.
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p. 841) — was an essential part. The new, real quantum theory of the future should then yield the answer to the entire complex of difficulties connected with the multiplet structure and the anomalous Zeeman effects of many-electron atoms. It would provide the deeper reason for the anomalous gyromagnetic factors, as well as for their observed independence of the relativistic mass increase. Pauli's decisive 'No' perturbed Kronig considerably, and when he received no positive response towards his intrinsic angular momentum hypothesis also in Copenhagen, for example from Hendrik Kramers and Werner Heisenberg, he gave up thinking seriously about it and did not publish even a hint of his idea. 19 His idea, however, recurred at another place, with more than half a year's delay: on 17 October 1925 George E. Uhlenbeck and Samuel Goudsmit in Leyden signed a note, entitled 'Ersetzung der Hypothese vom unmechanischen Zwang durch eine Forderung beziiglich des inneren Verhaltens jedes einzelnen Elektrons' ('Replacement of the Hypothesis of the Non-Mechanical Stress by a Requirement Concerning the Intrinsic Behaviour of Each Individual Electron'); it was sent to Naturwissenschaften, where it appeared in the issue of 20 November 1925 (Uhlenbeck and Goudsmit, 1925). They proposed to associate Pauli's fourth quantum number with 'an intrinsic rotation of the electron' ('erne innere Rotation des Elektrons,' Uhlenbeck and Goudsmit, 1925, p. 954). Paul Ehrenfest's Institute for Theoretical Physics in Leyden, whose members Uhlenbeck and Goudsmit were at the time, was certainly an appropriate place for the development of the idea of 'a proper or intrinsic rotation of the electron.' First, Ehrenfest had himself taken a deep interest in Bohr's theory of atomic structure at least since 1916. This interest extended especially to the principles of the theory, to which Ehrenfest had contributed through his adiabatic hypothesis. But at the Leyden Institute, the physicists were also fully informed — by Bohr personally and by Hendrik Kramers, a former member and frequent visitor to the Institute — about the details of the applications of the principles to describe the spectroscopic data. Second, in spite of the close connection with Copenhagen and other centres, where atomic theory was developed, Ehrenfest retained a certain independence from the mainstream of work on atomic theory. He did not follow slavishly all moves of the experts, partly because atomic spectroscopy was not his central field of interest and partly because he knew — through critical discussions with his friend Albert Einstein — about the great difficulties inherent in the existing atomic theory, to which nobody, as yet, had any clue. Thus Ehrenfest's attitude towards atomic theory in the 1920s was that of a critical observer, who occasionally participated himself in the analysis of specific problems: for example, he wrote with Einstein on the Stern-Gerlach effect (Einstein and Ehrenfest, 1922) or on the equilibrium between 19 Kronig recalled later on: 'I then went to Copenhagen, where I reported my considerations among others to Heisenberg and Kramers, but I did not find resonance with them' (Kronig to van der Waerden, 22 June 1959). In Copenhagen Bohr exerted the dominant influence, and he rather believed in 'non-mechanical stress' when dealing with multiplet spectra. Later on, when the concept of electron spin was established, the little ditty went round among the young physicists: 'Der Kronig hdtt' den Spin fast endeckt/hatt' Pauli ihn nicht abgeschreckt.' (Kronig had almost discovered spin/If Pauli had not frightened him.)
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radiation and matter (Einstein and Ehrenfest, 1923). To Ehrenfest's relative independence of the current fashion in atomic theory was added another independence: in Holland there existed several centres of experimental spectroscopy, such as Pieter Zeeman's Institute in Amsterdam, Leonard Ornstein's Institute in Utrecht, or even the Research Laboratory of Philips at Eindhoven (where scientists like Gustav Hertz made important observations). Since all these places were not far from Leyden, a theoretician could obtain detailed information about data by paying a few hours visit to an expert experimentalist. While Ehrenfest remained a bit outside of what one would call work on the burning questions of atomic theory — though in August 1925 he submitted a paper dealing with the energy fluctuations in a radiation field, in which he tried to illuminate certain paradoxes of Bose's statistical hypothesis (Ehrenfest, 1925a) — he did encourage his student Samuel Abraham Goudsmit in his theoretical analysis of spectroscopic data. Goudsmit had begun his studies at the University of Leyden in 1919 and soon joined Ehrenfest's Institute. 20 He had immediately become interested in spectroscopy and started to write papers in that field.21 Ehrenfest, who recognized Goudsmit's special talent in working closely with experiments, sent him away from Leyden to Amsterdam, to work as assistant to Pieter Zeeman for three days a week. Goudsmit concentrated so much on spectroscopy that Ehrenfest once — in the presence of the visiting Albert Einstein — told him: 'The trouble with you is I dont's know what to ask you, all you know is spectral lines. Can I ask you Maxwell's equations and things about that?' (Goudsmit, AHQP Interview, 5 December 1963, p. 12). The 'trouble' concerned an examination, which Goudsmit tried to postpone as long as possible. While not being the most dutiful student, he eagerly published contributions to the theory of atomic spectra, the first of which was contained in a short note on 'Relativistische Auffassung des Dubletts' ('Relativistic Interpretation of the Doublet'), which appeared in the issue of Naturwissenschaften of 9 December 1921 (Goudsmit, 1921). The problem had arisen from reading Sommerfeld's Atombau und Spektrallinien, and Goudsmit recalled later: 'On the basis of Sommerfeld's book I discovered, or thought I discovered, the Z 4 -law for the [optical] doublets and triplets. With that I went to Ehrenfest Ehrenfest at least couldn't see it wasn't 20 S.A. Goudsmit was born in The Hague, the Netherlands, on 11 July 1902, into a middle-class family of Jewish merchants. After passing high school, where Tennis van Lohuizen was his teacher in physics, he studied physics from 1919 at the Universities of Leyden and Amsterdam, obtaining his doctorate with Ehrenfest (at Leyden) in 1927. In 1927 he went t o the University of Michigan as instructor; in 1928 he was promoted to associate professor, and in 1932 to full professor of physics. He remained there until 1946. During World War II he worked at the M.I.T. Radiation Laboratory and served as Chief Scientific Officer of the Alsos Project. Prom 1946 to 1948 he joined Northwestern University, then (in 1948) moved to the Brookhaven National Laboratory, being chairman of the Brookhaven Physics Department from 1952 t o 1960. Besides, he acted as the managing editor of the Physical Review and initiated the publication of the Physical Review Letters. Goudsmit died on 4 December 1978 at Reno, Nevada. He received many honours for his scientific work and public service, including the Max Planck Medal of the German Physical Society (1965) and the National Medal of Science (1977). 21 Goudsmit had been introduced to spectroscopy to some extent already by Lohuizen. Later he studied Alfred Fowler's Report on Series in Line Spectra (Fowler, 1922).
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new. So he let me publish it in a short note to Naturwissenschaften' (Goudsmit, AHQP Interview, loc. cit, p. 14). 22 Within a few years Goudsmit became an experienced spectroscopist, who was familiar even with the experimental techniques from an extended visit to Priedrich Paschen's Institute in Tubingen. 23 Thus he continued to publish on spectroscopy: in 1924 he submitted, for example, two notes to Naturwissenschaften, one on the Zeeman effect of the scandium spectrum (Goudsmit, 1924a), and the other on the organization of terms in the lanthanum spectrum (Goudsmit, 1924b). He loved to play with spectroscopic data and knew how to squeeze out detailed information from numbers and photographs (of spectra). Towards the end of 1924 he turned to a discussion of the intensity of spectral lines: with Ralph Kronig he tried to find formulae to fit the empirical intensity rules of Ornstein and collaborators (Goudsmit and Kronig, 1925), and with Dirk Coster of Haarlem he worked on the intensities of X-ray spectral lines (Coster and Goudsmit, 1925). Further he contributed to the investigation of the most interesting current problems: in a paper submitted in March 1925 he studied the ground state of neon, obtaining agreement with the branching rule of Lande and Heisenberg (Goudsmit, 1925a); in another paper, submitted two months later, he analyzed the complex structure of the same atom with the help of a scheme similar to Pauli's in which four quantum numbers characterized the electron states and which had the advantage over Pauli's in that it allowed one to determine the g-values of all singlet and triplet terms (Goudsmit, 1925b). During the second half of 1925 Goudsmit wrote a couple of papers together with a colleague, whose name had not yet made its appearance in the scientific literature: George Eugene Uhlenbeck. Although Uhlenbeck was one and a half years older than Goudsmit and had also begun his studies earlier than him, in 1925 he was somewhat behind in his scientific career. 24 He had been in Leyden since 1918, but during his In his note Goudsmit applied Sommerfeld's relativistic doublet formula to optical doublets (the 2p-states) and triplets and claimed that it fitted the data. His interpretation disagreed with the Bohr-Sommerfeld theory, for he noticed: 'According to the view sketched here the components of a given p-term belong to orbits with different azimuthal quantum numbers, whereas in the case of Sommerfeld-Bohr the different values of the azimuthal quantum number are already used up in the interpretation of the different series' (Goudsmit, 1921, p. 995). It should be mentioned that Goudsmit had visited Tubingen in summer 1921, where he met Friedrich Paschen, who had shown him the fine structure of the helium 4868-A line. At t h a t time a prize had been announced by Teyler's Foundation for solving the following problem: t o find an extension of a rule of Sommerfeld such that the alternate elements in t h e periodic table had either doublet or triplet spectra. Goudsmit had planned to take part in this competition. After he had obtained his result, he showed it to his former teacher Lohuizen, and Lohuizen took him to meet Ehrenfest. Goudsmit also performed spectroscopic experiments later on at Zeeman's institute in Amsterdam. 24 G . E . Uhlenbeck was born on 6 December 1900 at Batavia, Java, t h e son of a military officer in the Dutch East Indies. At the age of six he went to The Hague, where he received his school education. In 1917 he began to study chemical engineering at the Technical University of Delft, but the following year he changed to the University of Leyden to study physics and mathematics. He spent the period from fall 1922 to June 1925 in Rome as tutor to the youngest son of the Dutch ambassador (J.H. van Royen); he taught all subjects required in the Dutch Gymnasium except
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early years there he had only little contact with Paul Ehrenfest who did not teach introductory courses. In 1922 Ehrenfest had been instrumental in getting him the position of a tutor for the son of the Dutch ambassador in Rome. 25 During these somewhat irregular university studies, Uhlenbeck had learned a lot from books: he had read Hendrik Lorentz' Beginselen der Natuurkunde — a two-volume set containing introductory lectures on physics, which had first appeared in 1888 (Lorentz, 1888), Clemens Schaefer's Einfuhrung in die Maxwellsche Theorie der Elektrizitdt und des Magnetismus — an elementary textbook on electrodynamics (Schaefer, 1908), Ludwig Boltzmann's Vorlesungen iiber Gastheorie, Gustav Kirchhoff's lectures on theoretical physics, and last but not least Isaac Newton's Principia. Thus, by the time he returned to Leyden in summer 1925 and took an assistantship under Ehrenfest, he had become a self-taught but learned theoretician. 26 Ehrenfest now tried to provide the requisite balance to Uhlenbeck's education. Ehrenfest did not like very learned things. 'If you couldn't say it simply, if you couldn't be to the point, then he did not want to hear it,' Uhlenbeck recalled his discussions with Ehrenfest. 'And if anything was, so to speak, long-winded and learned, he made immediate fun of it' (Uhlenbeck, AHQP Interview, 9 December 1963, p. 10). Uhlenbeck, who had often to read new papers and to report their contents to Ehrenfest, soon adapted himself to the professor's method. Ehrenfest then decided that his new assistant should become acquainted with the recent advances in physics, especially in atomic theory. For this purpose he made a proposal. As Uhlenbeck recalled: Ehrenfest gave Sam [Goudsmit] the task of teaching me. That was one of Ehrenfest's pedagogic principles. He always wanted to have people work together in pairs.... So that summer from June on, Sam came. I think we got together twice a week [in The Hague]. [On] the other [days] I went to Leyden to start working with Ehrenfest on something quite different — partial differential equations, properties of the wave equation, which I also was very much interested in. Sam just lectured to me. We [sat] together in a room, and he classical languages and history, for which there was a second tutor (Private Communication). Then he returned to Leyden and obtained his doctorate in 1927 with Ehrenfest. After spending several months as Lorentz Fellow at the Universities of Copenhagen and Gottingen, he received — still in 1927 — an instructorship at the University of Michigan. In 1928 he was promoted to assistant professor and two years later to associate professor. In 1935 he returned to Holland as professor of theoretical physics at the University of Utrecht. At the outbreak of World War II he returned to the United States, becoming a professor at the University of Michigan (1939-1960). From 1943 to 1945 he did research on radar at the M.I.T. Radiation Laboratory. He served as president of the American Physical Society (1959-1960) and spent that year at the Institute for Advanced Study, Princeton. In January 1961 he joined the Rockefeller University. Uhlenbeck worked on atomic structure, quantum mechanics, nuclear theory and, especially, statistical mechanics. 25 Previously, Uhlenbeck had taught at the Leyden Gymnasium, for he needed money to support himself. However, he did not like his teaching job particularly, and had no intention of becoming a high school teacher. •^Of course, Uhlenbeck had attended Ehrenfest's advanced lectures on Maxwell's theory and on statistical mechanics (including some quantum theory). He had also attended the famous Monday morning lectures of Hendrik Lorentz, where the old master talked about the recent developments of physics in his own way. In addition, he had attended the Wednesday evening colloquia organized by Ehrenfest.
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told [me] about all these successive parts, and he did that very beautifully. (Uhlenbeck, AHQP Interview, 31 March 1962, p. 18)27 Uhlenbeck was acquainted with Samuel Goudsmit from earlier times in Leyden as a student — as one who was just the opposite of himself. He knew that Goudsmit had published several papers on spectroscopic topics and was acquainted with experts in his field from abroad, such as Heisenberg and Hund. In contrast, Uhlenbeck had no idea what the g-factor of a spectroscopic term was or who Alfred Lande was. But from Goudsmit he learned such things quickly. 'I remember that very soon came about all these questions of the old duality, nichtmechanischer Zwang,' Uhlenbeck recalled almost four decades later. 'There were papers by several people. We studied them, especially Pauli's paper on the exclusion principle Sam had studied those, and we always came back to it' (Uhlenbeck, AHQP Interview, 31 March 1962, p. 18). 28 Goudsmit's teaching of Uhlenbeck turned out to be immediately fruitful. Goudsmit found himself forced to organize the material diligently and this led him to reformulate the problems of atomic structure in a paper, which appeared in the October issue of Physica (Goudsmit, 1925c).29 And he discovered from Uhlenbeck's questions that more difficulties existed in atomic theory than he had ever thought of earlier. In fact, previously he had been rather negligent of the difficulties in the description of spectroscopic data in terms of quantum numbers and all that. He declared: 1 never had the feeling of difficulties, never. I always had the feeling that these were empirical rules which some people could interpret, if not now, then later on. My main interest was really the mechanics of the empirical rules. By empirical rules, I mean also quantum numbers. Even in the year when Pauli's principle came up, to me that was no more than an additional empirical rule like selection rules, like assigning quantum numbers. It was just an additional rule which explained a lot more and put order in this mess. (Goudsmit, AHQP Interview, 5 December 1963, p. 20) Uhlenbeck, who had a very good knowledge of classical theory, knew that there were fundamental difficulties in quantum theory; he, therefore, questioned thoroughly the naive standpoint of Goudsmit, his teacher in atomic theory. And Goudsmit admitted: 'Uhlenbeck's fresh and unprejudiced occupation with atomic problems and his many skeptical remarks and intelligent questions led us to a number of important results' [Goudsmit, 1965, p. 450). The initial results of the (Goudsmit-Uhlenbeck collaboration were published in a paper on the spectra of hydrogen and helium, which came out in the 2
Uhlenbeck published a paper on a mathematical problem in the issue of Physica commemorating the fiftieth anniversary of Hendrik Lorentz' doctoral thesis (Uhlenbeck, 1925). 2 ° Goudsmit and Uhlenbeck studied not only the problems of spectroscopy, like multiplet structure and anomalous Zeeman effects, but also the phenomena of resonance fluorescence and the polarization of resonance radiation, which were discussed in the current literature. 29 I n the paper on problems of atomic structure, Goudsmit discussed especially the application of Pauli's exclusion principle.
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August-September 1925 issue of Physica (Goudsmit and Uhlenbeck, 1925). They started from an investigation of Gregor Wentzel, who had analyzed all doublet spectra from the point of view of the relativity approach, adding at the same time a kind of averaging procedure for the angular momentum of the (nonrelativistic) electron orbit (Wentzel, 1925b).30 They now extended Wentzel's scheme slightly and applied it to obtain the fine structure of the hydrogen lines. The terms thus determined exhibited a different structure than the one assumed on Sommerfeld's theory: e.g., the n = 2 term split into three instead of two levels, of which two possessed the inner quantum number j = 1 and the angular momentum quantum numbers k = 1^ and | , respectively, and the other one had j = 2 and k = 1^; and similar multiplets arose for the principal quantum number terms n = 3, 4, 5, etc. Goudsmit recalled that the interest in Wentzel's scheme was due to Uhlenbeck, adding: 'What suited me was the fact that in our scheme the hydrogen spectrum became a special case of the alkali spectra and the X-ray spectra. I was especially pleased by the fact that we finally managed to explain a mysterious component of the 4686 A line of [ionized] helium, which had been observed by Paschen, but which was forbidden according to Sommerfeld's theory' (Goudsmit, 1965, p. 450). They further attempted to organize the neutral helium spectrum according to the same principles. 31 Encouraged by such success, Goudsmit and Uhlenbeck continued their common discussions and analysis of the problems of atomic theory throughout the summer of 1925. Especially, they concentrated more and more on the mysterious four quantum numbers, which Pauli had introduced earlier that year. 'I only remember the discussion with George [Uhlenbeck], where I tried to explain the Pauli principle to him, that there are four quantum numbers, and one quantum number is always a half,' Goudsmit said later (Goudsmit, AHQP Interview, 6 December 1963, p. 16), and Uhlenbeck completed the report by recalling: 'It came one afternoon.... It was always this question of the four quantum numbers, and I still remember that I said, 'If there are four quantum numbers, there must be four degrees of freedom.... And if there are four degrees of freedom, then there must be some kind of internal motion" (Uhlenbeck in Goudsmit, AHQP Interview, 7 December 1963, pp. 4-5). What happened next? On an afternoon at the end of September 1925, Uhlenbeck proposed to connect Pauli's fourth quantum number of the electron with a dynamical degree of freedom of the electron. Although Goudsmit had been concerned with the fourth quantum number earlier — he had even developed a modification The simultaneous treatment of optical and X-ray doublets was Wentzel's favourite subject in 1925, and he wrote several papers on it (Wentzel, 1925a,b,d). Goudsmit and Uhlenbeck assumed that the helium spectrum consisted of singlet and doublets. We should mention that a proposal similar to Goudsmit and Uhlenbeck's on the hydrogen and helium spectra was made by John Slater in a paper, submitted on 16 November 1925 to the Proceedings of the National Academy of Sciences (Slater, 1925d). Further, Arnold Sommerfeld and Albrecht Unsold wrote a paper on the spectrum of hydrogen, which was received by Zeitschrift fur Physik on 6 February 1926; their scheme also coincided formally with the earlier one of Goudsmit and Uhlenbeck (Sommerfeld and Unsold, 1926).
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of Pauli's scheme, in which the fourth quantum number always assumed the values ± ^ for arbitrary terms (Goudsmit, 1925b, pp. 795-796) — he had never tried to give an interpretation to it as a fourth degree of freedom.32 For Uhlenbeck, who knew classical statistical mechanics very well, the situation was clear, and he argued: 'Since (as I had learned) each quantum number corresponded to a degree of freedom of the electron, the fourth quantum number must mean that the electron had an additional degree of freedom — in other words, the electron must be rotating' (Uhlenbeck, 1976, p. 46). Goudsmit found Uhlenbeck's proposal rather nice because with the intrinsic angular momentum of the electron of magnitude J|/i/27r everything fell into place. As Goudsmit recalled: After his remark about the spin [i.e., the intrinsic rotation of the electron], we recognized immediately that it was now completely clear why ms [i.e., the quantum number replacing Pauli's mi in Goudsmit's scheme] was always | or — | . Further we saw that all Zeeman splittings could be explained if one assigned to the electrons a magnetic moment of an entire Bohr magneton [i.e., of e/i/47rmec]. Besides it became clear that the spin agreed completely with our new interpretation of the hydrogen [spectrum], (Goudsmit, 1965, p. 450)33 It was not so clear, however, whether the new hypothesis did not lead into serious difficulties, and why — though it was an obvious consequence of Pauli's scheme — it had not occurred in literature before. Goudsmit and Uhlenbeck's were, therefore, a little suspicious, and they tried to investigate further the consequences that followed from the assumption of an intrinsic angular momentum of the electron. Thus, for example, Goudsmit asked Uhlenbeck in a postcard — sent from Amsterdam, where Goudsmit still worked part-time in Zeeman's Institute — whether the gyromagnetic ratio of a charged particle did not depend on its linear extension. In any case, the two of them did not plan at first to publish anything about their hypothesis, for they felt it was too speculative. At that moment Ehrenfest, who had been informed about the idea, interfered; he encouraged and helped his assistants. When, for instance, Uhlenbeck showed him Goudsmit's above-mentioned postcard, he drew his attention to an old paper of Max Abraham on the magnetic properties of the extended electron (Abraham, 1902b). Uhlenbeck studied this paper very carefully, then applied Abraham's theory to his problem and was satisfied to find that if the electron possessed only surface charge, the gyromagnetic ratio indeed turned out to be e/ro e c, which corresponded to a g-factor of 2; it was exactly this result which the spectroscopic data demanded. With that result in hand, Uhlenbeck recalled, 'Ehrenfest told us to write a short modest letter to Naturwissenschaften and to give it to him' (Uhlenbeck, 1976, p. 47). Since the ^-factor calculation rested on 32
I n his talk on the discovery of spin at the German Physical Society, Goudsmit explained: At this point let me mention again the difference between Uhlenbeck and myself as physicists. It can best be illustrated by t h e following, overly simplified example. When I told him [Uhlenbeck] about Lande's p-factors, he asked, to my surprise, 'Who is Lande?;' and when he mentioned four degrees of freedom of the electron, I asked him: 'What is a degree of freedom?' (Goudsmit, 1965, p. 450). 33 T h a t is, when the electron was assigned an anomalous gyromagnetic factor g = 2, the good results from the atomic core-series electron model could be taken over.
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an application of Abraham's nonrelativistic electron model, Ehrenfest added the remark: 'And then we shall ask Mr. Lorentz' ('Und dann werden wir Herrn Lorentz fragen,' Uhlenbeck, 1976, p. 47). Uhlenbeck and Goudsmit composed the note requested by Ehrenfest. They first described the well-known atomic core-series electron model or Lande's vector model of 1923, with all its complications, extra rules and difficulties: i.e., Pauli's demonstration that the (noble gas) core could not have nonzero magnetic moment; the disagreement with the 'construction' or 'building-up principle' (' Aufbauprinzip1); the failure to provide the correct ground states in the case of several many-electron atoms. Then Uhlenbeck and Goudsmit referred to the removal of these difficulties if one assigned, with Pauli, four quantum numbers to the electron; but they also drew attention to the fact that Pauli's scheme did not explain the relativistic X-ray doublets and the alkali spectra, unless one assumed a rather hypothetical 'nonmechanical duplicity.' The latter assumption might be avoided, however, if one re-interpreted the four quantum numbers of the electron slightly: while n, k and j (or the n, K and J in Lande's vector model) retained their previous meanings in Lande's vector model (as principal, azimuthal and inner quantum numbers), Uhlenbeck proposed to connect R (or 2mi of Pauli) with an 'intrinsic rotation of the electron.' This re-interpretation, they argued, would be perfectly consistent with the spectroscopic data, provided one could prove that the gyromagnetic factor of the electron was indeed 2, as suggested by the calculation from Abraham's model, and provided one could obtain the correct formula for the 'relativistic' doublets. Ehrenfest, just as he had promised Uhlenbeck and Goudsmit, wrote immediately to Lorentz, informing him about the problem of his students (Ehrenfest to Lorentz, 16 October 1925). Hendrik Lorentz at this time was retired, but he still gave lectures every Monday morning at Leyden. Following Lorentz' next Monday morning lecture — it was 19 October — Uhlenbeck approached him personally on the matter. 'Lorentz was very kind and interested, although I got the idea that he was rather skeptical,' Uhlenbeck recalled fifty years later. 'He said that he would think about it and that we should talk again the next Monday' (Uhlenbeck, 1976, p. 47). The following Monday, Lorentz brought a stack of papers filled with detailed calculations, from which Uhlenbeck just understood that there were serious problems. For example, the surface velocity of the rotating electron came out to be ten times the velocity of light in vacuo. Thus Uhlenbeck got the impression that 'if one extended the Abraham calculation properly as Lorentz had apparently done, then our picture of quantized rotation of the electron could not possibly be reconciled with classical electrodynamics' (Uhlenbeck, 1976, p. 47). Uhlenbeck and Goudsmit immediately went to Ehrenfest and told him that the idea contained in the note for Naturwissenschaften was nonsense, and that it would be better not to publish it. But to his surprise Ehrenfest answered that he had sent off the note quite a while ago, and that it would appear in print pretty soon. Ehrenfest comforted the perplexed authors by remarking: 'You are both young enough to be able to afford a stupidity like that' (lSie sind beide jung genug um sich eine Dummheit leisten
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zu konnen,' Ehrenfest, quoted by Uhlenbeck, 1976, p. 47). The short note of Uhlenbeck and Goudsmit was published a few weeks later (Uhlenbeck and Goudsmit, 1925). 34 The note of Uhlenbeck and Goudsmit initiated a quick response: on 21 November 1925, just the day after its publication, Werner Heisenberg from Gottingen wrote to Goudsmit — whom he knew quite well — stating his essential agreement with the idea of the rotating electron but asking how he had got rid of the factor 2 in the doublet formula.35 This difficulty had not been noticed by Uhlenbeck and Goudsmit because they had not tried to calculate the doublet formula. 36 In fact they did not even know how to proceed with the calculation. Fortunately, a little afterwards Albert Einstein came to Leyden and provided the necessary hint. He suggested that the calculation should be made in the coordinate system in which the electron was at rest. By performing the calculation Uhlenbeck found that there was indeed a difficulty about the factor 2 in the doublet formula. The negative result was, however, soon balanced by the response which the Uhlenbeck-Goudsmit hypothesis received from Einstein and Niels Bohr, who attended the celebration of the fiftieth anniversary of Hendrik Lorentz' doctorate in Leyden in December 1925. 37 Before arriving in Leyden, Bohr had passed through Hamburg and met Pauli, who had warned him against accepting the hypothesis of the rotating electron during his visit to Holland. But then Bohr was completely won over, as he wrote several months later to Ralph Kronig: When I came to Leyden to the Lorentz festivals, Einstein asked the very first moment I saw him what I believed about the spinning electron. Upon my question about the cause of the necessity of the mutual coupling between spin axis and the orbital motion, he explained that this coupling was an immediate consequence of the theory of relativity. This remark acted as a complete revelation to me, and I have never since faltered in my conviction that we at last were at the end of our sorrows. (Bohr to Kronig, 26 March 1926) Thus Bohr, who had shown only little interest in the magnetic electron before, became 'completely like a prophet for the electron-magnet gospel' (Bohr to Ehrenfest, 22 December 1925). 38 34 Ehrenfest had asked Goudsmit to invert the alphabetical order of names, i.e., put Uhlenbeck's name first. As Goudsmit recalled: 'Since I had already published several papers on spectra, [Ehrenfest] feared that the readers might remember my name and forget Uhlenbeck's, and, after all it was Uhlenbeck who had thought of the spin' (Goudsmit, 1965, p. 451). 35 Heisenberg then worked in the following months — as did Pauli — on t h e problem of the factor 2 in the quantum-mechanical formalism. We shall discuss this point in detail in dealing with matrix mechanics. 3 In the note in Naturwissenschaften, Uhlenbeck and Goudsmit had mentioned only the difficulty with the faster-than-light velocity of the rotating electron according to Abraham's theory (Uhlenbeck and Goudsmit, 1925, p. 954, footnote 2). 37 Lorentz had received his 'Doctor in de Wis-en Natuurkunde' from the Technical University of Leyden on 11 December 1875 and the fiftieth anniversary celebrations were planned to be held on 11 December 1925. Kronig recalled later that he gave a seminar talk at the Physical Society of Copenhagen on Uhlenbeck and Goudsmit's note in Naturwissenschaften. Although it was attended by most physicists then in Copenhagen, he did not receive 'any response at all' (Kronig, 1960, p. 26).
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Wolfgang Pauli, who had introduced the fourth quantum number of the electron, did not agree — neither in spring nor in fall 1925 — with its interpretation as a rotational degree of freedom. Of course, his outlook on atomic theory had changed since spring: thus he now believed that Heisenberg had provided the key to the correct atomic mechanics, and he had himself just completed the nonrelativistic calculation of the energy states of hydrogen with striking success (Pauli, 1926a). But with respect to the problem of equivalent electrons in many-electron atoms the new mechanics had not yet provided a breakthrough. Therefore Pauli continued to search for the solution of this problem by a purely quantum-theoretical description of the electron. In view of such a goal, any classical model — like a spinning electron — had to be, what he called, 'false doctrine' {'Irrlehre'). When Bohr met him in Berlin later in December 1925 — on his way back to Copenhagen from Leyden, Bohr passed through Gottingen and then Berlin, to participate in the celebration of the twenty-fifth anniversary of quantum theory at the German Physical Society on 18 December — he was not able to convince him of the 'magnetic electron gospel.' The application of quantum mechanics to atomic models including the electron spin led, as Pauli and Heisenberg showed during the following weeks, to a 'big catastrophe' in the case of hydrogen fine structure ('grofle Katastrophe,'1 Pauli to Bohr, 5 February 1926). 39 While Pauli — and, to a lesser extent, Heisenberg — struggled with the difficulties arising from the hypothesis of the magnetic electron, Goudsmit and Uhlenbeck made rapid progress by using it to organize the complex spectra. In a paper, which was received by Zeitschrift fur Physik on 27 November 1925, they developed a vector model of the atom, in which they attributed to each orbiting electron, besides the angular momentum vector K, another vector R of magnitude j(/i/27r) (Goudsmit and Uhlenbeck, 1926). Although they did not introduce explicitly 'the quantized rotating electron' in the main text of the paper, they mentioned in an addendum that the vector R should be interpreted as the intrinsic angular momentum of the electron (Goudsmit and Uhlenbeck, 1926, p. 625). 40 In any case, the new vector scheme enabled them to discuss all possible couplings between the 'quantum vectors' that might occur in atoms, provided one could describe the atoms as being constituted of an ion with angular momentum vectors K i (for the orbital motion) and Rx (for the intrinsic motion) and an electron with K2 and R2. For example, the two vectors associated with the ion, on one hand, and the series electron, on the other, would add up to two resultants, J i = K i + R i and J2 = K2 + R2, respectively, and the coupling of J i and J2 would then yield the term energy (i.e., We shall discuss the role of the Uhlenbeck-Goudsmit hypothesis in quantum mechanics at another place. Here we just mention the results of Pauli's investigations. 40 I n the same addendum, Uhlenbeck and Goudsmit referred to the paper of Arthur Holly Compton on the hypothesis of the rotating electron (Compton, 1921d), apologizing for not having quoted it in their note in Naturwissenschaften. We might mention, in passing, that Paul Ehrenfest, in an addendum to Uhlenbeck and Goudsmit's note in Naturwissenschaften, had noted that Wander Johannes de Hass had mentioned to him the idea of looking experimentally for an inner rotation of the electron some time ago.
The Discovery of Electron Spin
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the correction to the energy of the Kepler orbit); or, the momenta K i and K2 could add up to the resultant K = K i + K2, and similarly the momenta R i and R2 to R = R i + R 2 , such that the energy term was given by the coupling between K, the sum of the angular momenta, and R, the sum of the intrinsic momenta, of the constituents of the atom. 41 In all cases, Goudsmit and Uhlenbeck calculated the <7-factors. So they found in the case of ( J i , J2)-coupling the result
° = with
2(J2
- \)
9{Jl)
2(J=» - \)
JI - i + *2--K\
9(Ji) = 1 + "l and
+
9{J2)
'
(2)
0 ? ^ ' ^\ " ' 2(J?-\)
(2a)
i+ J- - j + i? - n
(2b>
^'= " i&r2
where J, Ji, J2, K\, Ki, Ri and R2 denoted the absolute values (in units of h/2n) of the corresponding vectors J , J i , J 2 , K i , K2, R i and R 2 . 4 2 Evidently, Eq. (2) possessed the same structure as Lande's ^-formula of 1923; that is, it represented an extension of the earlier 3-formula, which accounted well for the spectroscopic <7-values observed in the Zeeman effects of 'multiplets of higher order' or supermultiplets QMultiplett hoherer Stufe').43 Other couplings led to different ^-factor formulae; e.g., the (K, R)-coupling led to a consistent description of the g-values of the anomalous terms analyzed by Russell and Saunders earlier in the year (Russell and Saunders, 1925).44 It was these coupling schemes and their obvious empirical success, which helped to persuade Niels Bohr that one was at last at 'the end of the sorrows' in atomic theory. The other piece of evidence, which contributed to Bohr's conversion to the magnetic electron, consisted in the demonstration that the fine structure of the Balmer lines of hydrogen could be derived from the above assumptions. That was shown by Uhlenbeck and Goudsmit in a note, submitted in December as a letter to Nature, a letter, in the formulation of which Bohr took an active part and to which he also 41 It should be kept in mind that the resultants must be taken in the sense of Lande's vector formalism; that is, K i and R i combined to form the resultant J i , and the absolute values of J i , K i and Rx satisfied the relation \Ri - Ki\ + \ < J\ < \R\ + Ki\ - | . 42 Goudsmit also published the results of the (J, J)-coupling scheme in a note to Physica and — in German translation — in a letter to Naturvrissenschaften, dated 31 October 1925 (Goudsmit, 1925d). ""For the spectra of the first order (Stufe) the state of the ion was a singlet, which for the spectra of higher orders it was a multiplet. 44 S . Goudsmit, together with Ernst Back of Tubingen, later analyzed the Zeeman effects of several multiplet spectra, such as neon, silicon and lead, to determine the particular coupling schemes for the individual terms. They found, for example, that only in the case of silicon and the ground states of neon and argon, did pure (K, J)-coupling (i.e., Russell-Saunders coupling) exist; in the case of lead terms they found either (J, J)-coupling or a more complex coupling, in which J i was first coupled to K 2 , and then the resultant J 1 + K 2 was coupled to R2 (Goudsmit and Back, 1926).
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The Golden Age of Theoretical k 3
1
Old
Physics K
New ?
T
J 3
1
Fig. 1. Organization of the Lowest Energy Terms of the Hydrogen Atom.
appended a note (Uhlenbeck and Goudsmit, 1926). In it they not only presented the hypothesis of the 'spinning electron' for the first time to the English reader, but also claimed that for the one-electron atom (like the hydrogen atom or the helium ion) there arose the organization of terms postulated earlier (see Goudsmit and Uhlenbeck, 1925; Slater, 1925d). The new term scheme differed considerably from the previous one, given by Sommerfeld in 1915 (Sommerfeld, 1915c), as shown in Figure 1 for terms having the principal quantum number n = 3. The left-hand side of the figure represented Sommerfeld's old organization of the terms n*, = 3i, 32, 33; the right-hand side denoted the new organization described by the quantum numbers K (— \, §, §) and J (= 1,2,3). The dotted levels (Sx/2, P3/2,^5/2) in the middle between the old and the new scheme represented the situation without taking into account the magnetic moment of the electron. These levels were considerably shifted with respect to the levels in the Bohr-Sommerfeld theory of the hydrogen atom. However, with the inclusion of the magnetic electron, Uhlenbeck and Goudsmit's S-term agreed with Sommerfeld's k = 1 term; the lower P-term coincided with the 5-term, while the upper P-term coincided with the lower Dterm and both agreed with Sommerfeld's k = 2 term; finally, the upper D-term agreed with Sommerfeld's k — 3 term. Thus the new scheme, applied to doublet terms (except the ground state), reproduced the previous levels completely, but in a different manner, which helped to remove some of the difficulties which existed in Sommerfeld's relativistic theory. Thus Uhlenbeck and Goudsmit noted:
The Discovery of Electron Spin
607
In particular the new theory explains at once the occurrence of certain components in the fine structure of the hydrogen spectrum and the helium spark spectrum which according to the old scheme would correspond to transitions where K remains unchanged. Unless these transitions could be ascribed to the action of electric forces in the discharge which would perturb the electronic motion, their occurrence would be in disagreement with the correspondence principle, which only allows transitions in which the azimuthal quantum number changes by one unit. In the new scheme we see that, in the transitions in question, K will actually change by one unit and only J will remain unchanged. Their occurrence is, therefore, quite in conformity with the correspondence principle. (Uhlenbeck and Goudsmit, 1926, p. 264) Uhlenbeck and Goudsmit immediately pointed out in their letter to Nature that the new scheme also explained the X-ray doublets fully: the doublets formerly called 'screening' doublets were pairs of states having the same angular momentum J and different quantum number K, hence the two states corresponded to electron orbits, which penetrated to different distances from the nucleus; the so-called 'relativistic' doublets, however, were now characterized by the same value of the quantum number K, but in the two terms the orientation of the spin axis relative to the orbital plane was opposite, hence they 'may more properly be termed 'spin' doublets' (Uhlenbeck and Goudsmit, 1926, p. 265). Also the analogy between the X-ray and the optical doublets could be understood because the optical doublets were spin-doublets like the relativistic doublets in the X-ray spectra. Finally, Uhlenbeck and Goudsmit addressed themselves in a qualitative way to the anomalous Zeeman effects, claiming that the introduction of the magnetic electron did indeed fully account for the observed violation of Larmor's theorem. Some of the same questions, e.g., the explanation of the relativistic doublets of X-ray spectra and of the anomalous Zeeman effects on the basis of the hypothesis of the magnetic electron, were treated by F. Russell Bichowski and Harold Urey in a paper, which they communicated to the Proceedings of the National Academy of Sciences, where it appeared in the issue of February 1926 (Bichowski and Urey, 1926). 45 ' 46 Bichowski and Urey referred to Uhlenbeck and Goudsmit's note in Naturwissenschaften and applied the conception of the magnetic electron to 45 Francis Russell Bichowski was born on 19 February 1889 at San Gabriel, California. He became director of the Naval Research Laboratory, Anacostia, D.C., in 1927. He died in 1951. 46 Harold Clayton Urey was born on 29 April 1893 at Walkerton, Indiana. After working as a school teacher from 1911 to 1914, he studied chemistry at the University of Montana (1914-1917), receiving his B.Sc. in 1917 and served as instructor there (1919-1921). Then he continued his studies at the University of California at Berkeley (1921-1923), receiving his doctorate under A.R. Olsen and G.N. Lewis in 1923. He spent the academic year 1923-1924 with Bohr in Copenhagen on a fellowship of the American-Scandinavian Foundation. On his return to America he became an associate in chemistry at Johns Hopkins University (1924-1929). Then he went to Columbia University as an associate professor (1929-1934), becoming a full professor (1934-1945). In 1945 he moved to the University of Chicago, becoming Martin A. Ryerson Distinguished Professor of Chemistry at the Institute for Nuclear Studies (1952-1958), and finally, in 1958, he joined the University of California at San Diego. Harold Urey worked on absorption spectra and the structure of atoms and molecules, on the separation of isotopes, and on questions related to the origin of the solar system. For his discovery of the heavy hydrogen isotope, he received the 1934 Nobel Prize in Chemistry.
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calculate explicitly the double separation due to different relative orientations of the electron's magnetic moment and the orbit. Thus, by assuming the magnetic moment to be (e/2m e c) • (h/2n) — i.e., half as big as the value assumed by Uhlenbeck and Goudsmit — they obtained the same expression as on the relativistic theory. While this result fitted the alkali doublets and the X-ray doublets, it seemed to Bichowski and Urey to be incompatible with the fine structure of the hydrogen lines: because in that case, they claimed, one had to take a superposition of Sommerfeld's relativistic separation (the relativistic doublet) plus the separation of Uhlenbeck and Goudsmit, arising from the interaction of the magnetic moment of the electron with the internal field.47 Like Bichowski and Urey, also Uhlenbeck and Goudsmit had trouble describing the fine structure of the hydrogen lines. The reasons, however, were different. Bichowski and Urey had simply suppressed the factor 2 in the doublet formula by assuming a normal gyromagnetic factor, i.e., g = 1, for the magnetic electron. Of course, this assumption could not be correct in view of the anomalous Zeeman effects. Therefore, the treatment of Uhlenbeck and Goudsmit had to be preferred, although their hope that a 'closer study of quantum mechanics and perhaps also of questions concerning the structure of the electron' might contribute to the 'final solution' of the entire complex of problems (Uhlenbeck and Goudsmit, 1926, p. 265) received no justification during the following months. The solution came from an unexpected source: in a letter to Nature from Copenhagen, dated 20 February 1926, Llewellyn Hilleth Thomas dealt with 'The Motion of the Spinning Electron' (Thomas, 1926). Thomas, who came to Copenhagen in fall 1925 as Isaac Newton Student from Cambridge, participated in the discussions of Bohr and Kramers — which were going on in late 1925 and early 1926 — concerning the difficulty with the factor 2 in the spinning electron model. 48 After Bohr explained the coupling of the spin and the transformed electric field of the nucleus and the resulting discrepancy, Thomas naively suggested: 'Well, why didn't people work that out relativistically,' to which Kramers replied that, 'that will only make a small correction' (Thomas, AHQP Interview, 10 May 1962). But Thomas did not really believe in what Kramers said As Bichowski and Urey pointed out, the theory of the magnetic electron solved a difficulty that had existed before in Heisenberg's theory of the alkali doublets (Heisenberg, 1922a). According to this theory the energy order of the p\ and p2 terms came out the wrong way (see Breit, 1923b); in the new scheme, however, the sign of the pi — p2 term difference was opposite to that in the previous scheme. 48 L.H. Thomas was born on 21 October 1903 in London. He studied physics at the University of Cambridge from 1921 to 1925. He received his B.A. in 1924. During the winter of 1925-1926 he spent several months in Copenhagen as Isaac Newton Student. On his return to Cambridge, he completed his doctorate in 1927. From 1927 to 1929 he was an 1851 Exhibition Scholar at Cambridge. Then he went to the United States, where he joined Ohio State University: assistant professor (1929-1930), associate professor (1930-1936), and full professor of physics (1936-1943). In 1946 he became a member of the senior staff at Watson Scientific Laboratory in New York City; in 1950 he was appointed as a professor of physics at Columbia University. Thomas contributed to questions of the structure of atoms and molecules and to the use of computing machines in scientific problems.
The Discovery of Electron Spin
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and, being familiar with general relativity from Arthur Stanley Eddington's lectures at Cambridge, he thought about using it in the present problem. In particular, he remembered the calculation of the general relativistic effects on the moon's nodes, which was contained in Eddington's book on The Mathematical Theory of Relativity (Eddington, 1923b). Thomas consulted the book and adapted the method to his needs, and within three days he found the answer: the reasoning which had led to Eq. (1) was erroneous because in transforming the atomic system, consisting of a moving electron and a nucleus at rest, to a system, consisting of an electron at rest and a moving nucleus, one had neglected an important effect. This effect arose from the acceleration of the electron, and it reduced the inner magnetic field Hi entering in Eq. (1). In particular, Thomas found that the angular velocity w of the intrinsic rotation of the electron was not (—e/m e c 2 )(E x v) — (E x v) being the vector product of the electric field E (acting on the electron) and the velocity v of the electron — but rather
" = S ( E X V ) -2> X f >'
<3>
where f was the acceleration of the electron. Since f was given in the first approximation by the Lorentz force on the electron, i.e., by the expression (—e/m e )E, he arrived at the result W
= ^
E X V
) "
<3'>
Since w gave, apart from the factor 2n, the doublet separation, Ai/, in the Uhlenbeck-Goudsmit model of the magnetic electron, Thomas had indeed provided the factor | , which was necessary in order that Eq. (1) agree with the empirical data. The above calculation, which Thomas presented in greater detail in a paper published in the Philosophical Magazine (L.H. Thomas, 1927), immediately convinced Niels Bohr. It made the way clear for the satisfactory description of the fine structure of hydrogen lines and the anomalous Zeeman effects, which was achieved soon afterwards in a quantum-mechanical calculation by Werner Heisenberg and Pascual Jordan (Heisenberg and Jordan, 1926). And soon there followed other applications of the rotating electron hypothesis, which established the spin as a firm property of the electron. Only Pauli hesitated. For several weeks he doubted the correctness of Thomas' calculation; then, after accepting it, he tried to incorporate the electron spin into the deeper quantum-mechanical description of the electron. 49 Within this formalism Bohr would be able to show (Bohr, 1932) to Pauli's great satisfaction, 'that the electron spin cannot be measured by classically describable 4
When, ultimately, Pauli accepted the correctness of Thomas' calculation and thus the concept of electron spin, he wrote to Bohr: 'Now all t h a t remains for me is to capitulate completely' (Pauli to Bohr, 12 March 1926). To Goudsmit he wrote: 'I am writing to you today to tell you, first of all, that I have meanwhile — on the basis of recent reports from Copenhagen — finally come to the conviction t h a t I was wrong in my objections against Thomas and that his relativistic consideration can be cast in a perfectly correct and unobjectionable form. The problem of fine structure is thus really satisfactorily explained now' (Pauli to Goudsmit, 13 March 1926).
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The Golden Age of Theoretical Physics
experiments (as, for instance, deflection of molecular b e a m s [of electrons] in external electromagnetic fields) and must therefore be considered a n essentially q u a n t u m mechanical p r o p e r t y of the electron' (Pauli, 1948, p . 134). A p a r t from these fundamental questions, which remained to be clarified in later years on t h e basis of wave mechanics, t h e conception of electron spin h a d indeed ended in large p a r t t h e sorrows of atomic theory. References Abraham, M. (1902b), Nachrichten Ges. Wiss. Gottingen, pp. 20-41. Back, E. (1925a,b), Ann. d. Phys. (4) 76, 317-372; Z. Phys. 15, 206-243. Bichowski, F.R., and H.C. Urey (1926), Proc. Nat. Acad. Sci. (U.S.A.) 12, 80-85. Bohr, N. (1932), La Theorie Atomique et la description des phenomenes, Paris: GauthierVMars. Bohr, N. (1977), Collected Works 4: The Periodic System (1920-1923), Amsterdam-New York-Oxford: North-Holland. Breit, G. (1923b), Nature 112, 396. Compton, A.H. (1921d), J. Franklin Inst. 192, 145-155. Coster, D., and S. Goudsmit (1925), Naturwiss. 13, 11-12. Eddington, A.S. (1923b), The Mathematical Theory of Relativity, Cambridge (England): Cambridge University Press. Ehrenfest, P. (1925a), Z. Phys. 34, 262-373. Einstein, A., and P. Ehrenfest (1922), Z. Phys. 11, 31-34. Einstein, A., and P. Ehrenfest (1923), Z. Phys. 19, 301-306. Fowler, A. (1922), Report on Series in Line Spectra: The Physical Society of London. Geel, W.C. Van (1925), Z. Phys. 3 3 , 836-842. Gotze, R. (1921), Ann. d. Phys. (4) 66, 285-292. Goudsmit, S. (1921), Naturwiss. 9, 995. Goudsmit, S. (1924a,b), Naturwiss 12, 743-744; 851-852. Goudsmit, S. (1925a,b,c,d), Z. Phys. 32, 111-112; 794-798; Physica 5, 281-292; Naturwiss. 13, 1090-1091. Goudsmit, S. (1965), Phys. Bl. 21, 445-453. Goudsmit, S., and E. Back (1926), Z. Phys. 40, 530-538. Goudsmit, S., and R. De Laer Kronig (1925), Naturwiss. 12, 90. Grotrian, W. (1921), Z. Phys. 8, 116-125. Heisenberg, W. (1922a), Z. Phys. 8, 273-297. Heisenberg, W. (1925a,b), Z. Phys. 3 1 , 617-628; 32, 841-860. Heisenberg, W., and P. Jordan (1926), Z. Phys. 37, 263-277. Hund, F. (1925c,d,e), Z. Phys. 3 3 , 345-371; 855-859; 34, 296-308. Jordon, P. (1925c), Zs. Phys. 3 3 , 563-570. Kronig, R. De Laer (1925a,b), Z. Phys. 3 1 , 885-897; 3 3 , 261-272. Kronig, R. De Laer (1960), in Theoretical Physics in the Twentieth Century (eds. Fierz and Weisskopf), pp. 5-38. Kronig, R. De Laer (1926a), Nature 117, 550. Lande, A. (1923d), Z. Phys. 17, 292-294. Lande, A. (1924c), Z. Phys. 27, 149-156. Lande, A. (1925a,b), Ann. d. Phys. (4) 76, 273-283; Z. Phys. 3 1 , 339. Lande, A., and W. Heisenberg (1924), Z. Phys. 2 5 , 279-286. Laporte, O., and G. Wentzel (1925), Z. Phys. 3 1 , 335-338. Lorentz, H.A. (1888), Beginselen der Naturkunde, 2 Volumes, Leyden: E.J. Brill.
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Nagaoka, H., Y. Sugiura and T. Moshima (1924), Nature 113, 459-460. Ornstein, L.S., and H.C. Burger (1925), Z. Phys. 31, 355-361. Ornstein, L.S., H.C. Burger and W.C. Van Geel (1925), Z. Phys. 32, 681-683. Paschen, F . (1919), Ann. d. Phys. (4) 60, 405-453. Pauli, W. (1924c), Naturwiss. 12, 741-743. Pauli, W. (1925a), Z. Phys. 31, 373-385. Pauli, W. (1926a), Z. Phys. 36, 336-363. Pauli, W. (1948), Nobel lecture, Nobel Lectures: Physics 1942-1962, Amsterdam-LondonNew York: Elsevier, pp. 27-43. Russell, H.N., and F.A. Saunders (1925), Astrophysical Journal 6 1 , 38-61. Rydberg, J.R. (1894), Ann. d. Phys. (3) 52, 119-131. Saunders, F.A., and H.N. Russell (1923), Phys. Rev. (2) 22, 201. Schaeffer, C. (1908), Einfuhrung in die Maxwellsche Theorie der Elektrizotdt und des Magnethismus, Leipzig: B.G. Teubner. Slater, J.C. (1925d), Proc. Nat. Acad. Sci. (U.S.A) 11, 732-738. Sommerfeld, A., and H. Honl (1925), Sitz. ber. Preuss. Akad. Wiss. (Berlin), pp. 141-161. Sommerfeld, A., and A. Unsold (1926), Z. Phys. 36, 259-275. Thomas, L.H. (1926), Nature 117, 514. Thomas, L.H. (1927), Phil. Mag. (7) 3, 1-22. Uhlenbeck, G.E. (1925), Physica 5, 423-428. Uhlenbeck, G.E. (1976), Phys. Today 29, No. 6, pp. 43-48. Uhlenbeck, G.E., and S. Goudsmit (1925), Naturwiss. 13, 953-954. Uhlenbeck, G.E., and S. Goudsmit (1926), Nature 264-265. Wentzel, G. (1923a), Phys. Zs. 24, 104-109. Wentzel, G. (1924b), Ann. d. Phys. (4) 73, 647-S50. Wentzel, G. (1925b,e), Ann. d. Phys. (4) 76, 803-828; Z. Phys. 34, 730-735.
18 The Discovery of the Fermi-Dirac Statistics* In Heisenberg's treatment of many-body systems, and especially of the helium atom, the problem of statistics entered his work probably for the first time. 1 Heisenberg arrived at two conclusions: first, in atomic systems consisting of several identical particles, two classes of states are possible in principle, which can be distinguished by different symmetry properties either of the radiation intensity for transitions or of the wave functions for stationary states (see Heisenberg, 1926b) with respect to permutations of two particles; second, nature selects only one of these classes, which corresponds to the statistical description proposed earlier by Satyendra Nath Bose and Albert Einstein. Heisenberg later recalled 'that this symmetry would provide a clue to Pauli's exclusion principle and one could say that 'if two things are always connected symmetrically, then there can only be one electron in each state"' (Heisenberg, Conversations with Mehra, April 1960 and June 1968, p. 385; cf. AHQP Interview with Heisenberg, 27 February 1963, p. 12). He further recalled: 'For a long time I continued to mix up Bose-Einstein and Fermi-Dirac statistics. I did not know Fermi-Dirac statistics at that time [at the time of writing the papers on helium (Heisenberg, 1926b,c)]; I knew only the Pauli exclusion principle. I was always confused between Bose-Einstein statistics and the Pauli exclusion principle which produce different ways of counting states. When I wrote the equations for two identical electrons, there were two solutions, one symmetrical and the other anti-symmetrical. First I thought that I had to take the anti-symmetrical solution to obtain the Bose statistics, and that [i.e., Bose-Einstein statistics] must be the one which gives the Pauli principle. Later I saw that it was the other way round. One must take the symmetrical solution to get Bose statistics, and the anti-symmetrical solution to get Pauli's exclusion principle' (Heisenberg, Conversations with Mehra, Lectures delivered at the University of Cambridge, April 1985, Scuola Normale, Pisa, 1990, and UNESCO, Paris, 1991. This revised and complete version was published (with Helmut Rechenberg) in The Historical Development of Quantum Theory (Springer-Verlag New York, 1987). 1 Of course, the last section (4.3) of the three-man paper (Born, Heisenberg and Jordan, 1926) also dealt with the quantum-theoretical statistics of light-quanta; but this section was introduced by and worked out exclusively by Pascaul Jordan.
612
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Statistics
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April 1960 and June 1968, pp. 385-386; cf. AHQP Interview with Heisenberg, 27 February 1963, p. 12). 2 Actually, neither in the first paper on the many-body systems (Heisenberg, 1926b) nor in that dealing with the application to the helium atom (Heisenberg, 1926c) did Heisenberg make any statement of the above type; this occurred for the first time in a later work, namely in the paper entitled 'Mehrkorperprobleme und Resonanz in der Quantenmechanik, IF ('Many-Body Problems and Resonance in Quantum Mechanics IF) and submitted in December 1926 (Heisenberg, 1927a). In that paper he then referred in particular to the clarification of the problem recently achieved by Paul Dirac (1926f) and to the 'Fermische Theorie der Gasentartung' ('Fermi's Theory of Gas Degeneracy,' Heisenberg, 1927a, p. 239), citing a note already presented by Fermi in February 1926 to the Rome Academy (Fermi, 1926a). Hence, although Heisenberg claimed that Fermi's statistical method was not known to him in the summer of 1926, it already existed in print; indeed, it was even available in a German paper published in a May issue of Zeitschrift fur Physik (Fermi, 1926b) at about the time when Heisenberg became involved in the helium problem. We have to conclude that he overlooked the paper at that time, in spite of the fact that he was familiar with its author. In the years before 1926, the name of Enrico Fermi had occasionally been attached to contributions on atomic and quantum theory. The young Italian had even stayed abroad for some time, notably in Gottingen (with Max Born) and Leyden (with Paul Ehrenfest) in 1923 and 1924. Still, his new work on the theory of an ideal gas did not make an immediate impression even on people who knew him personally. The reasons for this delayed reception may be sought in his background and the particular type of early scientific work that he did. Fermi was born on 29 September 1901 in Rome, where he also grew up and attended primary and secondary schools.3 From the autumn of 1918 he studied at the University of Pisa as a fellow of the Scuola Normale Superiore, receiving his doctorate in July 1922. On returning to his hometown he worked as a private tutor (1922-1923) in mathematics and physics, but he was associated with the University of Rome. In the winter of 1923 Fermi was awarded a scholarship by the Italian Government to study abroad and he chose to go to Max Born in Gottingen; in the autumn of 1924 he went to Paul Ehrenfest in Leyden on a Rockefeller fellowship. The period in Gottingen proved to be a difficult one for the young Italian, despite 2 P a u l Dirac, like Heisenberg, made use of Schrodinger's wave mechanical approach in treating quantum systems of several identical particles; without taking notice of Fermi's papers, he rediscovered Fermi (-Dirac) statistics a few months later (Dirac, 1926f; see our discussion on 'Symmetry Properties of Wave Functions and Quantum Stastistics' following the description of Fermi's work). Pauli, who, based on his record at that time, was a greater expert on statistical mechanics than either Heisenberg or Dirac, also did not at all recognize the connection established by Fermi and Dirac between the statistics of identical particles, the many-electron problem and the exclusion principle. 3 For details of Enrico Fermi's background we refer to Laura Fermi's and Emilio Segre's biographies (L. Fermi, 1954; Segre 1970), and the latter's 'Biographical Introduction' in Enrico Fermi: Collected Papers, Vol. I (Segre 1962).
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physics
the fact that he spoke German fluently. 'In Germany his old shyness returned and hampered his social relations,' reported his widow thirty years later, 'he could never shed the feeling that he was a foreigner and did not belong in the group of men around Professor Born' (L. Fermi, 1954, p. 31). Heisenberg, then a member of Bom's inner circle, essentially confirmed this point; he recalled that he saw Fermi only rarely, and: 'I had some discussions with him. [In Gottingen] he was always a bit shy and kept himself to himself; I found it difficult to get into contact with him. He probably had personal problems; it was not a good time for him. I liked him as a rather different type of physicist' (Heisenberg, Conversations with Mehra, April 1960; cf. AHQP Interview with Heisenberg, 15 February 1963, pp. 12-13). The type of physics treated in the winter of 1922/1923 at Born's Seminar, i.e., the perturbation theory of multiply periodic systems, did not attract Fermi. 'He disliked proofs of convergence and mathematical subtleties of all kinds He felt, 'that is not physics," recalled Heisenberg (Heisenberg, Conversations with Mehra, April 1960; cf. AHQP Interview, 15 February 1963). 4 Finally, the whole status of atomic theory at that time, when the old Bohr-Sommerfeld theory had been shown to break. down in characteristic examples and the experts in Copenhagen and Gottingen were groping for a new mathematical and philosophical foundation, contradicted Fermi's 'preferences... for concrete problems' which 'gave him the immediate sensation of the importance, and a check of the correctness of his work' (Segre, 1962, p. XXVI). Compared with Gottingen, Fermi's visit to Leyden turned out to be more fruitful: Ehrenfest, in particular, provided him with much needed encouragement. 5 Back in Italy, Fermi became a lecturer in 1925 and moved to the University of Florence to teach mathematics and physics. After an unsuccessful competition in February 1926 for a chair of mathematical physics at the University of Cagliari (Sardegna), he was offered (in November of the same year) the newly created professorship for theoretical physics at the University of Rome. 6 From his early studies at Pisa, Enrico Fermi had become acquainted with topics of modern physics — not all from his teachers, but through his own personal initiative. He read widely, e.g., Walther Nernst's Theoretische Chemie, Owen Willans Richardson's The Electron Theory of Matter (1914), Ernest Rutherford's It is of interest to note that Fermi had obtained a much better education in mathematics than in physics; among his early friends at the University of Rome he counted the renowned senior mathematicians Guido Castelnuovo, Frederigo Enriques and Tullio Levi-Civita. 5 George E. Uhlenbeck, Ehrenfest's student, became acquainted with Fermi during his stay in Rome (1923-1925). He recalled: 'After this exam [of Uhlenbeck in Leyden] Ehrenfest said that I should look up Fermi [in Rome]. He gave me a letter and also a series of questions... in connection with the paper by Fermi on the proof of the ergodic theorem [Fermi, 1923a] Ehrenfest was so impressed by that paper I looked him up, of course, and Fermi was clearly very alone. He had two people who were at least about his own age.. .[A.] Pontremoli a n d . . . [Enrico] Persico I talked quite a lot with Fermi I told him in detail about Leyden very much, and t h a t , I am sure, made him decide to use the half year of his fellowship [in Leyden]' (Uhlenbeck, A H O P Interview, 4 May 1962, pp. 10-11). "Fermi's main promoter in Rome was Orso Maria Corbino (1876-1937), the director of the Physics Institute at the University of Rome from 1918 — he was a very influential physicist and politician in Italy.
The Discovery of the Fermi-Dirac Statistics
615
Radioactive Substances and their Radiations, and especially Hermann Weyl's Raum-Zeit-Materie and Max Planck's Vorlesungen uber Warmestrahlung. 'Fermi was already very much acquainted with modern physics, especially relativity,' his Pisa classmate Enrico Persico remarked more than forty years later, adding: 'I learned of the existence of relativity through Fermi' (Persico in AHQP Interview with Persico and Rasetti, 8 April 1963, p . 1). Franco Rasetti, another former colleague of Fermi's continued: 'Fermi was the first person ever to mention to me Planck's constant and the whole [quantum] theory.. .[L.] Puccianti [the director of the Pisa Physics Institute and renowned spectroscopist]... knew [about] the existence of the Bohr atom' (AHQP Interview with Persico and Rasseti, 8 April 1963, p. 8). However, Fermi read original papers as well, such as that by Niels Bohr on the constitution of atoms and molecules (Bohr, 1913b,c,e), and: 'He had the Zeitschrift fur Physik in his hands practically all day, and he [remarked], 'See, they have understood the orbits of the electrons in the atom' (AHQP Interview with Persico and Rasetti, 8 April 1963, p. 11). Evidently, this knowledge of modern physics soon went into the early papers, which Fermi began to submit from January 1921 to scientific journals, the first of these being concerned, in particular, with the problems of special and general relativity. His earliest experimental investigations were on X-ray physics, notably the use of a curved crystal to obtain images with monochromatic radiation; Fermi presented the results of this work in his doctoral dissertation of 1922. During his stay at Gottingen he submitted in April 1923 a paper on a 'Beweis, daft ein mechanisches Normalsystem im allgemeinen quasi-ergodisch ist' ('Proof that a Mechanical System is Quasi-Ergodic in General' to Zeitschrift fur Physik (Fermi, 1923a). This paper on the mechanical foundation of the second law of thermodynamics aroused the interest of Paul Ehrenfest, as we have mentioned earlier (in Footnote 190). Fermi devoted another paper from Gottingen (Fermi, 1923b) to Ehrenfest's adiabatic hypothesis. The note 'Sopra la teoria di Stern della costante assoluta delt entropia di un gas perfetto monoatomico' ('On Stern's Theory of the Absolute Constant of Entropy for a Perfect Monatomic Gas'), submitted in December 1923 (Fermi, 1923c), marked the beginning of Fermi's interest in a topic which continued during the next few years (see e.g., Fermi, 1924a). He also investigated the theory of collisions between atoms and electrically charged particles (Fermi, 1924b), and in Florence he entered into experimetnal work with Rasetti on the influence of an external magnetic field on the polarization of resonance radiation. All this preceded the next, most important, contribution by the young Fermi to physics, the papers containing a new statistical method for a monatomic gas (Fermi, 1926a,b). The problem of the appropriate form of quantum statistics had been around since the beginning of quantum theory, already showing up in Planck's derivation of his radiation law (Planck, 1900f). The question of a theoretical foundation of the statistics involved in Planck's derivation had led Albert Einstein to supplement his light-quantum concept (Einstein, 1905b) by a strange assumption: radiation exhibited simultaneously features that could be ascribed to independent quanta and
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physics
to electromagnetic waves (Einstein, 1909a). Fifteen years later, the Indian physicist Satyendra Nath Bose had been able to resolve the puzzling situation by proposing a new, systematic, consistent counting method for light-quanta, based on a peculiar occupation of quantized phase cells (Bose, 1924a). A quantized phase space, on the other hand, had at least been familiar to physicists since the early 1910s when the quantization of the translational motion of atomic particles was proposed. Otto Sackur and Hugo Tetrode had thus achieved descriptons in which the so-called chemical constant (i.e., a constant which occurred in the expression for the absolute entropy of a gaseous substance) could be calculated (Sackur, 1911, 1912; Tetrode, 1912). In their treatment, the behaviour of gases at low temperatures exhibited, even in the case of ideal monatomic substances, deviation from the classical equation of state. By using similar arguments one had tried to explain the behaviour of electrons in metals, especially their negligible contribution to the specific heats (see, e.g., Sommerfeld, 1914b). More than a decade later, Albert Einstein applied Bose's statistical method to establish the first fundamental quantum theory of ideal gases, suggesting in particular a relation to the theory of metal electrons (Einstein, 1924c; 1925a,b). Further, he connected Bose's statistics with an apparently quite different novel idea, the matter waves of Louis de Broglie (Einstein, 1925a). Fermi's early contributions to the theory of ideal gases preceded the lastmentioned developments. In the first paper on the Stern-Tetrode theory of the (absolute) chemical constant — it was presented on 2 December 1923 to the Accademia Lincei in Rome — Fermi analysed the derivation of the formula S = Nk{llnT-l»P
+
l n [ V ^ k ^ } ,
(1)
describing the entropy 5 of a monatomic gas of N atoms (having mass m) at temperature T and pressure p (with k and h denoting Boltzmann's and Planck's constant, respectively). Stern had derived it (in 1919) from a comparison of the kinetic theory of gases and solids, assuming the existence of a finite zero-point energy (of magnitude hv/2 connected to each vibrational degree of freedom of the solid; see Stern, 1919). In Stern's treatment the expression for the saturated vapour pressure also became modified as "H
(27rm) 3 / 2 ^ - 3 / w + \hv . TT£— 3 2— e x p (fcT) / V kT
J
(2)
(with v the geometrical average frequency of the solid, and the w the energy required to vaporize an atom, i.e., to liberate it from the solid). 7 Fermi now argued that the same equation could be obtained without the explicit assumption of a zeropoint energy, 'by a slight modification of the kinetic calcualtion' ('percio modificare leggermente la deduzione cinetica,' Fermi, 1923c, p. 396). 8 8
T h e usual expression contains an exponential exp(—w/RT), only. T h e main point of Fermi's demonstration involves the expansion of the sum,
$2^°=oexp
(—njhvj/kT). Such an evaluation somehow automatically takes care of a zero-point energy of the right magnitude.
77ie Discovery of the Fermi-Dirac
Statistics
617
The topic of the second paper, which Fermi submitted to Nuovo Cimento in January 1924, was a specialization of Sommerfeld's phase-integral quantization for the cases of systems having several identical particles (Fermi, 1924a). Starting from the simplest example, the two-electron helium atom, he argued that the path of integration extends from a given electron configuration to the next configuration that cannot be distinguished from the former: in particular, two situations, in which only identical particles have changed places, are indistinguishable. Translating this prescription to the case of an ideal gas consisting of n (identical) atoms, Fermi showed that the usual quantization procedure (involving the quantization of the translational motion of the gas atoms and the introduction of quantum cells) yielded the correct entropy expression (by Stern and Tetrode) only if one considered subsystems (space cells or volume elements), each containing not more than one atom. In all other treatments additional entropy terms arose that had the order of magnitude of the mixing entropy. 9 The advent (in 1924) of Bose's statistics, and especially its application by Einstein to the ideal gas theory (1924-1925), changed the situation in a qualitative way. That method made it possible to obtain an expression for the entropy of the gas which, unlike that of the classical kinetic theory, immediately satisfied Nernst's heat theorem. Further, Bose and Einstein treated the particles (i.e., light-quanta and atoms of an ideal gas) as indistinguishable objects. These results certainly interested Fermi very much when they became available in print. But he must have paid even greater attention to a result expounded by Wolfgang Pauli in early 1925: the exclusion principle for electrons, i.e., the ban on two electrons occupying identical states in an atom (Pauli, 1925b). This principle allowed Fermi to justify his own quantum-theoretical treatment of the ideal gas of the previous year. Still, it took him a while to digest the two major advances in the field of quantum statistics, probably caused partly by the fact that he was engaged in moving to Florence and starting experimental spectroscopic research there. Then he assembled the various pieces of his own and others' work and proposed a new gas theory, first in an Italian publication, the note 'Sulla quantizzazione del gas perfetto monoatomico' ('On the Quantization of the Ideal Monatomic Gas') which was presented by Antonio Garbasso to the Reale Accademia Lincei in Rome on 7 February 1926 (Fermi, 1926a), and soon afterwards in an extended German translation, lZur Quantelung des idealen einatomigen Gases,' which was received by Zeitschrift fur Physik on 24 March 1926 (Fermi, 1926b). Fermi's goal was the same as that which he had already tried to achieve more than two years earlier, namely to derive a theory describing the ideal gas which, in the limit of absolute zero temperature (T = 0), satisfies Nernst's heat theorem. In particular, Fermi wanted 'to present a method for the quantization of an ideal gas, The problem treated by Fermi in the above paper (Fermi, 1924a) was discussed by several physicists in those years, including Paul Ehrenfest, Max Planck and Erwin Schrodinger. Fermi essentially supported Planck's point of view on the statistical treatment of identical particles, which was opposed by Ehrenfest and Schrodinger.
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The Golden Age of Theoretical
physics
which in our opinion is as independent as possible from arbitrary assumptions concerning the statistical behaviour of gas molecules' (' eine Methode fur die Quantelung des idealen Gases darzustellen, welche nach unserem Erachten moglichst unbhdngig von wilkurlichen Annahmen uber das statistische Verhalten der Gasmolekiile ist,' Fermi, 1926b, p. 902). 10 Fermi referred in his paper to the recent work on the quantum theory of ideal gases by Albert Einstein (1924c; 1925a,b) and Max Planck (1925a). 11 These theories exhibited deviations from the ideal gas theory at very low temperatures and high pressures, as Fermi's own would do; however, the magnitude, and partly the sign of these deviations, would be different. Unfortunately, the experiments, under the conditions mentioned, did not yet make it possible to decide in favour of any one of the proposed theories. Nevertheless, Fermi felt confident enough to present his own ideas. In the quantum theory with which Fermi was so far familiar, there existed several methods of quantizing an ideal gas. 12 For example, one could enclose the gas in a vessel with parallel walls which reflect the molecules elastically, such that their motion becomes periodic; another method would be to have the molecules exposed to an external field of such a type as to establish similar periodic boundary conditions. However, whatever procedure one selects, it still does not suffice to determine the quantization completely, since the latter still depends crucially on the size of the vessel — hence for large, macroscopic distances between the walls the discontinuity of the quantum states practically vanishes. Recalling his earlier result (see Fermi, 1924a), Fermi now concluded that 'one obtains a degeneracy [of the gas] of the expected order of magnitude, if one chooses the vessel to be so small as to contain, on average, only one molecule' (Fermi, 1926b, p. 903). This extra hypothesis, which had earlier provided the Stern-Tetrode value for the entropy of an ideal gas, he called 'a supplementary rule to be added to the Sommerfeld quantum conditions' ('eine Zusatzregel zu den Sommerfeldschen Quantenbedingungen,' Fermi 1926b, p. 904). The supplementary rule seemed, as Fermi noticed, to be supported by Wolfgang Pauli's rule for the occupation of the electron orbits in atoms. Hence he concluded: 'We shall demonstrate... that the application of the Pauli rule allows us to present a complete, consistent theory of the degenracy of ideal gases' (' Wir werden zeigen,... dafidie Anwendung der Paulischen Regel uns erlaubt, eine vollstandig konsequnete Theorie der Entartung idealer Gase darzustellen,'1 Fermi, 1926b, p. 904). Evidently, Fermi believed that Pauli's rule determined not only the statistical behaviour of electrons but of gas molecules as well. In the case of a quantum gas, then, the rule stated that only one atom existed in the entire volume having a given 10
I n the following, we present and analyse the content of the German paper (Fermi, 1926b), which is a slightly more extended and detailed version of the original Italian note (Fermi, 1926a). 11 See Fermi, 1926b, p. 902, footnote 2. 12 A s we have mentioned, Fermi hesitated in following the work on the formulation of quantum mechanics in Gottingen.
The Discovery of the Fermi-Dirac
Statistics
619
set of quantum numbers, which comprised in general the internal quantum numbers as well as the translational numbers; of course, for an ideal or perfect monatomic gas in the ground state only the translational quantum numbers had to be considered. As the appropriate model for demonstrating the consequences of this extended exclusion principle, Fermi chose a system of N gas molecules (atoms) of mass m, each of which was attracted to a fixed point, the origin, by an elastic central force whose potential was given by the expression U = 27rVmr2 ,
(3)
with v the eigenfrequency of the oscillating molecules. 13 The molecules of a monatomic gas in the ground state are described by the three quantum numbers, si, S2, and S3, of the translational motion, and the energy of each molecule becomes w — hv(si + S2 + S3) = shu,
(4)
i.e., an integral multiple of the energy-quantum hv. On the other hand, a given total energy, say shu, can be obtained by quite a few different distributions of the individual quantum numbers (si = 0 , 1 , 2 , . . . , «2 = 0 , 1 , 2 , . . . , and S3 = 0,1,2,...), namely altogether in
o, = ( s + 1 > 2 " + 2 >
(5)
possible ways. Then Fermi's 'Pauli rule' states: for any given s only Qs molecules can occur in the gas model. This means, in particular, that in the case of absolute zero temperature — for which the energy must be a minimum — the lowest energy states are fully occupied: i.e., one molecule possesses energy zero, three molecules possess the energy hu, six the energy 2hv, etc. Hence, Fermi concluded that one obtains for the gas 'a kind of shell-like constitution, which exhibits a certain analogy to the shell-like organization of the electrons in a many-electon atom' ('eine Art schalenformigen Aufbau, der tine gewisse Analogie zur schalenartigen Anordnung der Elektronen in einem Atom mit mehreren Elektronen aufweist,' Fermi, 1926b, p. 9 0 6 ) . u If the temperature rises above zero, the distribution of a given total energy E(= Ihv, with I an integer) of a monatomic ideal gas of N molecules (in the ground state) can be found by copying a procedure by Einstein from his papers on gas theory. Clearly, the relations
52N. = N,
(6)
J2sN3 = E,
(7)
and
ld I n this model the motion is obviously periodic — with a single period. The density of the gas is not uniform (as is usually assumed) with most of the molecules assembled close to the origin (r = 0). 14 Fermi was quite impressed by Pauli's success in arranging the electrons in shells and t h e fruitfulness of this arrangement in subsequent analyses of the structure of many-electron atoms — he referred especially to Friedrich Hund's paper on complex atoms (Hund, 1925c).
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The Golden Age of Theoretical physics
with the subsidary condition Ns < Qs
(8)
had to be satisfied (with Na denoting the number of molecules having energy shu). Fermi further demanded: 'Two distributions should then be considered to be alike, if the places occupied by the molecules are identical' ('Zwei Anordnungen sollen dabei als gleich angesehen werden, wenn die von den Molekulen besetzten Platze dieselben sind,' Fermi, 1926b, p. 906). 15 Since the number of possible distributions of Na atoms among Qs places is ($*)' the total number of distributions, P, is simply given by the product ( ^ ° ) ( ^ ) ( ^ ) ^his product is then determined — by using Stirling's approximation formula which is valid for the large numbers involved — by the equation
N.J
VV
Qs-Ns
' ~°a
Q
The maximum P satisfying the subsidiary conditions (6) and (7) is then determined for ^L-=aexp(-l38), (10) with a and (3 denoting constants independent of the individual distributions associated with the integer s (and the energy shu). By inserting Eq. (10) into Eqs. (6) and (7) one arrives at expressions for the total number of molecules and the energy as functions of a and /?. In order to determine the physical significance of the constants a and /3 one should think of using the well-known formulae of statistics and thermodynamics (i.e., the Planck-Boltzmann relation S = klnP for the entropy and the relation T = dE/dS). Fermi, however, wished to avoid using them, as they might not really hold in the quantum theory. Instead he turned to a different calculation which employed a correspondence argument: namely, in the limit of r, the distance from the origin, tending to infinity the density of the gas becomes negligible, hence the classical statistical behaviour (average energy of each molecule equal to \kT; Maxwellian distribution) must be valid. To apply this limiting consideration, he calculated the density of the s-molecules (i.e., of molecules with energy shv) as a function of the distance r from the origin, obtaining for r —• oo the Maxwellian distribution n{L)dL = const.vL(£L • exp I — — I ,
(11)
with L the kinetic energy of a molecule, provided the constant (3 assumed the value
15 Fermi also realized the fact that if he considered two permuatations of the type mentioned (i.e., a situation with the same set of distribution numbers N„, in which two molecules were exchanged) as different distributions, the total number of distributions would be multiplied by a factor iV!; however, this factor would not influence the later result.
The Discovery of the Fermi-Dirac
Statistics
621
For very large but finite r, the distribution was given by the equation (27r)(2m)3/2 rj n{L)dL
_
_
Aexp(-L/kT)
SLdL . _ _ _ _ _ _ _ _ ,
(13)
where A denotes the quantity 2ir2v2mr'2 A = a e x p I —kT
(13a)
Hence the constant a can be related to the density n ( = /0°° n(L)dL) of the molecules at a distance r, and therefore ultimately to the number of molecules in the gas model. 16 The equation of state now followed in a straightforward manner as hW3 P
~
P(0)
2irmkT/h2n2/3 — is given by
1
1 + 25/203/2
32/3^1/3
5 • 2^3
•
(14)
V h2™2/3 ) '
2nm
where the function P ( 0 ) — with 6 = 0
(2TrmkT\
for very large ©,
(14a)
for very small 0 .
(14b)
.22/3_4/3
5
1 + 3
V3
@ 2 +
-
Thus for weak degeneracy, i.e., for large T and small n (i.e., large 0 ) , the equation for the pressure p assumes the form p = nkT
1 + 16
h3n {irmkTf2
+ •
(15)
That is, the gas pressure in Fermi's theory of ideal monatomic gases turns out to be larger than that given by the classical equation; e.g., for helium at T = 5° absolute and p — 10 atmospheres, the difference amounts to 15 percent. 17 For very low temperatures and high densities a large degeneracy resulted, and in the case of zero temperature a finite zero-point pressure Po =
1
20 \n)
/6\2/3h2n5'3
-ZT
(16a)
and a zero-point energy (per molecule)
L =_(_y/3___ 40 \7rJ 16
(16b)
m
Fermi gave a detailed expression for the density of the s-molecules (i.e., the molecules having energy shv), which he showed would go over into the right-hand side of Eq. (13) for suitably large s ( = L/hv + {2-n2vm/h)r2). In the case of Einstein's gas theory the pressure is smaller than the classical one (see Einstein, 1925a, p. 14).
622
The Golden Age of Theoretical
physics
was obtained. 18 Finally, the specific heat at low temperatures v
°~
dl _ dT~
2 4 / 3 TT 8 / 3 2 3
3 /
mk2T /i 2 n 2 /3 '
(17)
indeed vanishes at absolute zero, in agreement with Nernst's heat theorem. Since, for high temperatures, there also followed Stern's and Tetrode's entropy expression, Eq. (1), Fermi had completely fulfilled the programme he had set up for himself. It took some time before a reaction to Fermi's paper came, but it did not come directly. George Uhlenbeck, who knew Fermi at that time, recalled: 'It [Fermi's theory] looked strange to u s . . . he [Fermi] did it so strangely at that time because as a vessel he took a harmonic oscillator' (Uhlenbeck, AHQP Interview, 9 December 1963, p. 15). Only later did Uhlenbeck learn (from the papers of Heisenberg and, in particular, that of Dirac) of the connection between the statistics and the symmetry properties of the wave functions, and then he went back to Fermi's original paper. Even Pauli, whose exclusion principle had provided the basis of the new statistics, did not immediately take positive notice of the result. As late as 19 October 1926 he confessed in a letter to Heisenberg: 'I now think considerably more favourably of the Fermi-Dirac statistics, and at present it appears to me that several arguments support it. Between a crystal lattice and radiation there already exists a difference, namely that of zero-point energy hu/2.19 It is certainly reasonable to assume that in the case of an ideal gas there also exists a zero-point energy. Such a concept can only be put on top of the Einstein-Bose theory in a very artificial manner... and this speaks from the beginning against that theory and in favour of the Fermi-Dirac theory.' Pauli then sketched a few conclusions from Fermi's theory which we shall present a little later, since they are more closely connected with Paul Dirac's treatment of the quantum statistical methods. This treatment will be discussed next. 1. Symmetry Properties of Wave Functions and Quantum Statistics During the first half of 1926, when Schrodinger's papers on wave mechanics were submitted and published, Paul Adrien Maurice Dirac was concerned with working out his own scheme of quantum mechanics. The mathematical objects which he employed, the q-numbers, exhibited a close relation to the matrices used by the Gottingen physicists; however, their range of application seemed to be greater, since they could be easily applied to describe all kinds of multiply periodic atomic systems (Dirac, 1926c), and even to aperiodic relativistic atomic processes such as the Compton effect (Dirac, 1926f). Being completely absorbed in these investigations, which he also used to obtain his doctoral degree in May 1926, Dirac — the independent and lonely physicist — did not pay too much attention to other new 18 Evidently, the relation between L, the average kinetic energy of the molecule, and the pressure of the ideal gas is given by p = \nL. 19 P a u l i assumed t h a t such an energy did exist in the case of the vibrations of a crystal but not for the radiation (as it would then be infinite in a cavity).
The Discovery of the Fermi-Dime
Statistics
623
approaches in atomic theory, although he followed the literature in general. Of course, one might wonder whether he regularly looked at the Annalen der Physik where Schrodinger's communications appeared, beginning in March 1926, but where not much work on atomic or quantum theory was otherwise published at that time. If however, he had not seen the first communication (Schrodinger, 1926c), he soon afterwards received a strong hint about it from Werner Heisenberg, who wrote on 9 April from Munich: 'A couple of weeks ago a paper by Schrodinger came out in the Annalen der Physik (Volume 79, issue No. 4, p. 361, 1926), whose content must, in my opinion, be connected mathematically with quantum mechanics.' Heisenberg went on to ask Dirac: 'Have you reflected upon the question of how far this Schrodinger treatment of the hydrogen atom is related to the quantum mechanical one?... I am particularly interested in this mathematical problem, since I believe that one can gain quite a lot from it for the physical, fundamental interpretation of the formalism' (Heisenberg to Dirac, 9 April 1926). Heisenberg had indeed addressed his letter to the right person, because Dirac — in his second paper on quantum mechanics — had treated the problem of the hydrogen atom within the q-number theory, a scheme that generalized the previous Gottingen matrix methods and was much closer to Schrodinger's differential equation approach in many respects (Dirac, 1926a). We note with some curiosity Heisenberg's remark that the undulatory theory might help to solve the problem of the physical interpretation of quantum mechanics, a point of view which he later on — when Schrodinger tried to provide such an interpretation — would refute completely. In any case, Heisenberg now hoped to learn from Dirac's mathematical expertise, and he therefore ended his letter with a couple of questions: 'Couldn't you come during the next [few] months to Copenhagen? I would so much like to discuss with you all the new problems. Or will you still be in Cambridge in July? Then perhaps, as I did last year, I would have a great desire to come to England for a short time' (Heisenberg to Dirac, 9 April 1926). Dirac could not go to Denmark until later in the fall of 1926;20 still he wrote back to Heisenberg, and their correspondence stimulated his own work which indeed took a turn towards wave mechanics. The letter, in which Dirac replied to Heisenberg's letter of 9 April 1926, seems to have been lost. 21 On 26 May 1926, however, Heisenberg answered Dirac from Copenhagen. He reported first about 'the connection between Schrodinger's theory and the quantum mechanics,' referring to the results that 'are given in a third paper of Schrodinger [1926e], which may come out in the next weeks. 22 Heisenberg may have heard of this paper in April, while he was at Sommerfeld's Institute in 2 In the spring of 1926 Dirac was occupied in assembling the g-number theory of quantum mechanics for the purpose of a doctoral dissertation; he obtained the Ph.D. degree from Cambridge University in May. In the Heisenberg Nachlafl no scientific correspondence before 1930 is contained. It was probably lost towards the end of World War II. 22 Heisenberg wrote from Copenhagen in English, while his previous letter (of 9 April) had been written in German.
624
The Golden Age of Theoretical
physics
Munich, as Schrodinger had already sent it in March to Wilhelm Wien. On the other hand, Heisenberg certainly heard the news of the independent equivalence proof by Wolfgang Pauli, which the latter had formulated in his letter of 12 April to Pascual Jordan, on his passage through Gottingen around 20 April. Indeed, Heisenberg essentially sketched for Dirac the content of Pauli's letter. Thus, for example, he gave the formula for the matrix element of a physical quantity in wave mechanics, /mn=
/ fMmdto,
(18)
where tpn and ipm are two eigenfunctions of the system and f dil denotes the integration over its position variables. He further showed that with the definition (18) the matrix multiplication rule followed for two such quantities, / and g, i.e., {fg)nm = ]T) fnigim •
(19)
Assuming that the definition ^ -
=- ^ / ^
d
"
(20)
yielded the matrix elements for the momentum component pk of the system, Heisenberg could prove that the quantum mechanical commutation relations did hold automatically. He concluded: 'In a similar way you get from [the Schrodinger equation] the Hamiltonian function as a function of the matrices p and q, and then we know, that [Eqs. (18) and (20)] really give the solution of the quantum mechanical equations. I think, it is not necessary that I write it down' (Heisenberg to Dirac, 26 May 1926). Heisenberg further presented in his letter a relativistic form of the Schrodinger equation. Then he discussed specific questions, which had essentially been raised before by Dirac. 'I quite agree with your criticism of Schrodinger's paper with regard to the wave theory of matter. This theory must be inconsistent,' he wrote. Nevertheless he also mentioned a more positive aspect by saying: 'I see the real progress made by Schrodinger's theory in this: that the same mathematical equation can be interpreted as point mechanics in a non-classical kinematics and as wave theory according to Schrodinger. I always hope, that the solution of the paradoxes in the quantum theory later would be found this way' (Heisenberg to Dirac, 26 May 1926). He then reported on the interest taken by the Copenhagen physicists in Dirac's quantum-mechanical treatment of the Compton effect (Dirac, 1926c). Finally, he mentioned his own investigation of the helium problem, especially his success in explaining the large singlet-triplet separation as a consequence of the quantum-mechanical resonance effect. 'Really one gets in this way a qualitative explanation of the spectrum with regard as well to the frequency as to the intensities. And I hope, the quantitative agreement is only a question of long numerical work,' he concluded (Heisenberg to Dirac, 26 May 1926).
The Discovery of the Fermi-Dirac
Statistics
625
The last point must have alerted Dirac who had for years been interested in obtaining a satisfactory theory of the helium atom. Actually, Dirac seems to have had (for some time in early 1926) a suitable quantum mechanical action-angle scheme for dealing with that task (Dirac 1926b). Probably the preparations for obtaining his Ph.D. degree and the work on the Compton effect, a problem of fundamental importance, had delayed Dirac's final attack on the helium problem. Or did another difficulty exist? The definite, positive answer to this question can be derived from the introduction to a paper by Dirac that his supervisior Ralph Fowler had communicated to the Proceedings of the Royal Society of London in late August 1926 (Dirac, 1926f). There he first described the characteristic feature of the Gottingen-Cambridge approach to quantum mechanics as: 'One can build up a theory [of atomic systems] without knowing anything about the dynamical variables except the algebraic laws that they are subject to, and can show that they may be represented by matrices whenever a set of uniformising variables for the dynamical system exists.' He added: 'It may be shown, however (see Section 3), that there is no set of uniformising variables for a system containing more than one electron, so that the theory cannot progress very far on these lines' (Dirac, 1926f, p. 661). The proof of this observation, which Dirac presented in Section 3 of his new paper, ran like this: Consider in quantum mechanics a system consisting of two electrons without any interaction. Evidently, its properties must be described by matrices: the particular element (mn;m'n') then denotes the transition from a state (mn) — the first electron being in the staionary state m and the second in the state n — to the state (m'n'). The question now arises as to whether the two states of the system, (mn) and (nm), which differ only by the fact that the two electrons have exchanged their individual states, are taken to be identical or not. The assumption that they have to be counted as separate states immediately leads to a contradiction, because only the sum of the intensities due to the transitions (mn) —> (m'n1) and (nm) —> (n'm1) can be determined experimentally. The other alternative, however, results in the identity, xi(mn;m'n')
= X2(nm;n'm'),
(21)
holding for xi and X2 (denoting corresponding co-ordinates for the electrons 1 and 2) and all values of (mn) and (m'n'). This would now demand the identity of the two matrices x\ and X2, which is obviously nonsensical. Dirac concluded: 'We must infer that unsymmetrical functions of the co-ordinates (and momenta) of the two electrons cannot be represented by matrices' (Dirac, 1926f, p. 667). 23 Since Dirac had not addressed the above argument either in his March paper (Dirac, 1926b) or in his April paper on the relativistic extension of the ^-number theory (1926c), he must have noticed the difficulty later in the spring or early summer of 1926. He remarked many years later: 'Things which are not pretty obvious 23
I n the case t h a t the variables were symmetric in both particles, 1 and 2, equations similar to Eq. (21) resulted; but then no contradiction arose.
626
The Golden Age of Theoretical physics
are things which take me years before there is a breakthrough. I don't think it's any puzzle how someone gets the idea that leads to the break' (Dirac, Conversations with Mehra, June 1968). Evidently, the many-electron problem was solved only with the help of symmetry arguments, and the latter were indeed available at that time. For example, Heisenberg made use of them in the months from May to July 1926, when he interpreted the results that emerged from his calculation of the helium term-system (Heisenberg, 1926b,c). We are not certain whether the physicists in Cambridge discussed similar ideas; but on 29 July 1926 Max Born arrived from Gottingen to speak at the Kapitza Club on the wave-mechanical collision problem. 24 Dirac, a member of the Kapitza Club, met Born and received some information on Heiseinberg's work on the symmetry properties of wave functions, as he pointed out in a footnote in his later paper: 'Professor Born has informed me that Heisenberg has independently obtained results equivalent to these' (Dirac, 1926f, p. 670, footnote f)- This statement confirms that at that time of Born's visit to Cambridge — i.e., before Heisenberg's first paper on the many-body problem appeared in print (in the Zeitschrift fur Physik, issue of 10 August 1926) — Dirac had already found the decisive results of his own wave mechanical treatment of many-electron systems. After recognizing the advantage of Schrodinger's theory, Dirac systematically analysed wave mechanics from a point of view that was more familiar to his own way of thinking. 25 That is, he recast the scheme first into the general form he had given previously to the g-number theory, notably in the paper on 'relativity quantum mechanics' (Dirac, 1926c, Section 2). 26 So he began by associating with the position variables qr and the time variable t of a quantum mechanical system the canonically conjugate momentum variables, now in the form of differential operators,
and
(22b)
-'•41-
Evidently, all properties of the system could now be expressed in terms of qr and t and their derivatives, provided they depended only in a rational way on the momenta pr and the energy W. Further, Dirac now noted that to the classical HamiltonJacobi equation, i.e., H(qr,Pr,t) 24
-W
= F(qrtPr,t,W)
= 0,
(23)
See the Minute Book of the Kapitza Club, filed on AHQP Microfilm No. 38. A talk on Schrodinger's theory at the Kapitza Club by Thomas MacFarland Cherry, presented on 8 June 1926 — apart from the correspondence with Heisenberg — may have contributed in stimulating Dirac's interest in wave mechanics. 26 Dirac presented this step in Section 2 of his August paper, entitled 'General Theory' (see Dirac, 1926f, pp. 662-666). In this section, a far-reaching parallel to Section 2 of the April paper (Dirac, 1926c, pp. 407-409) is to be found. Especially, the inclusion of the time variable on the same footing as the position variable is characteristic of both treatments. 25
The Discovery of the Fermi-Dirac
Statistics
627
there corresponded the wave equation F(V0 = 0
(23')
which is nothing other than Schrodinger's equation. Because of the linearity of Eq. (23') in a wave function ij), a general solution might be written as
V> = ^ c"^" >
(24)
n
with c n being arbitrary constants and ijjn constituting a complete set of independent solutions, called eigenfunctions of the wave equation. 27 As a second step, Dirac went on to 'show that any constant of integration of the dynamical system... can be represented by a matrix whose elements are constants, there being one row and column of the matrix corresponding to each eigenfunction •tpn' (Dirac, 1926f, p. 664). Since any such constant a commuted with the 'Hamiltonian' F of the system (aF = Fa), the function aipn was also a solution of Eq. (23'). As a consequence, its expansion arfn = y ^ m Q m n , m
(25)
in terms of the complete set of eigenfunctions, {ipm\, provided the elements, amn, of the matrix corresponding to a.28 Since arbitrary functions, x, of the dynamical variables, pr,qr,W and t, at a given time to, provided special examples of such constants of motion, Dirac further concluded that every dynamical variable of the system, x(t), might 'be represented by a matrix whose elements are functions of t only' (Dirac, 1926f, p. 665). In the restricted case that F does not depend explicitly on the time variable t, W is a constant of motion; and the eigenfunction ipn can be chosen to make the matrix representing W a diagonal matrix. Since any such constant a commuted with the 'Hamiltonian' F of the system (aF = Fa), the function atjjn was also a solution of Eq. (23'). As a consequence, its expansion •Emn — amn GXp
~(Wm n
- Wn)t
(26)
Consequently, the equivalence of the wave-mechanical scheme and the matrix theroy of Born, Heisenberg and Jordan was more or less established. 29 Dirac further concluded that for constants of integrarion, a, b, c,..., which commuted, the same set of 27 Dirac was aware of the fact that besides eigenfunctions belonging to discrete eigenvalues, labelled by integers n, there also existed those belonging to continuous eigenvalues, labelled by a parameter a varying between certain limits. Then, in general, the right-hand side of Eq. (24) contained an additional integral of the type J caip(a)da. 28 Heisenberg had given Eq. (28) in his letter of 26 May. Dirac, following Heisenberg's lead, now showed that this definition of the matrices satisfied the rules of matrix multiplication. 29 Dirac did not succeed at that time in proving that all matrices representing dynamical variables had to be Hermitean.
628
The Golden Age of Theoretical
physics
eigenfimctions produced diagonal matrices (with elements an, bn, cn,... representing the corresponding 'eigenvalues'), i.e., 'we can have eigen functions representing stationary states of an atomic system with definite values for the energy, angular momentum, and other constants of integration' (Dirac, 1926f, p. 666). In applying the wave-mechanical scheme to systems containing several identical particles, Dirac observed that the difficulties of the previous quantum-mechanical formulation (i.e., of matrix mechanics and his g-number theory) actually disappeared. This could be recognized immediately in the simplest example, namely a two-electron atom, in which one neglects the interactions between the electrons, the eigenfunctions, ipmn, of this system should be basically given by the product of the eigenfunctions of the individual electrons, •0 m (l) • V'(2). However, due to the indistinguishability of the two electrons, the states (mn) and (nm) of the whole system should be identical; hence ipmn must rather assume the general form rpmn = a m n V m ( l ) V > n ( 2 ) +
ftm„V'm(2)V»(l)
>
(27)
with constants amn and bmn such that the two states described by ipmn and ipnm are the same. This implies that every symmetrical function A of the two electrons may be expanded as Atpmn = ^ A Tn'n1,77171
*
(28) m'n'
Two possible types of coefficients satisfy this requirement, namely, 30 amn = bmn
(27a)
and amn = -bmn
•
(27b)
The choice of the second solution (27b) leads to a change of sign in ipmn when the two electrons are permuted, i.e., (mn) —>• (nm). Still, in quantum theory this change does not imply observable consequnces. Dirac concluded: 'Thus the symmetrical eigenfunctions alone or the anti-symmetrical eigenfunctions alone give a complete solution of the problem,' but also that 'the theory at present is incapable of deciding which solution is the correct one' (Dirac, 1926f, p. 669). The result of the case of two electrons could easily be extended, as Dirac immediately noticed, to cases with any number of electrons. For example, the symmetrical eigenfunctions of systems containing r non-interacting electrons were represented by the sum ]T
ipni(ai)ipn2(a2)
• • • ipnr(ar),
(29a)
ai,...,ar 30
I n the case of symmetrical eigenfunctions, given by Eq. (27a), the left-hand side of Eq. (28) is symmetrical; hence the right-hand side can be expressed by symmetrical eigenfunctions. In the other case, given by Eq. (27b), the left-hand side is antisymmetrical, hence in the expansion of the right-hand side only antisymmetrical functions must occur.
The Discovery of the Fermi-Dirac
with ai,...,ar denoting any permutation of the integers 1,2, ...,r. symmetrical eigenfunctions could be written as the determinant
(V£i ' " - "r)anti =
^ni(l),
^(2),
•••
'4>ni{r)
^n2(l),
V-n 2 (2),
•••
Tpn2(r)
Statistics
629
The anti-
(29b)
Vv(l), V-n ••• electrons, V „ » there will still be r(2), the Dirac further stated: 'If there is interaction between symmetrical and antisymmetrical eigenfunctions, although they can no longer be put into these simple forms. In any case the symmetrical ones alone or the antisymmetrical ones alone give a complete solution of the problem' (Dirac, 1926f, p. 699). Dirac's conclusions agreed with what Heisenberg had obtained several weeks before in his papers on the many-body problem and the helium atom (Heisenberg, 1926b,c). As we have mentioned earlier, Dirac was not aware of Heisenberg's work when he composed his own paper, in which he also rediscovered another of Heisenberg's results, namely: since antisymmetrical eigenfunctions vanish identically if two particles occupy the same orbit, they describe systems satisfying Pauli's exclusion principle. Consequently, Dirac argued, the solution with the symmetrical eigenfunctions that allows any number of electrons to be in the same orbit 'cannot be the correct one for the problem of electrons in an atom' (Dirac, 1926f, p. 670). 31 Unlike Heisenberg, who had restricted the application of the treatment to the problem of many-electron atoms. Dirac now proceeded to discuss another elementary many-particle problem; an assembly of molecules of the ideal gas, for which either symmetrical or anti-symmetrical eigenfunctions would be appropriate. 32 For this purpose, Dirac first considered a single relativistic molecule and constructed its eigenfunctions by solving the wave equation, 33
a2
dx2
+
&_
By2
+
_^__i_f_
dz2
c2 dt2
m2c2 "• V» = 0. (h/2n)2
(30)
The solution was provided by the following expression,
Vw
2iri . exp - jn— {atix + a2y + a3z -
ht)
(31)
where the quantities a i , ct2, and a^ denoted the momentum components and E the energy of the molecule. If the molecule were enclosed between ideally reflecting 3
At this point Dirac referred, in a footnote, to the information given to him by Born on Heisenberg's similar conclusions, which we have already cited. 32 Dirac presented the 'Theory of the Ideal Gas' in Section 4 of his paper (Dirac, 1926f, pp. 670-673). 3 At that time, in the summer of 1926, Eq. (30) was proposed and discussed in publications by several authors, including Schrodinger himself.
630
The Golden Age of Theoretical
physics
boundary walls, the momenta became restricted; e.g., in the case where such walls are posted at x = 0 and x = 27r (in units of relative length), one finds a1—=n + r,
(32)
with n denoting an integer and r any real number. Hence the number of waves of the form (31) existing in a three-dimensional box of volume (27r)3 — enclosed within a region 0 < x, y, z < lix — can easily be counted to be
Nr
^ = ^Kkr{E-m2ci)1/2EdE
(32)
for a molecule of mass m having an energy between E and E + dE. In the nonrelativistic limit (E\ = E — mc2) this number reduces to
"*"*
=
Ji^{2m)V2E^dEl
•
(32 }
'
Finally, for a box of volume V expressions (32) and (32') have to be multiplied by the factor V7(2TT) 3 . After these preliminaries Dirac turned to a gas consisting of N molecules contained in a box of volume V. He assumed that, (i) its eigenfunctions were either of the symmetrical or of the anti-symmetrical type, and (ii) 'all stationary states of the assembly (each represented by one eigenfunction) have the same a priori probability' (Dirac 1926f, p. 671). He then claimed: 'If now we adopt the solution of the problem that involves symmetrical eigenfunctions, we should find that all values for the number of molecules associated with any wave have the same a priori probability, which gives just the Einstein-Bose statistical mechanics. On the other hand, we should obtain a different statistical mechanics if we adopted the solution with antisymmetrical eigenfunctions, as we should then have either 0 or 1 molecule associated with each wave' (Dirac, 1926f, pp. 671-672). While Dirac referred, for the first part of the claim, just to the publications of Bose (1924a) and Einstein (1924c; 1925a), he worked out the case with antisymmetrical wave functions in detail. For this purpose, he divided the waves in sets, each of which is associated with molecules of the same kinetic energy Es. If As denotes the number of waves in the sth set available, then the probability of a distribution in which Na molecules have the energy Es will be p
° n - w ( A.-*,)r
(33
>
Dirac now proceeded like Einstein (1924c, Section 3). 3 4 The logarithm of P, if multiplied by Boltzmann's constant k, gives the entropy S, which has to be a In the symmetrical case (the Bose-Einstein statistics) nearly the same expression is valid; the denominator term {As — JV3)! is replaced by (A3 + Ns)\.
The Discovery of the Fermi-Dirac Statistics 631 maximum under the subsidiary conditions of a given energy E(= J^ s ESNS) and a given total number of molecules N(= ^ Ns). The variational calculation then yielded the relation In(^--l)
=a + {3Es
(34)
for each energy Es, with a and /3 being energy-independent constant. Hence, in the case of the equilibrium particle distribution, there followed the equation N
°
=
(35)
exP(a + 0Es) + V
where As is determined by Eq. (32'), i.e., V_ N tel ° = 7^3 (27T) E?
A
(36a)
•
The constant /? is obtained by using the thermodynamical relation defining the absolute temperature T(= SE/6S, with S the above-defined entropy) as
(36b)
*=h-
Hence the gas of N particles of mass m in volume V will satisfy the two characteristic relations £ N3 = 2*V (^)3 V \ h
N:
f J Jo
, W?' exp{a + E3/kT)
+ l
(37) '
K
and ~
\
h
J
Jo
exp(a + ££33/fcT) /fcT) + 1 '
(38)
Dirac noted: 'By eliminating a from these two equations and using the formula pV = | £7, where p is the pressure, which holds for any statistical mechanics, the equation of state may be obtained.' He concluded further: 'The saturation phenomenon of the Einstein-Bose theory does not occur in the present theory. The specific heat can easily be shown to tend steadily to zero as T —*• 0, instead of first increasing until the saturation point is reached and then decreasing, as in the Einstein-Bose theory' (Dirac, 1926f, p. 673). The above-mentioned observations evidently confirmed Dirac's earlier statements: 'The solution with symmetrical eigenfunctions must be the correct one when applied to light-quanta, since it is known that the Bose-Einstein statistical mechanics lead to Planck's law of blackbody radiation. The solution with antisymrnetrical eigenfunctions, though, is probably the correct one for gas molecules, since it is known to be the correct one for electrons in an atom, and one would expect molecules to resemble electrons more closely than light-quanta' (Dirac, 1926f, p. 672). Enrico Fermi had also tended towards a similar opinion several months earlier when
632
The Golden Age of Theoretical physics
he applied Pauli's exclusion principle to his quantum theory of an ideal gas. This, and the identity of the other results, seem to suggest that Dirac knew about Fermi's work. When he was asked about it several decades later, he remarked: 'I had read Fermi's paper on Fermi statistics [probably the one in Zeitschrift fur Physikr. Fermi, 1926b] and forgotten it completely. When I wrote my work on the anti-symmetric wave functions, I did not refer to it at all. Then Fermi wrote and told me and I remembered that I had previously read about it' (Dirac, AHQP Interview, 7 May 1963, p. 4). This recollection of Dirac's can be substantiated for two reasons. First, a letter from Fermi to Dirac exists, stating: In your interesting paper 'On the Theory of Quantum Mechanics' [Dirac, 1926f], you have put forward a theory of the Ideal Gas based on Pauli's Exclusion Principle. Now a theory of the ideal gas that is practically identical to yours was published by me at the beginning of 1926 (Zs. f. Phys. 36, p. 902; Lincei Rend. February 1926). Since I suppose that you have not seen my paper, I beg to attract your attention on it. (Fermi to Dirac, 25 October 1926) Second, although Dirac had indeed seen Fermi's paper, the whole derivation was not to his way of thinking and he quickly forgot about the results, recreating them later by his own methods. In any case, Fermi's letter had the effect that Dirac later on never forgot to mention the priority of his Italian colleague when referring to the statistics obeyed by electrons and the like. In spite of this admitted priority of Fermi, it was essentially Dirac's paper that helped the physicists tremendously in understanding the meaning of the new statistical methods. This occurred first in Leyden, where Paul Ehrenfest and his student George Uhlenbeck, stimulated by Schrodinger's wave-theoretical derivation of Einstein's gas theory (Schrodinger, 1926b), had analyzed the BoseEinstein statistics. 35 In a note entitled lFuhrt die Bose-Einsteinsche Statistik zu einer Entartungscondensationf ('Does the Bose-Einstein Statistics Lead to a Degenracy-condensation?'), they arrived at the conclusion that it did not. They attributed Einstein's result, obtained earlier (Einstein, 1925a, p. 4), to an error in his evaluation of the sum
°° 7V
1
= Eexp(a + /3Es)-l'
(39)
expressing the number of molecules in a given volume, by means of an integral. They added jokingly: And one understands how happy were the Fermi-Dirac gases, that they did not even get into this embarrasment because of their positive sign [in the denominator of their equation corresponding to Eq. (39)] (' Und man begreift wie froh die Fermi-Dirac-Gase waren, daflsie wegen ihres Pluszeichens gar nicht erst 35 Uhlenbeck recalled that Schrodinger's paper 'made a great impression on Ehrenfest and me, because it was so clear' (Uhlenbeck, AHQP Interview, 10 May 1962, p. 1).
The Discovery of the Fermi-Dirac
Statistics
633
in diese Verlegenheit kamen').36 Einsten, to whom Ehrenfest had sent the draft, quickly responded (in a letter to Ehrenfest, dated 24 November 1926), pointing out that the condensation phenomenon also followed from a direct evaluation of the sum, Eq. (39). However, its very existence and the strange behaviour of the specific heat at low temperatures — which did not decrease monotonically to zero — made people quite unwilling to accept Bose's statistics as the method applying to massive microscopic particles. Another problem, which bothered the Leyden physicists, concerned the role of the old (classical) Boltzmann statistics within the quantum mechanical scheme. After the appearence of Dirac's paper, they thought in particular of how to express this statistics in terms of wave functions. Quickly the idea became 'very clear that . . . you had to take all the eigenfunctions, not only the symmetric ones,' recalled Uhlenbeck. He added: 'Well, it was a little exercise to show that it is equivalent to Boltzmann's statistics, and then Ehrenfest and I wrote a little note about it' (Uhlenbeck, AHQP Interview, 10 May 1962, p. 2). In that note, which was received by Zeitschrift fur Physik on 15 December 1926, Ehrenfest and Uhlenbeck demonstrated for a one-dimensional model of the ideal gas, 'how wave mechanics also allows, in a completely natural manner (ungezwungen), the interpretation of the older Boltzmann-Planck counting method [besides those of Bose-Einstein and Fermi-Dirac],' and that 'wave mechanics does not yet per se imply the refutation of Boltzmann's method' (Ehrenfest and Uhlenbeck, 1927a, p. 24). In a more mathematical language this means: if for a system of N identical particles in a one-dimensional tube of extension — 7r/2 < x < n/2 all eigensolutions ip = sin(AiiXi) sm(k2x2) • • • sin(fcjvarjv),
(40)
with ki, k2, • • • , k?tf as integral numbers, are taken for a given energy E(— ^2S NSES, where Es = (h2 /%ir2m)(k\ + k\ -\ h kjf), m the mass of the particles), then the number of different solutions is provided by Boltzmann's expression,
Ehrenfest and Uhlenbeck concluded: 'Not before certain definite solutions of the Schrodinger equations are selected, and the others ultimately rejected, would we be forced to give up Boltzmann's counting method' (Ehrenfest and Uhlenbeck, 1927a, p. 24). While Ehrenfest tried as much as possible to define the particular statistical method of his venerated teacher Ludwig Boltzmann, he did not fail to realize that Fermi's statistics seemed to offer a very nice feature. Again Uhlenbeck recalled: 'Then [i.e., after completing the embedding of Boltzmann's statistics into the wave mechanical scheme] Ehrenfest went away. I think he was in Paris... and I got 36 A copy of this draft-manuscript is contained in the Einstein-Ehrenfest correspondence in Einstein Archives.
634
The Golden Age of Theoretical
physics
suddenly a typical Ehrenfest postcard He says, 'Fermi statistics means the impenetrability of matter.' Exclamation sign; exclamation sign. 'See you in Leyden. Come ten o'clock in the morning.' (Uhlenbeck, AHQP Interview, 10 May 1962, p. 2). When Uhlenbeck showed up in Ehrenfest's office, the professor again developed the new idea in the one-dimensional gas model discussed above, whose eigenfunctions were of the type given by Eq. (40). Then, taking care of the 'reciprocal impenetrability of the molecules... by means of the additional 'diagonal restriction,' which in wave mechanics corresponds to the impossibility of two molecules occupying the same point in space at the same time' (Ehrenfest, 1927a, p. 196), he arrived at the anti-symmetrical solution of Heisenberg and Dirac (see Eq. (29b) above). 'If you . . . require that the wave function has to be zero if the co-ordinates of the two points are the same, then they [i.e., the molecules] cannot penetrate [each other],' Uhlenbeck reported and continued: 'Then he [Ehrenfest] thought that the Fermi statistics followed from that, which is true in one dimension but not in two or three' Uhlenbeck, AHQP Interview, 10 May 1962, p. 2). 37 The withdrawal of the 'proof of Fermi-Dirac statistics on the basis of the impenetrability of molecules, recalled above by Uhlenbeck, affected parts of the results obtained in a joint paper of Ehrenfest and Uhlenbeck, on Einstein's mixing paradox (Ehrenfest and Uhlenbeck, 1927b; see our Footnote 37). This paradox which seemed to exist in both Bose-Einstein and Fermi-Dirac statistics, claimed that at low temperatures the pressure and the average density of an ideal gas differs from those derived for a gas mixture containing two types of gas molecules whose properties are infinitesimally close.38 In a wave-mechanical treatment the paradox could be shown to disappear for anti-symmetric wave functions. Since the mutual impenetrability of the molecules does not suffice, however, in order to exclude the symmetrical wave functions, the Einstein paradox could still show up in nature, namely in the case of gases obeying Bose-Einstein statistics. Viewed from the final outcome of the development of the quantum-statistical methods during those years, the endeavours of Ehrenfest and his close associate Uhlenbeck — who obtained his doctorate soon afterwards — did not lead to important, lasting results. They served, however, to illustrate beautifully the difficulty of grasping the physical content of these somehow singular, rather unexpected, revolutionary theories of Bose-Einstein, on the one hand, and those of Fermi and Dirac, on the other hand. Although it ultimately did not hold, the close relationship of the impenetrability of matter and Fermi-Dirac statistics seemed to support the 3
Later that spring Ehrenfest published an addendum to his note in Nature, in which he admitted: 'It is not true t h a t the reciprocal impenetrability of the molecules allows only the HeisenbergDirac determinant solutions, and excludes all others. On the contrary, all the symmetrical and antisymmetrical characteristic solutions which existed for absolutely penetrable molecules remain for a (not one-dimensional) gas with molecules having a very small radius compared with the mean distance.... If, therefore, the Pauli principle is valid... for the translatory motion of such gas molecules, with radius almost zero, then such a remarkable relation between t h e molecules cannot simply be explained by wave mechanics as my mistake led me to believe' (Ehrenfest, 1927b, p. 602). 38 See Einstein, 1924c, p. 267.
The Discovery of the Fermi-Dirac Statistics
635
view expressed by several physicists during that period, namely that all material particles, whether atoms or electrons, obeyed Fermi-Dirac statistics. As a typical example of this view, we quote Wolfgang Pauli, who wrote in December 1926: 'Since it is hardly possible to select consistently, for different material particles, the quantum mechanical solution realized in nature from all the possible solutions according to the different points of view, we shall adopt here the viewpoint, defended by Dirac, that for a material gas Fermi's statistics is valid, and not that of Bose and Einstein' (Pauli, 1927a, p. 84). In his paper entitled 'Uber Gasentartung und Paramagenetismus' ('On Gas Degeneracy and Paramagnetism'), Pauli, the inventor of the exclusion principle, did not use the argument of the impentrability of matter; he rather proposed two other arguments in favour of applying the Fermi-Dirac statistics namely: (i) material systems must be different from radiation because they possess zeropoint energy (which, in Pauli's opinion, could not be associated with BoseEinstein systems); (ii) light-quanta and material particles exhibit a fundamental difference, insofar as the former can be represented by de Broglie waves in the ordinary threedimensional space but not the latter, and therefore they must obey different quantum statistics. Pauli's paper (Pauli, 1927a) appeared in print almost simultaneously with that by Paul Dirac on 'The Qunatum Theory of the Emission and Absorption of Radiation,' communicated at the beginning of February to the Proceedings of the Royal Society of London (Dirac, 1927b). In the latter, Dirac investigated especially the interaction of atoms with electromagnetic waves and showed: 'If one takes the energies and phases of the waves to be g-numbers satisfying the proper quantum conditions instead of c-numbers, the Hamiltonian function takes the same form as in the lightquantum treatment. The theory leads to the correct expression for Einstein's A's and B's' (Dirac, 1927, p. 265). That is, the quantum-mechanical formulation of the electromagnetic waves with g-numbers or operators allowed one to derive the well-known emission and absorption coefficients that Einstein had postulated fully a decade before (Einstein, 1916d). Dirac further showed that the same result also followed from treating light-quanta according to Bose's statistical method. Again half a year later, in July 1927, Pascual Jordan went a step further. He first interpreted Dirac's conclusion as a demonstration of 'how Einstein's idea, namely to represent the ideal material gas in analogy to the light-quantum gas by quantized waves in the usual three-dimensional space, can be carried out rigorously in quantum mechanics' ('wie der Einsteinsche Gedanke, das ideale materielle Gas analog zum Lichtquantengas darzustellen durch gequantelte Wellen im gewohnlichen dreidimensionalen Raume, quantenmechanisch exakt durchgefurt... werden kann,' Jordan, 1927, p. 473). The main content of his paper, 'Zur Quantenmechanik der Gasentartung' ('On the Quantum Mechanics of Gas Degeneracy'), however, was to develop 'a corresponding theory for the ideal Fermi gas' ('erne entsprechende Theorie fur
636
The Golden Age of Theoretical physics
das ideale Fermische Gas,' Jordan, 1927, p. 473). Actually he found a method of quantizing waves that could be associated with electrons obeying Fermi's statistics, from which he concluded: 'The results obtained hardly seem to leave any doubt that — despite the validity of Pauli's, rather than Bose's, statistics for the electrons — a quantum mechanical wave theory of matter can be developed, in which the electrons are described by quantized waves in the ordinary three-dimensional space The Schrodinger eigenfunctions of the matter waves constructed by Dirac and Heisenberg play within this picture a role which in no way lets them appear as an analogue to the electromagnetic waves. They rather turn out to be a special case of the general probability amplitudes, which have to be used as a mathematical device to describe the statistical behaviour of the quantized light and matter vibrations' (Jordan, 1927, p. 480). This conclusion, of course, proceeds far beyond the developments that took place in 1926; it rather illustrates the situation in the physical interpretation of quantum mechanics at an advanced state — which we shall present later. Here we restrict ourselves to stating that the papers of Fermi, and especially those of Dirac, led to the establishment of a statistical method, different from the one invented earlier by Bose and explored further by Einstein. Moreover, in his above-mentioned paper, Pauli drew conclusions by applying Fermi statistics to the electron gas in metals, thus arriving 'at least [at a] qualitative understanding of the [empirical] fact that, in spite of the existence of the intrinsic angular momentum of the electron [i.e., the electron spin], many metals — in particular, the alkali metals — exhibit no paramagnetism, or only very weak paramagnetism, which is approximately independent of the temperature, in their solid state' (Pauli, 1927a, p. 81). Pauli's work constituted the beginning of the quantum theory of metal electrons, which was continued soon afterwards with even more spectacular success by Arnold Sommerfeld and Enrico Fermi. 39 During the couple of years following the discovery of Fermi statistics, the relation between the symmetry of wave functions and the statistical methods became firmly established. In particular, it was shown that protons, like electrons, obeyed the exclusion principle (see, e.g., Dennison, 1927b), and the detailed analysis of the collision processes of microscopic particles was used to determine the symmetry of wave functions: for example, the a-particles seemed to obey Bose statistics (Mott, 1930). Ultimately, the observation was confirmed that particles with half-integral spin obeyed Fermi statistics, and those with spin zero or one (in units of h/2n) Bose statistics. The work of Fermi and Dirac would provide the basis for numerous future discoveries in atomic, nuclear and subnuclear physics during the following decades, spreading the fame of the two physicists. 40 39 The earliest application of Fermi's statistics to matter in general was made by Ralph Fowler, who tried to explain with its help the properties of the dense matter in the interior of stars (Fowler, 1926). 40 Both Fermi and Dirac would make important contributions to many of these discoveries, which we shall discuss later on. In Fermi's perhaps greatest theoretical work, his theory of /3-decay, particles obeying Fermi statistics also played a major role.
The Discovery of the Fermi-Dirac Statistics
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The Golden Age of Theoretical Physics The Golden Age of Theoretical Physics brings together 37 selected essays. Many of these essays were first presented as lectures at various universities in Europe and the USA, and then published as reports or articles. Their enlarged, final versions were published in the joint work of Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, while the other essays were published as articles in scientific journals or in edited books. Here they are published together as a tribute to the Mehra-Rechenberg collaboration sustained for several decades, and cover various aspects of quantum theory, the special and general theories of relativity, the foundations of statistical mechanics, and some of their fundamental applications. Two essays, 'Albert Einstein's "First" Paper' (Essay 1) and 'The Dream of Leonardo da Vinci' (Essay 37), lie outside the major themes treated in this book, but are included here because of their historical interest. The origin of each essay is explained in a footnote. This book deals with the most important themes developed in the first 40 years of the twentieth century by some of the greatest pioneers and architects of modern physics. It is a vital source of information about what can veritably be described as 'the golden age of theoretical physics.'
Jagdish Mehra Hoping lo become a writer, Jagdish Mehra pursued studies in order to write about the historical and conceptual development of modern physics, especially quantum theory, on the advice of his hero and guide Aldous Huxley from his early youth. He was trained as a theoretical physicist in the research schools of Wemer Heisenberg and Wolfgang Pauli. and later came under the strong influence of P.A.M. Dirac. From the beginning of his scientific career he has devoted himself to his chosen field. Through his close contact with nearly all the major architects of quantum theory and their collaborators and successors, who made fundamental applications of the emerging field to all branches of modern physics, he developed extensive new materials that have been used in his investigations. A widely published author, Jagdish Mehra has written and given lectures extensively on the historical development of the quantum and relativity theories, and edited books honoring the work of P.A.M. Dirac (The Physicist's Conception of Nature), Joseph Maria Jauch (Physical Reality and Mathematical Description, cocdited with Charles P. Enz), and Eugene Paul Wigner (The Collected Works of Eugene Paul Wigner, coedited with Arthur S. Wightman in eight volumes). Early in 1970. with the active encouragement and warm recommendation of Werner Heisenberg, Mehra arranged to bring Helmut Rechenberg to the University of Texas in Austin, where together they organized Mehra's extensive archives, developed a detailed outline of a major work on The Historical Development of Quantum Theory, and initiated their research partnership. Beginning in January 1975, they worked together at the University of Geneva. Switzerland, where Mehra had been appointed a Swiss National Science Foundation Invited Professor in December 1974. and Rechenberg, with the direct intervention of Heisenberg, was supported by the Deutsche Forschungsgemeinschafr, their joint researches and writing were continued at the International Solvay Institutes of Physics and Chemistry (Universite libre de Bruxelles), Brussels. Belgium, where Mehra was also appointed as an
Institute Professor, and then in Munich, Germany, and Houston. Texas, USA. where they worked in close collaboration. Their work The Historical Development of Quantum Theory was published in six volumes (nine books) by Springer-Verlag New York (1982, 1987.2000). During this period they also worked on other books and articles. Jagdish Mehra worked (with Arthur S. Wightman) on The Collected Works of Eugene Paul Wigner (SpringerVerlag, 1993-2000), and wrote the books The Beat of a Different Drum: The Life and Science of Richard Feynman (Oxford University Press, 1994), Einstein, Physics and Reality (World Scientific, 1999), and Climbing the Mountain: The Scientific Biography of Julian Schwinger (with Kimball A. Milton; Oxford University Press, 2000). Helmut Rechenberg coedited the multivolume Collected Papers of Werner Heisenberg (Springer-Verlag and R. Piper Verlag. 1984-1993); he also authored several books, including a biography of Hermann von Helmholtz (1994) and The Origin of Nuclear Forces (1996, with Laurie Brown). Theirs has been a close, loyal, and intense collaboration, and Mehra has dedicated this book of essays lo Rechenberg in his honor. Jagdish Mehra has held prestigious academic positions. Apart from those mentioned, in 1978 he was appointed University Distinguished Professor of Sciences and Humanities at the University of Houston. He also served as Andrew D. White Senior Scholar at the Society for the Humanities at Cornell University, Regents' Professor at the University of California at Irvine, and UNESCO-Sir Julian Huxley Distinguished Professor of Physics and the History of Science in Paris, France, and Trieste, Italy (1989-1991). He has lectured extensively at universities in Switzerland, Italy, Germany, Scandinavian countries and Spain, but most of all at Cambridge and Oxford in the UK, apart from many of the most prestigious academic institutions in the USA, where he continues to be associated with the University of Houston, Texas. In 1976. the Alexander von Humboldt Foundation honored him with the U.S. Distinguished Senior Scientist Award (the Humboldt Prize).
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