Early Theories of the Electron Gas

Early Theories of the Electron GasAuthor(s): Walter KaiserSource: Historical Studies in the Physical and Biological Sciences, Vol. 17, No. 2 (1987), pp. 271-297Published by: University of California PressStable URL: http://www.jstor.org/stable/27757584 .

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WALTER KAISER*

Early theories of the electron gas

Although heinrich hertz' discovery of electromagnetic waves seemed to have confirmed Maxwell's electrodynamics, in fact, it left an astonishing number of questions unanswered. The material basis of the theory?electric current, electric resistance, and the reaction of

matter to rapidly changing fields, as in the dispersion of light in dielectrics and the absorption of light by metals remained vague.1 Physicists like Oliver Heaviside and Hertz were well aware of this

incompleteness despite their appreciation of Maxwell's theory. In letters to Hertz, Heaviside spoke of the "embryonic state" and "skeleton-framework" of Maxwell's theory. In Heaviside's opinion,

*Fontanestrasse 26, D-6500 Mainz 31, Federal Republic of Germany. This research was supported by a grant from the Johannes Gutenberg-Universitat Mainz and by the

visiting scholar program of the University of California, Berkeley. Without the kind

hospitality of the Office for History of Science and Technology and stimulating discus

sions with John L. Heilbron this paper would not have been possible. i also thank Da

vid Cahan, Michael Eckert, Karl Hufbauer, Bruce R. Wheaton, and Norton Wise for

valuable information and criticism.

The following abbreviations are used: AHES, Archive for history of exact sciences;

AHQP, Archive for history of quantum physics, University of California, Berkeley; AP, Annalen der Physik; BCW, Niels Bohr, Collected Works, vol. 1 (Amsterdam and New

York, 1972); BSC, Bohr Scientific Correspondence in AHQP; DM, Deutsches Museum,

Munich; EMW, Felix Klein, ed., Encyklopddie der mathematischen Wissenschaften; JRE, Jahrbuch der Radioaktivitat und Elektronik; LCP, Hendrik Antoon Lorentz, Col

lected Papers (9 vols.; The Hague, 1934-39); LTZ, Hendrik Antoon Lorentz, Correspon

dence, Algemeen Rijksarchief, Den Haag; PM, Philosophical magazine; PR, Physical re

view; PZ, Physikalische Zeitschrift; RDN, O.W. Richardson Collection, University of

Texas, Austin; SB, Akademie der Wissenschaften, Berlin, Sitzungsberichte, SW, Aka

demie der Wissenschaften, Vienna, Mathematisch-Naturwissenchaftliche Classe, 2 Abt.,

Sitzungsberichte; WWN, Wilhelm Wien-Nachlass, Staatsbibliothek Preussischer Kultur

besitz; ZPC, Zeitschrift fur physikalische Chemie; ZSP, Zeitschrift fur Physik Microfilms of LTZ, RDN, and WWN are available at the Office for History of Science

and Technology, Berkeley, and other depositories of AHQP. 1. J.C. Maxwell, A treatise on electricity and magnetism (2 vols., 3rd. ed., Oxford,

1904), 216-218 (current); 362-363 (resistance); 437-438 (dispersion of light), 446-447 (metal optics).

HSPS, 17:2 (1987)



272 KAISER

electrodynamics had to be a molecular theory as well as the macros

copic description Maxwell tried to sustain.2 Nonetheless, British elec

trodynamics only gradually freed itself from Maxwell's favorite idea of

continuously distributed "electricity." Joseph Larmor's long and com

plicated path toward electron theory is an indication of the hold Maxwell's idea had on the first generation of Maxwellians.3

In contrast to the British school, Hendrik Antoon Lorentz worked to develop Wilhelm Weber's electrodynamics of moving charges.4 Lorentz' first step, completed in 1892, moved toward an explanation of the dielectric constant, of dispersion, and of light propagation in a

moving dielectric substance.5 It did not concern electrical conduction and resistance. But Weber himself had already developed an ela borate microphysical model of electrical resistance, which was

equivalent to a limitation on the free paths of carriers by their colli sions with the molecules of conductors.6 More influential at first, how

ever, was Weber's famous?or notorious?model of the fundamental interactions of electrodynamics: An electric current consists of oppo site electric charges moving with equal but opposite velocities. An

example of its wide dissemination is Boltzmann's use of it in 1886 to

explain the different signs of the Hall coefficient in different conduc tors.7

2. Heaviside to Hertz, 13 Jul and 14 Aug 1889 (DM/2924-2925). 3. Jed Z. Buchwald, "The abandonment of Maxwellian electrodynamics: Joseph

Larmor's theory of the electron," Archives internationales d'histoire des sciences, 31

(1981), 135-180, 373-438; From Maxwell to microphysics: Aspects of electromagnetic

theory in the last quarter of the nineteenth century (Chicago and London, 1985), 141

173.

4. H.A. Lorentz, "La theorie electromagnetique de Maxwell et son application aux

corps mouvants," Archives neerlandaises des sciences exactes et naturelles, 25 (1892), 363-553.

5. Ibid., 463-527.

6. W. Weber, "Ober die Bewegung der Elektricitat in Korpern von molecularer Con

stitution," AP, 156 (1875), 1-61, on 34-41, 49-55.

7. W. Weber, uElektrodynamische Maassbestimmungen. Ober ein allgemeines

Grundgesetz der elektrischen Wirkung [1846]," in Weber, Werke, 3 (Berlin, 1893), 25

214, on 132-136; L. Boltzmann, "Zur Theorie des von Hall entdeckten elektromag netischen Phanomens," SW, 94 (1886), 644-669, on 645-647.



ELECTRON GAS THEORY OF METALS 273

1. THE LEGITIMACY OF THE ELECTRON GAS MODEL

The theories of Riecke and Drude

In 1898 Eduard Riecke tried to bring together for the first time in a mathematical theory the qualitative attempts toward a theory of

conductivity made since the work of Weber, who had been his teacher. His approach, which he designed to supersede Maxwell's, rested primarily on the theory of electrolysis and on experimental results in the field of gas discharge.8

Riecke combined Weber's microphysical explanations of resistance and conduction. He assumed that both positive and negative electri cal particles move "in the space between the ponderable molecules

[the metal atoms];" a redundancy that seemed necessary to him on the evidence of electrolysis and experiments with cathode rays and canal

rays.9 Though redundant in character, these carriers were few enough in number that Riecke ignored their interactions: Only their collisions

with metal atoms had to be taken into account. His theory was not

yet an electron gas theory, but rather a "geometry of particle trajec tories."10

Riecke divided these trajectories into straight free paths and curved "molecular paths," or collisions. He assumed that the mean kinetic energy of the charge carriers is proportional to the tempera ture, but he could not, on his very general theory, determine the value of the constant of proportionality. His theoretical expressions there fore remained much too general or flexible for experimental testing. This is true even for that very important quantity in the electron gas theory of metals, namely the ratio of the expressions for the conduc tivities of heat and electricity, i.e., the theory of the famous law of Gustav Wiedemann, Rudolf Franz, and Ludvig Valentin Lorenz.11

8. E. Riecke, "Zur Theorie des Galvanismus und der Warme," AP, 66 (1898), 353?

389, 545-581, on 352-356, 569-572. Riecke himself had given a kinetic treatment of

conductivity and diffusion in electrolytes: E. Riecke, "Molekulartheorie der Diffusioi} und Elektrolyse," ZPC, 6 (1890), 564-572.

9. E. Riecke, "Galvanismus" (ref. 8), on 355-356, 569-571. At first Riecke

identified positive carriers with metal ions, but soon changed his mind; Riecke, "1st die

metallische Leitung verbunden mit einem Transport von Metallionen?," PZ, 2 (1900

01), 639.

10. R. Seeliger, "Elektronentheorie der Metalle (1921)," in EMW, 5:2 (Leipzig,

1904-22), 777-878, on 782.

11. E. Riecke, "Galvanismus" (ref. 8), 379-381; cf. E. Riecke, "iCber die Elek

tronentheorie des Galvanismus und der Warme," JRE, 3 (1906), 24-47, on 32-34, and

Seeliger (ref. 10), 784-785.



274 KAISER

The breakthrough to an electron gas theory came two years later, in 1900, in two lengthy papers by Paul Drude. For some time Drude had hesitated to accept Maxwell's theory. When he did accept the

electromagnetic theory of light, he combined it with the kinetic ele ments of the optical theory he had developed. In his electron theory of metals, he aimed at a still higher level of integration, of the kinetic

theory of gases with electrodynamics.12 In adopting Maxwell's theory of light, he had to resolve one of the puzzles Maxwell's continuity theory had left unsolved, namely the mechanism of reflection at metal surfaces. His solution had been based on a reaction of free ("conduc tivity") electrons and of bound ("isolating") electrons to rapidly changing electromagnetic fields.13 From this approach, his free elec tron gas theory of metals developed easily; indeed, he designed it as a

means of investigating the relative numbers of free and bound elec trons to further his theory of metallic reflection.14

With the success of the extension of the kinetic theory of gases to

electrolytic phenomena in mind, Drude inferred the possibility of a full transfer of the theory to the carriers of metallic conduction. The crucial point in the transfer was taking over Boltzmann's equipartition theorem

l/2ra1v12+l/2m2v22+ + l/2rnnvn2 = aT

and the numerical value of the "universal constant" a:15

These laws for gases obviously have now proven valid also for the osmotic pressure exercised by ions in electrolytes, and not only in a for mal sense but also with the same numerical constants. If a metal is immersed in an electrolyte in "temperature equilibrium," the free electrons.. .in the metal have the same kinetic energy as the ions in the electrolyte. Therefore we also have.. .[in the case of free electrons] the same constant a that occurs in the gas laws.

For the physicists of the time this was probably the decisive idea in Drude's theory. That was certainly the basis of Arnold Sommerfeld's

optimism as we learn from some of his lecture notes for the summer semester of 1912. Drude's "transfer of the numerical value furnished

by the gas theory" appears there as the "salient point" of the early

12. P. Drude, Physik des Aethers auf elektromagnetischer Grundlage (Stuttgart, 1894); P. Drude, "Zur Elektronentheorie der Metalle," AP, 1 (1900), 566-613, and AP, 3

(1900), 369-402. 13. P. Drude, "Zur Ionentheorie der Metalle," PZ, 1 (1900), 161-165. 14. Drude, "Elektronentheorie" (ref. 12), 568, 584. The derivation of the number of

free electrons from the optical properties of metals gave only a very rough numerical ap

proximation. 15. Drude "Elektronentheorie" (ref. 12), 572.




electron gas theory of metals.16 Drude's own calculations were based on a very general dual

current, whose carriers might have arbitrary integer multiples of the

elementary charge. Using a diffusion equation from Boltzmann's "gas theory,"17 Drude derived an expression for the flow of heat carried by the moving electrons down a temperature gradient. Next, taking into account the consequences of collisions between electrons and metal

atoms, he calculated the constant drift velocity of electrons in an elec tric field. Finally, specializing to singly charged positive or negative carriers, he derived a ratio of the coefficients of thermal and electrical conductivities that did not involve the typically "kinetic" quantities mean free path, concentration or number of carriers, velocity of car riers. The outcome was a theoretical expression for the empirical

Wiedemann-Franz law: k/<t = 4/3(a/e)2 T. Here k and a are respec tively the coefficients of thermal and electrical conductivity, a is 3R /2, R is the gas constant, e the elementary charge, and T the abso lute temperature.18

It was of course no surprise to find so close a relationship between the conductivities of heat and electricity. What was exciting in Drude's theory was the possibility of relating the constant of the

experimentally established Wiedemann-Franz law to the gas constant and to the value of the elementary charge. With Drude's theory it became possible to explain an important property of solid matter with the help of a basic thermodynamic constant, which had a distinct

microphysical meaning, or, if one preferred, to deduce the value of a constant in the theory of gases from the electric behavior of metals.19 Drude's expression for the Wiedemann-Franz law agreed brilliantly with the measurements of Wilhelm Jaeger and Hermann Diesselhorst at the Physikalisch-Technische Reichsanstalt in Berlin.20

16. A. Sommerfeld, "Elektronenth[eorie] d[er] Metalle und verwandte statistische

Fragen" (AHQP 21:8). The date given in AHQP is summer semester 1908; Michael

Eckert (ref. 34), 202, has pointed out that the correct date must be summer semester

1912. For the importance of the transfer of the gas constant to the electron gas, see E.

Riecke, "Uber das Verhaltnis der Leitfahigkeiten der Metalle fur Warme und fur

Elektricitat," AP, 2 (1900), 835-842, on 835, and "Die jetzigen Anschauungen iiber das

Wesen des metallischen Zustandes," PZ, 10 (1909), 508-518, on 509.

17. Drude, "Elektronentheorie" (ref. 12), 573. The passage in Boltzmann's lectures cited by Drude precedes a chapter on electric conductivity and the viscosity of gases. L.

Boltzmann, Vorlesungen iiber Gastheorie (2 parts, Leipzig, 1896-1898), 1, 77-80. 18. Drude, "Elektronentheorie" (ref. 12), 577-578.

19. H.A. Lorentz, "Le mouvement des electrons dans les metaux," Archives neerlandaises des sciences exactes et naturelles, 10 (1905), 336-371, reprinted in LCP, 3, 179-214.

20. W. Jaeger and H. Diesselhorst, "Warmeleitung, Electricitatsleitung, Warmecap acitat und Thermokraft einiger Metalle," SB (1899), 719-726. Jaeger and Diesselhorst were members of the division for electric measuring at the Physikalisch-Technische



276 KAISER

The experimental confirmation of the theoretically derived Wiedemann-Franz law in turn reemphasized the outstanding impor tance of the gas constant. It is not surprising that Max Planck, who

always thought in terms of universal constants, called it the "Boltzmann-Drude constant."21 As a further reward, Drude was made an ordinary member of the Prussian Academy of Sciences. His nomi nation was drawn up by Emil Warburg and signed by Max Planck

among others. It stressed Drude's "fundamental result" that "the ratio of thermal conductivity and electrical conductivity is a universal constant proportional to the absolute temperature."22 The surprising beauty and simplicity of the fundamental result were often extolled.23

It was a great strength, or perhaps a great weakness, in the model that for elementary calculations like those implicated in the deduction of the Wiedemann-Franz law the same numerical relationships resulted whether one assumed positive or negative carriers. Still, a natural philosopher might demand to know whether positive electri

city was similar to negative in all respects but sign. Experiment seemed to say no. More and more precise values of the specific charge elm of the negative electrons obtained from bending cathode rays or from measuring the Zeeman effect always came out to be over a thousand times the specific charge of the positive carriers occurring in

gas discharges.24 Contemporary uncertainty and discomfort about the nature of

positive electricity may easily be documented, for example, in the

experimental papers of Julius Edgar Lilienfeld of Leipzig or in the ter

minology in the publications of Willy Wien.25 And Planck tried to

Reichsanstalt from 1887 to 1927 and from 1894 to 1910, respectively. Their paper was

presented to the Prussian Academy by Friedrich Kohlrausch, then president of the PTR, as a supplement to his own paper entitled "Uber den stationaren Temperaturzustand eines von einem elektrischen Strome erwarmten Leiters," ibid., 711-718.

21. |M. Planck, "Uber die Elementarquanta der Materie und der Elektrizitat," AP, 4

(1901), 564-566.

22. Emil Warburg, "Wahlvorschlag fur Paul Drude," in Christa Kirsten, Hans Giinther Korber, and Hans-Jurgen Treder, eds., Physiker uber Physiker, Wahlvorschlage zur Aufnahme von Physikern in die Berliner Akademie 1870-1929 (2 vols., Berlin, 1975), 1, 164.

23. E.g., M. Reinganum, "Theoretische Bestimmung des Verhaltnisses von Warme und Elektricitatsleitung der Metalle aus der Drude'schen Elektronentheorie," AP, 2

(1900), 398-403, on 399, 403. 24. J.J. Thomson, Die Korpuskulartheorie der Materie (Braunschweig, 1908), 25-26. 25. J.E. Lilienfeld, "Uber neuartige Erscheinungen in der positiven Lichtsaule der

Glimmentladung," Deutsche Physikalische Gesellschaft, Verhandlungen, 9 (1907), 125

135, on 132, and "Die Elektrizitatsleitung im extremen Vakuum (Leipziger Habilita

tionsschrift), AP, 32 (1910), 673-738, on 675, 736-737; W. Wien, "Untersuchungen tiber die elektrische Entladung in verdiinnten Gasen," AP, 8 (1902), 244-266, and "Uber die Natur der positiven Elektronen," AP, 9 (1902), 660-664, on 660.

8.




stimulate Wien's experimental research into positive rays on the

ground that "almost nothing at all" could be deduced about positive electricity by theory alone. Similarly, Alfred Bucherer complained to Lorentz that lack of a "clear insight into the nature of positive electri

city" made derivation of Planck's radiation formula from electron gas theory especially difficult.26 Even after Thomson's atomic model of 1906 one could not do entirely without the positive electron, which seemed implicated in experiments on the Faraday effect.27 As late as 1908 and 1909, J.J. Thomson had not ruled out the existence of some

thing analogous to the extraction of negative electrons from different substances "in the case of positive electricity."28

2. LORENTZ' ELECTRON GAS THEORY OF METALS

The primary reason for retaining carriers of both signs in the elec tron gas theory of metallic conduction was the Hall effect. The lead

ing theorists, however, preferred simplicity, downgraded the Hall

effect, and, like J.J. Thomson and Arthur Schuster, usually admitted

only negative corpuscles.29 Lorentz too based his theories on negative electrons despite his own lingering doubts and Drude's objections drawn from galvanomagnetic effects.30

26. Planck to Wien, 16 Mar 1903 (WWN/5); Bucherer to H.A. Lorentz, 31 Mar 1912

(LTZ). 27. As a tentative explanation of the opposite rotation of polarized light in a magnet

ic field in the region of absorption bands, R.W. Wood suggested negative and positive

electrons; Wood, "On the existence of positive electrons in the sodium atom," PM, 15

(1908), 274-279. See also Jean Becquerel, "On the dispersion of magnetic rotatory power in the neighbourhood of bands of absorption in the case of rare earths;" ibid., 16

(1908), 153-161, on 155, in which he reported results that, he wrote to Lorentz, "give the existence of a positive electron a degree of probability close to certainty." J. Bec

querel to Lorentz, 11 Jul 1908 (LTZ). Cf. Lilienfeld (ref. 25), 675.

28. J.J. Thomson, "Positive electricity," PM, 18 (1909), 821-845, on 821. For objec tions against J.J. Thomson's views see W. Wien, "On rays of positive electricity," ibid., 14 (1907), 212-213; O.W. Richardson, "The specific charge of the ions emitted by hot

bodies," 76 (1908), 740-767.

29. J.J. Thomson, "Some speculations as.to the part played by corpuscles in physical

phenomena," Nature, 62 (1901), 31-32, and "Indications relatives a la constitution de

la materiere fournies par les recherches recentes sur la passage de l'electricite a travers

les gaz," Congres international de Physique, Rapports, (4 vols. Paris, 1900-1901), 3,

138-150; A. Schuster, "On the number of electrons conveying the conduction currents in metals," PM, 7(1904), 151-157, on 157.

30. P. Drude, "Optische Eigenschaften und Elektronentheorie," AP, 14 (1904), 677

725, 936-961, on 679, 939. As early as 1884 H.A. Lorentz had pointed out that any

theory of the Hall effect based on moving "electric particles" must assume an asym

metry in the behavior of "positive and negative electricity." H.A. Lorentz, "Le

phenomene decouvert par Hall et la rotation electromagnetique du plan de polarisation de la lumiere," LCP, 2, 136-163, on 141-142; cf. Lorentz, "Weiterbildung der



278 KAISER

In a most important paper of 1905, Lorentz introduced almost

apologetically the assumption that "the metal only contains a single kind of free electrons all of which possess the same charge e and the same mass m." He was still looking over his shoulder at the Hall effect and declared himself ready to face the difficulties of free carriers of both signs should they be required. Among the difficulties he foresaw was creation of energy in the recombination of the oppositely charged particles, a process in apparent violation of the second law of

thermodynamics.31 Lorentz' clean if brutal choice of the unitary theory eventually won

the support of Riecke. In his important encyclopedia article of 1921,

Seeliger considered the choice and its consequences the only exception to his characterization of the early electron theory of metals as "stag nant." He felt that "the bold idea" of assuming only a single kind of carrier and of identifying this carrier with the otherwise familiar elec tron simplified and unified "the whole physical world view and, in

general, should be considered a great success."32 Another influential contribution of Lorentz' theory of metals was

his use of advanced methods of statistical mechanics. Riecke and Drude had considered only mean velocities in their calculations.

Lorentz, however, assumed that the particles in his electron gas pos sess the Maxwell-Boltzmann distribution of velocities. By an addi tional small term he was able to take external electric fields, tempera ture gradients, and, eventually, contact phenomena into account. His efforts produced, among much else, a Wiedemann-Franz law in which, however, his numerical factor of 8/9 gave a poorer fit than Drude's factor of 4/3. The better theory gave less precise agreement with

experiment. Lorentz described the deviation between his theory and the experi

mental Wiedemann-Franz law as "not inconsiderable." It, and other

difficulties, made him doubt the legitimacy of handling electrons in solid matter according to the kinetic theory of gases, the justification of taking over the equipartition theory from the theory of the gaseous state, and the ascription of enormously high velocities to charged par ticles in densely packed matter. Nonetheless, it was not the shortfall or the doubts that he emphasized when discussing the electron gas

theory in 1905, but the astonishing concordance of numbers related to the absolutely separate subjects of ideal gases and metallic conduc tion.33

Maxwellschen Theorie. Elektronentheorie [1903]," EMW, 5:2 (Leipzig, 1904-1922),

145-280, on 222.

31. H.A. Lorentz (ref. 19), 180, 206-208, 214.

32. Riecke, "Anschauungen" (ref. 16), 509; Seeliger (ref. 10), 779.

33. H. A. Lorentz (ref. 19), 103-104, 109, 194.




It appears that, like Drude, Lorentz came to join these separate subjects over the bridge of the theory of dilute solutions. In 1909 he

justified the transfer of the gas model, including the equipartition theory, to the liquid and solid states by arguing from success. The kinetic theory had worked in van't HofFs approach to osmotic pres sure, in Nernst's theory of concentration cells, and in Jean Perrin's test of Einstein's and Smoluchowsky's theories of Brownian motion.

Why then feel surprise at its reappearance as a "successful first

approximation" in the theory of metallic conduction?34 The closeness of the tie between the kinetic theory of dilute solu

tions and the electron gas theory may be seen in the details of termi

nology. In his influential theory of the Hall effect, which he based on

Lorentz' statistics, Richard Gans explained the limitation of the transverse flow of electricity owing to the growing concentration of carriers in terms of an "osmotic force" acting on the metal electrons. Another good example is Paul Hertz' suggestion that Lorentz' electron

gas theory of metals might be used to explain transport phenomena in

electrolytes. He saw an analogy so strict that it could be used to deduce consequences for the original domain from refinements of the

theory introduced for application in the new domain.35 The mutual reinforcement of the kinetic theories of gases, solutions, and metals transformed them all, in the opinion of Henri Poincare, from the realm of the ingenious but doubtful to that of the probable.36 A most

interesting echo of this opinion occurs in a letter by Hermann Haga nominating Perrin for the physics prize for the year 1918:37

In support of this proposal I may be allowed to quote the following pas sage from Prof. Ernst Mach's Die Mechanik in ihrer Entwicklung: "[Newton].. .had the habit.. .of clinging as far as possible to any idea he had once conceived even in cases where the original circumstances did not hold, and of preserving in his ideas the same uniformity that nature shows us in her workings. What has once shown itself somewhere to be

34. H.A. Lorentz, "On the molecular motion of dissolved substances [1889]," LCP,

6, 112-113; "Sur la theorie moleculaire des dissolutions diluees [1892]," ibid, 114-133; "The methods of the theory of gases extended to other fields [1909]," LCP, 8, 159-182, on 176; The theory of electrons.. A course of lectures delivered at Columbia University, New York, in March and April 1906 (Leipzig, 1909), 10.

35. R. Gans, "Zur Elektronenbewegung in Metallen," AP, 20 (1906), 293-326, on

310; Hertz to Lorentz, 4 and 20 Mar 1908, 7 Apr 1911 (LTZ). 36. H. Poincare, "Les rapports de la matiere et de Tether [1912]," Oeuvres, 9 (Paris,

1954), 669-682, on 670.

37. H. Haga to Comite Nobel pour la physique, Stockholm, 2 Jan 1918 (LTZ). In

the Berkeley microfilm the copy is misfiled under "Lan, Perrin." Cf. J. Perrin, "Discon

tinuous structure of matter," Nobel lectures.. .Physics, 1922-1941 (4 vols., Amsterdam,

London, New York, 1965), 3, 138-164, on 145.



280 KAISER

a property of Nature recurs, even if not equally obviously, always and

everywhere." That is the principle of continuity that has guided M. Per rin in his researches_It is.. .the principle of continuity that caused him to compare a droplet of emulsion in a beaker 0.1 millimeter in

depth with the earth's atmosphere.

A new element: the electron vapor

The electron gas soon gave birth to an electron vapor. The idea

originated in the work of Harold A. Wilson, Owen W. Richardson, J. J.

Thomson, Friedrich Kriiger, and Karl Baedeker. Thomson,38 Kriiger,39 and Baedeker40 dealt with the transition of the electron

vapor from one solid phase to another. For example, Baedeker evaluated the thermoelectric work as the work necessary to transfer an electron vapor in a reversible way from one metal in a thermocouple (with a characteristic vapor pressure) to the other metal (with a different vapor pressure) and back to the starting point of the thermo

dynamic cycle. Wilson and Richardson concentrated on the thermo

dynamics of the system composed of the solid phase and "free" elec tron vapor. Richardson pictured the emission of electrons from hot

metal surfaces as "an electron gas evaporating from the hot source;" using a kinetic approach, he derived the well-known exponential law for the thermionic current.41 Wilson restricted his considerations from

38. Thomson (ref. 24), 71-75.

39. F. Kruger, "Uber die Anwendung der Thermodynamik auf die Elektronentheorie

der Thermoelektrizitat," PZ, 77 (1910), 800-808, and 72 (191 1), 360-368. The publica tion was delayed because Planck, although he thought the electron vapor theory and the

analogy to Nernst's theory of electrolytes a promising idea, found Kriiger's treatment

faulty and rejected it for AP; Planck to Wien, 18 Mar 1908 (WWN). Apparently Kruger submitted this paper, or a realted one, to PZ, but withdrew it. When Baedeker submit

ted his theory, Kruger, who was now an editor of PZ, published his preliminary paper. Baedeker was very careful to acknowledge Kriiger's results; Baedeker to Richardson, ca.

1912 (ref. 40). 40. K. Baedeker, "Zur Elektronentheorie der Thermoelektrizitat," T^Z, 77 (1910),

809-810; K. Baedeker, Die elektrischen Erscheinungen in metallischen Leitern (Braun

schweig, 1911), 91. O.W. Richardson, "The electron theory of contact electromotive

force and thermoelectricity," PM, 23 (1912), 263-278, on 277, criticized Baedeker, who

defended his preference for a macroscopic theory of the electron vapor over conjectures about micro-mechanisms in a letter to Richardson, 11 Feb 1912 (RDN R-000060). Cf.

W. Schottky, "Bericht uber thermische Elektronenemission," JRE, 72 (1915), 147-205, on 150-167; Seeliger (ref. 10), 836-837.

41. O.W. Richardson, "Thermionic phenomena and the laws which govern them," in

Nobel lectures (ref. 37), 3, 224-236, on 226; "The electrical conductivity imparted to a

vacuum by hot conductors," Royal Society of London, Philosophical transactions, 201A

(1903), 497-549, on 499-503; "Thermionics," PM, 17 1909), 813-833; "Notes on the

kinetic theory of matter," PM, 18 (1909), 695-698.




the beginning to a mere "thermodynamical" treatment, applying the

Clausius-Clapeyron equation for the vapor pressure above the surface of a liquid to the vapor pressure of electrons above the surface of the

metal.42

The idea of an electron vapor proved to be extraordinarily fruitful. In retrospect Richardson felt that in the years between 1900 and 1913

physicists could have learned much more about the behavior of elec trons in the electron gas than about molecules in an ordinary gas: One could heat cold bodies by absorption of electrons or cool hot matter

by emission of electrons.43 But doubts about the legitimacy of apply ing the equation of a perfect gas to electrons remained. Richardson raised this question when he ventured to derive an expression for the Peltier effect by a thermodynamic analysis similar to Baedeker's. The small size of electrons compared with gas molecules and the low con

centration of external electrons, to which he applied the gas law, seemed to him to make the electron gas an even "closer dynamical approximation to the ideal gas" than "any real gas under comparable conditions."44

Richardson's most important contributions to the analogy between the kinetic theory of gases and the electron gas theory were his meas urements of the velocity distribution of electrons emitted by hot metals. Although not immune from criticism, these data provided evidence enough that the velocities of the electrons outside the source

obey the Maxwell-Boltzmann distribution, and, by an obvious infer

ence, that the velocities of the electrons inside the metal do so too.45

Although the electron vapor theory was most promising, in the

opinions of Planck and Lorentz among others,46 it had its difficulties.

42. H.A. Wilson, "On the discharge of electricity from hot platinum," Royal Society of London, Philosophical transactions, 202A (1904), 243-275, on 258-259.

43. Richardson, "Thermionic phenomena" (ref. 41), 226-227; O.W. Richardson and

H.L. Crooke, "The heat developed during the absorption of electrons by platinum," PM, 20 (1910), 173-206; "The absorption of heat produced by the emission of ions

from hot bodies," PM, 25 (19L3), 624-643, and 26 (1913), 472-476.

44. Richardson (ref. 40), 273.

45. O.W. Richardson and F.C. Brown, "The kinetic energy of the negative electrons

emitted by hot bodies," PM, 16 (1908), 353-376. Richardson claimed, as Otto Stern

did later for his measurements of silver atoms, that he had been the first to demonstrate

"Maxwell's law for any gas, although the law was enunciated by Maxwell in 1859;" Richardson (ref. 41), 227, and O.W. Richardson, "The kinetic energy of the ions emit

ted by hot bodies-II," PM, 18 (1909), 681-695, on 688, fig. 3, a diagram comparing the

calculated and the observed velocity distributions; and Richardson (ref. 40), 263-264.

46. Planck to Wien (ref. 39); H.A. Lorentz, "Anwendung der kinetischen Theorien

auf Elektronenbewegung [1913]," LCP, 8, 214-243, on 216, 224-225. Lorentz' paper was first published in David Hilbert, ed., Vortrage uber die kinetische Theorie der Ma

terie und der Elektrizitat. Gehalten.. .auf Einladung.. .der Wolfskehlstiftung (Leipzig and

Berlin, 1914), 169-193, on 170-171, 174-178.



282 KAISER

Where did the electrons that appeared in thermionic processes come

from? Perhaps from chemical reactions in the residual gas or the filament of the vacuum tube?47 This question was answered negatively

with the help of newly developed filaments for incandescent lamps. W.R. Whitney and I. Langmuir of the General Electric Company sup plied Richardson with specimens of ductile tungsten,48 which with stands very high temperatures and furnishes large currents: With it, Richardson was able to rule out this objection quantitatively in 1913.49

Another problem, debated by Richardson, Bohr, and Walter

Schottky, was the precise definition of the thermionic work function, which was important in the application of electron gas theory in the

technology of vacuum tubes. Should the energy required to liberate the already free electron gas from the lattice and the metal surface or rather the energy required to liberate electrons still bound to metal atoms be considered as the true thermionic work function?50 Bohr and Richardson also discussed the influence of a surface charge dependent on temperature. Both the complex process of liberating an electron from an atom and the occurrence of a temperature dependent surface

layer contradicted the assumption of a constant specific heat for elec trons in the metal. Such questions therefore touched upon the vali

dity of the basic picture of electrons evaporating from a hot liquid source.51

Bohr's attempt to save the phenomena

Although Niels Bohr's dissertation of 1911 marks the final step in the development of the classical electron gas theory, it does not fit

exactly into the picture drawn above. Bohr's Studies on the electron

theory of metals do not reflect the old need to justify the transfer of the gas model to the solid state. But they are very important for the

analogy of the kinetic theory of gases and the electron gas theory of metals. Bohr's theory rested on Lorentz' statistical approach, which it extended by a much more general treatment of the interaction of the electrons and the metal atoms.52 The move from Lorentz to Bohr

47. W. Schottky (ref. 40), 150-167.

48. O.W. Richardson, "The emission of electrons from tungsten at high tempera tures: An experimental proof that the electric current in metals is carried by electrons," PM, 26 (1913), 345-350, on 350.

49. Richardson, "Thermionic phenomena" (ref. 41), 227.

50. Bohr to Richardson, 29 Sep 1915, and Richardson to Bohr, 9 Oct 1915, BCW, 1,

482-488; Schotty (ref. 40), 170-185.

51. Bohr to Richardson, 15 Oct 1915, BCW, 1, 489-491; F.A. Lindemann to

Richardson, 30 Jun 1915 (RDN R-000 753). 52. N. Bohr, Studier over metallernes elektrontheori (Copenhagen, 1911), tr. J. Rud

Nielsen, BCW, 1, 291-393, on 302-317.




consequently shows a structural similarity to the historical develop ment of the kinetic theory of gases.

Maxwell had tried to reconcile the kinetic theory of gases with

experiments on the temperature dependence of the viscosity of gases assuming repulsive forces between gas molecules proportional to the inverse fifth power of the separation of their centers.53 Boltzmann pro ceeded in a similar manner. He considered attractive forces propor tional to r~5 and in addition very short range repulsive forces in order to take collisions into account.54 The explanation of the viscosity of

gases was the early success of the kinetic theory of gases; the deriva tion of the Wiedemann-Franz law was the triumph of the electron gas theory of metals. Bohr attempted to combine the two: To improve the agreement between Lorentz' theoretically derived expression of the

Wiedemann-Franz law and the measurements of Diesselhorst and

Jaeger, he considered a special law for the repulsive force between electrons and metal atoms in collisions, namely a force proportional to 7.-3

55

This approach went in the right direction, according to Peter

Debye, but only half way. Debye declared that two different kinds of force laws were necessary to bring the expression for the Wiedemann Franz law into conformity with experiment and with the expressions for conductivity of heat and for conductivity of electricity taken

separately.56 Bohr himself had recourse to other force laws in a

desperate struggle to account for black-body radiation on the electron

gas theory. Moreover, James Jeans had had to assume different force laws "to reconcile experiment with electron-theory," that is to recon cile the the equipartition theorem with the electric conductivity of metals and a radiation curve that falls off for high frequencies.57

53.i James Clerk Maxwell, "On the dynamical theory of gases [1866]," in Maxwell, 77k? scientific papers [1890] (2 vols., New York, 1965), 2, 26-78, on 29, 40-43. Cf.

Stephen G. Brush, The kind of motion we call heat (2 vols., Amsterdam, 1976), 2, 386

418; Karl von Meyenn, "Dispersion und mechanische Athertheorien im 19. Jahrhun

dert," in P.L. Butzer and F. Feher, eds., E.B. Christoffel. The influence of his work on

mathematics and the physcial sciences (Basel, Boston, Stuttgart, 1981), 680-703, and

"Engpasse in der Atomtheorie des friihen 19. Jahrhunderts," in Charlotte Schonbeck, ed., Atomvorstellungen im 19. Jahrundert (Paderborn, 1982), 35-55.

54. L. Boltzmann, "Uber die Moglichkeit der Begriindung einer kinetischen

Gastheorie auf anziehende Krafte allein," AP, 24 (1885), 37-44, and L. Boltzmann,

Wissenschaftliche Abhandlungen [1909] (3 vols., New York, 1968), 3, 101-109.

55. Bohr (ref. 52), 341.

56. Debye to Bohr, 30 May 1911, BCW, 1, 400-402; Bohr to Debye, undated, ibid., 402-403.

57. Bohr (ref. 52), 364-365, 379; J.H. Jeans, "The motion of electrons in solids. Part II," PM, 20 (1910), 209-226, on 224.



284 KAISER

The Tolman-Stewart experiment

The questions whether moving electricity possesses mass at all, and if so, whether this mass is observable or not date back to Maxwell and Boltzmann.58 Lorentz' calculations of the consequences of the mass

transported by electrons suffering centrifugal accelerations predicted only a very small effect, possibly beyond the reach of experiment.59 This prediction seemed to be confirmed by measurements made by Ernest Fox Nichols in 1906. Although Nichols was encouraged by Lorentz and Nernst, both of whom had lectured in Nichols' backyard at Columbia University, he left off his "whirling disk experiments" without achieving definitive results.60

The first successful experiments were performed in Berkeley in 1916 by Richard Chase Tolman and Thomas Dale Stewart. In a quite imaginative way, Tolman and Stewart hoped to show the inertia of

metal electrons subject to high accelerations: They tried to measure

potential differences by abruptly slowing down rotating metal coils. What is particularly interesting from the point of view developed in this paper is that they started with the intermediate step of measuring potential differences in accelerated electrolytes by an experiment designed as early as 1888 by Walther Nernst.61

The success of the Tolman-Stewart experiments seemed to several

leading theorists of the electron gas to vindicate the supposition of a

single sort of charge carrier in metals:62

[Lorentz'] assumption [of a single carrier] was in fact very bold. It has been confirmed by results proceeding from a consistently developed theory that prove it indirectly by observation. It is, however, remark able that this assumption also has had a direct confirmation by

58. J.C. Maxwell (ref. 1), ?574, on 216-218; H. Hertz, "Versuche zur Feststellung einer oberen Grenze fur die kinetische Energie der elektrischen Stromung," AP, 10

(1880), 414-448, and "Obere Grenze fur die kinetische Energie der bewegten

Elektricitat," AP, 14 (1881), 581-590; L. Boltzmann, Vorlesungen uber Maxwells

Theorie der Elektricitat und des Lichtes, 2 parts [1891-1893], in W. Kaiser, ed., Ludwig Boltzmann, Gesamtausgabe, 2 (Graz, 1982), part 1, 18.

59. Lorentz (ref. 46), 238-240.

60. E.F. Nichols, "Die Moglichkeit einer durch zentrifugale Beschleunigung erzeugten elektromotorischen Kraft," PZ, 7(1906), 640-642; Nichols to Lorentz, 21 Oct 1906, 17

Jul 1909 (LTZ). The problem in measuring may have been that the small potential differences sought, which are around 10~8 volts, are masked by thermoelectric

phenomena at the sliding contacts.

61. R.C. Tolman and T.D. Stewart, "The electromotive force produced by the ac

celeration of metals," PR, 8 (1916), 97-116; W. Nernst, "Zur Kinetik der in Losung befindlichen Korper," ZPC, 2 (1888), 613-637, on 636-637, and "Die elektromotor

ische Wirksamkeit der lonen "

ZPC, 4 (1889), 129-181, on 130-131.

62. Seeliger (ref. 10), 788.




experiment. Apparently R.C. Tolman and T.D. Stewart have succeeded

in determining the mass of the charge carriers and in finding a value that agrees well with values derived from different experiments, namely 1/1,900 to 1/1,940 of the hydrogen atom.

Lorentz himself considered the Tolman-Stewart experiment to be a

capital point for the electron gas theory: "There are many difficult and unsolved questions in the theory of electrons, but instead of speaking of these I shall.. .point out to you some of its triumphs. Among these we may reckon in the first place the beautiful experiment performed by Tolman and Stewart, in which an electric current was produced simply by changing the velocity of the conductor."63 It is likely that Lorentz stimulated the repetitions of the Tolman-Stewart experiment undertaken by John Ambrose Fleming and his coworkers.64

3. THREE SEVERE PROBLEMS

The equipartition theorem

The equipartition principle made grave difficulties when carried over from the gas theory to the electron theory. If every electron has a mean kinetic energy of 3kT/2, then all the electrons N contained in a unit volume of the metal should contribute 3kN/2 to its specific heat. Nothing is clearer, however, than the fact that specific heats of metals do not equal the sum of the Dulong-Petit value and the contri bution of the electron gas.

Max Reinganum, who worked with Kamerlingh Onnes in Leyden, pointed out this difficulty even before Drude published the second

part of his basic paper of 1900.65 After several physicists had exam ined the problem, they reached the consensus expressed by Nernst in 1911: Either the equipartition theorem does not hold for the electron

gas or the assumed number of electrons does not agree with the contri bution of the electrons to the specific heat of metals.66

63. H.A. Lorentz, "Physics in the new and the old world [1926]," LCP, 8, 405-417, on 409; "The motion of electricity in metals [1925]," ibid., 307-332, on 312-316.

64. Fleming to Lorentz, 1 Dec 1925, 22 Jan and 4 Feb 1926 (LTZ). 65. M. Reinganum, "Theoretische Bestimmung des Verhaltnisses von Warme- und

Elektricitatsleitung der Metalle aus der Drude'schen Elektronentheorie," AP, 2 (1900),

398-403, on 401.

66. J.J. Thomson (ref. 24), 83; Arnold Eucken, "Neuere Untersuchungen iiber den

Temperaturverlauf der spezifischen Warme," JRE, 8 (1911), 489-534, on 509-511; W.

Nernst, "Untersuchungen iiber die spezifische Warme bei tiefen Temperaturen," SB

(1911), 306-315, on 310; Seeliger (ref. 10), 853-854.



286 KAISER

A typical way out of the dilemma was the construction of conduc tion mechanisms that reduced the number of free electrons. Thom

son, Bohr, and Lorentz each discussed the possibility.67 It was, how

ever, if anything too powerful a method, and Lorentz came to use it as a reductio ad absurdum. He observed that to compensate for the reduction in the number of free electrons, the mean free path A had to be extended because the conductivity is proportional to NX; but the resultant enormous free path conflicted with estimates of the separa tion of molecules based on the densities of metals and on x-ray diffraction. These considerations had given separations of the order of small multiples of the radius of the Bohr atom; calculations based on the electron gas theory had furnished mean free paths of metal electrons 10 to 100 times as large.68 The contradictions only worsened if one took the super conductivity discovered by Kamerlingh Onnes in 1911 into account.69

The equipartition theorem also failed in electron gas theory when

applied to black body radiation. This failure was particularly distress

ing because of the expectation, in which Wien and Drude concurred, that the kinetic theory of gases and the theory of black body radiation would confirm or support one another.70

As early as 1903 Lorentz had calculated the emission and absorp tion of electromagnetic radiation from free electrons in metals. He obtained an expression that for long wavelengths agreed with Planck's radiation law, which seemed to be experimentally confirmed; and Riecke and Baedeker, among others, rated the agreement an important success of the electron gas theory.71 Further investigations by Wilson

67. J.J. Thomson (ref. 24), 84-87. See also Bohr (ref. 52), 325; Bohr to C.W. Oseen, 1 Dec 1911, BCW, 1, 426-431; Bohr to S.B. McLaren, 17 Dec 1911, ibid., 432-434;

Lorentz (ref. 46), 216; Lorentz (ref. 63), 323-325.

68. F.A. Lindemann, "Untersuchungen iiber die spezifische Warme bei tiefen Tem

peraturen, IV," SB (1911), 316-321, on 317-318; Baedeker, "Erscheinungen" (ref. 40),

18; letters of Bohr to Oseen and to McLaren (ref. 67); Seeliger (ref. 10), 853.

69. H. Kamerlingh Onnes, "Investigations into the properties of substances at low

temperatures, which have led, amongst other things, to the preparation of liquid heli

um," Nobel lectures (ref. 37), 1, 306-336, on 335; J.J. Thomson, "Conduction of electri

city through metals," PM, 30 (1915), 191-202, on 193; Per F. Dahl, "Kamerlingh Onnes and the discovery of superconductivity: The Leyden years, 1911-1914," HSPS, 15:1 (1984), 1-37.

70. W. Wien, "Uber die Energievertheilung im Emissionsspectrum eines schwarzen

Korpers," AP, 58 (1896), 662-669, on 664; P. Drude, Lehrbuch der Optik (Leipzig,

1900), preface. 71. H.A. Lorentz, "On the emission and absorption by metals of rays of heat of great

wave-lengths [1903]," LCP, 3, 155-176; E. Riecke, "Uber die Elektronentheorie des Galvanismus und der Warme," JRE, 3 (1906), 24-47, on 47; Baedeker,

"Erscheinungen" (ref. 40), 135.




and Jeans, and Bohr's dissertation, confirmed the result.72 But Bohr reevaluated the success by stressing the disagreement for short

wavelengths. In 1911 McLaren offered a calculation of emission and

absorption based on the statistics of electron gas theory that resulted in the Rayleigh-Jeans law for all frequencies.73 Contrary to experimen tal results, this law indicated that, in Planck's words, there may be no

equilibrium in the distribution of energy between radiation and matter.74

The hope that the electron gas theory of metals might yield a satis

factory theory of black body radiation seemed fully justified. Black

body radiation did characterize the radiation of metallic cavities;75 and it seemed only reasonable to describe the radiating cavities by the most successful theory of the solid state. Although Bohr felt that the radiation revealed the limitations of classical electrodynamics for solid

matter and high frequencies, others hoped to find an appropriate mechanism for the interaction of electrons and metal atoms that would explain the shape of the experimentally established radiation curve at high frequencies.76 In this respect, the tentative interaction of the theories of black body radiation and of the electron gas reflects and explains the painful process of realizing the break with classical

72. H.A. Wilson, "The electron theory of optical properties of metals," PM, 20

(1910) , 835-844, on 841-844; J.H. Jeans, "The motion of electrons in solids. Part I.

Electric conductivity, KirchhofFs law and radiation of great wave-length," PM, 17

(1909), 773-794. In 1910 Jeans compared an easily intelligible theory of radiation, which depends on continuous motion and on equipartition of energy, with a quantum

theory "to many still unthinkable," which assumed not only a system of vibrators with

"definite multiples of a fixed unit of energy" but also indivisible "atoms" of energy in

the aether. J.H. Jeans, "On non-Newtonian mechanical systems, and Planck's theory of

radiation," PM, 20 (1910), 943-954, on 943, 953; Bohr (ref. 52), 357-359.

73. S.B. McLaren, "The emission and absorption of energy by electrons," PM, 22

(1911) , 66-83, on 66-72.

74. Planck to Lorentz, 1 Apr 1908 (LTZ); printed in part in Hermann,

Frilhgeschichte der Quantentheorie, 1899-1913 (Mosbach, 1969), 47-48.

75. Hans Kangro, Early history of Planck's radiation law (New York, 1976), 75-77,

159, 168, 171, 175.

76. Lorentz to Wien, 6 June 1908, in Hermann (ref. 74), 46; W. Wien, "Theorie der

Strahlung [1909]," EMW, 5:3 (Leipzig, 1909-1926), 282-357, on 326-333; J.L. Heil

bron and T.S. Kuhn, "The genesis of the Bohr atom," HSPS, 1 (1969), 211-290, on

217. G.H. Livens traced Lorentz' result to the assumption that the period of the in

cident radiation is great compared with the time between the collision of the electrons, an assumption that seemed to apply only to high temperatures. Livens to Lorentz, 9 Jul

1914 (LTZ). Otto Sackur's objections led to a result similar to Livens'. Sackur, "Die

universelle Bedeutung des sog. elementaren Wirkungsquantums," AP, 40 (1913), 67-86, on 85. C.W. Oseen argued that Lorentz' derivation of the radiation law for long

wavelengths is only valid if one allows long periods of time to observe the calculated mean values of emitted energy; C.W. Oseen, "Zur Kritik der Elektronentheorie der

Metalle," AP, 49 (1916), 71-84, and Oseen to Lorentz, 13 Jan 1916 (LTZ).



288 KAISER

theory that had occurred in Planck's and even more in Paul Ehrenfest's and Albert Einstein's derivations of the radiation law.77

Planck himself preferred to restrict discontinuity to the absorption of radiation by his very abstract "resonators" than to allow discon tinuities in the ether. Or, he preferred to ascribe discontinuity only to the process of emission of radiation. This "second theory" included the idea that colliding electrons in metals might obey his "new quan tum hypothesis" rather than be reflected classically. Because they emitted only discrete amounts of energy, metal electrons might exchange only a part of their kinetic energy and thus add only a small fraction to the specific heat of solid matter.78

Wien felt quite uncomfortable about the apparent link between the

equipartition theorem in electron gas theory and the Rayleigh-Jeans law in the theory of black body radiation. By 1913 he was willing to

give up Drude's explanation of the empirical Wiedemann-Franz law.79

By application of Planck's radiation law, Wien derived the propor

tionality of electrical resistance to the absolute temperature, and so had a hearing for his heterodox views. He did not persuade Lorentz, however, who in his Wolfskehl lecture in 1913 complained about

Wien's rejection of the free electron gas theory. Lorentz found it par

ticularly grievious that Wien "even went so far as to give up com

pletely the picture of a real heat movement since he assigned the elec trons a velocity independent of the temperature."80

Lorentz gave strong arguments against Wien's assumption of an electron velocity independent of temperature. He referred to the suc cessful kinetic approach in the theory of electrolytes and to the most

striking of the experimental confirmations of the electron gas theory of

metals, Richardson's measurements of the Maxwell-Boltzmann distri bution of the velocities of electrons emitted by hot metals.81 As

77. Planck to Wien, 27 Feb 1909 (WWN). In 1905 Jeans had claimed that in a sys tem consisting of a gas and radiation "enclosed by an ideal perfectly reflecting boun

dary" the "energy accumulates in the aether," i.e., that there is no radiation equilibri um; J.H. Jeans, "On the partition of energy between matter and aether," PM, 10 (1905),

91-97, on 97.

78. Planck to Lorentz, 21 Nov 1908 and 10 Jul 1909 (LTZ); Planck to Wien, 14 Jan 1911 (WWN); M. Planck, "Eine neue Strahlungshypothese," Deutsche Physikalische

Gesellschaft, Verhandlungen, 13 (1911), 138-148; "Zur Hypothese der Quantenemis sion," SB (1911), 723-731. In 1913, Planck discussed a thermodynamic equilibrium between "oscillators" and radiation mediated by free electrons; M. Planck, "Uber das

Gleichgewicht zwischen Oszillatoren, freien Elektronen und strahlender Warme," SB

(1913), 350-363. 79. W. Wien, "Zur Theorie der elektrischen Leitung in Metallen," SB (1913), 184

200, on 186.

80. Lorentz (ref. 46), 214.

81. Lorentz (ref. 46), 216.




mentioned earlier, Richardson inferred from those measurements that the electrons within the metal must possess the Maxwell-Boltzmann distribution of velocities. Hence the velocities of the electrons must

depend on temperature, and the equipartition theorem should hold. Nor was Planck pleased with Wien's theory. Planck had spent some

time trying to reconcile the quantum theory of radiation (or at least of the emission of radiation) with the Maxwell-Boltzmann distribution

among free electrons and with a thermodynamic equilibrium between oscillators and electrons.82

The inference from Richardson's measurements to a Maxwell Boltzmann distribution among the electrons within the metal had been made by Jeans and also by McLaren, who intended thereby to

destroy the hope of reconciling the theory of the classical free electron

gas and Planck's radiation law.83 It was also made by Paul and Tatiana Ehrenfest in their famous analysis of the conceptual founda tions of statistical mechanics. For the Ehrenfests, the glow discharge was a "direct and almost complete confirmation" of the propriety of

applying the kinetic theory in electron gas theory without including the ether or radiation.84 C.G. Darwin argued similarly in a manuscript of 1912.85 Bohr's successful quantum theoretical treatment of Rutherford's atom did not at first have an important effect on the electron gas theory. Among those who tried to incorporate Bohr's ideas were Fritz Haber and Owen W. Richardson.86 They treated

superconductivity in terms of electrons occupying Bohr orbits with common tangents. Thus they pictured a movement of electrons without resistance. Ehrenfest thought Bohr's advanced theory of intertwined electron orbits might be extended to explain the general behavior of electrons in metals.87 Another extravagant possibility, dis cussed at the Wolfskehl conference of 1913, also attracted moderate attention. This was the concept of an electron gas obeying quantum statistics.88 An early and important adherent to the degenerate gas

82. Planck, "Uber das Gleichgewicht" (ref. 78), 352. Arguing against Wien's rejec tion of free electrons, Planck referred to superconductivity, which implied very large

mean free paths for metal electrons, Planck to Wien, 10 Jan 1913 (WWN). 83. Jeans, "The motion" (ref. 72), 792; McLaren (ref. 73), 68.

84. Paul Ehrenfest and Tatiana Ehrenfest, "Begriffliche Grundlagen der statistischen

Auffassung in der Mechanik [1911]," EMW, 4 (Leipzig, 1907-1914), Article 32, 3-90, on 74.

85. C.G. Darwin, "The theory of radiation," Aug 1912 (AHQP 36:2). 86. Seeliger (ref. 10), 862-863; Per F. Dahl, "Superconductivity after World War I

and circumstances surrounding the discovery of a state of B = 0," HSPS, 16:1 (1986),

1-58, on 6.

87. P. Ehrenfest to Lorentz, 5 Feb 1922, erroneously dated 1912 (LTZ). 88. Michael Eckert, "Propaganda in science: Sommerfeld and the spread of the elec

tron theory of metals," HSPS, 17:2 (1987), 191-233.



290 KAISER

theory was Richardson, who almost immediately abandoned his con

clusion that inside the metal the Maxwell-Boltzmann distribution holds.89

The Hall effect

The Hall effect was discovered in 1879 at Johns Hopkins by E.H.

Hall, a student of Henry Rowland.90 Lorentz and Boltzmann referred the changing sign of the effect to opposite forces that act on oppositely charged moving electricity in the transverse magnetic field, i.e., they appealed to dual mechanisms of conductivity.

As we know, the first electron gas theories of metals, those of Riecke and Drude, also assumed carriers of different signs, with different "drift velocities" or "mobilities." The Hall effect fit nicely into the framework of electron gas theory; what deviation existed seemed to be experimental problems. But when Lorentz restricted electron gas theory to a single carrier, the Hall effect became a prob lem of principle.

In 1907, in his Habilitationsschrift, Karl Baedeker reported some

disconcerting facts about cuprous iodide. With fresh preparations, Baedeker obtained a very small value for the specific resistance, which was rather "strange" for a transparent substance.91 Further investiga tions with samples in equilibrium with iodine vapor showed an addi tional and marked drop in resistance. By varying the strength of the

iodine, Baedeker eventually was able to maintain a variable and com

paratively high electrical conductivity in cuprous iodide. He then combined his variable semiconductor with platinum in order to make thermoelectric measurements. He explained the variable thermoelec tric power in terms of the electron vapor theory.92 By far the most

interesting of Baedeker's experiments was a measurement of the Hall effect in the doped cuprous iodide. The result: "the sign of the effect was positive according to the accepted nomenclature, i.e., it was

89. Richardson (ref. 50). 90. See Jed Z. Buchwald, "The Hall effect and Maxwellian electrodynamics in the

1880s," Centaurus, 23 (1979), 51-99, 118-162; Buchwald, Maxwell to microphysics,

(ref. 3), 73-108.

91. K. Baedeker, "Uber die elektrische Leitfahigkeit und die thermoelektrische Kraft

einiger Schwermetallverbindungen," AP, 22 (1907), 749-758, 765-766; K. Baedeker and E. Pauli, "Das elektrische Leitvermogen von festem Kupferjodur," PZ, 9 (1908), 431

(quote). 92. K. Baedeker, "Uber eine eigentumliche Form elektrischen Leitvermogens bei

festen Korpern," PZ, 9 (1908), 431-433. See also Herbert Geotzeler, "Zur Geschichte der Halbleiter, Bausteine der Elektrotechnik," Technikgeschichte, 39 (1972), 31-50, on

36; W. Kaiser, "Karl Baedekers Beitrag zur Halbleiterforschung," Centaurus, 22 (1978), 187-200.




opposite to the effect in bismuth."93

Although the follow up experiments by Baedeker's graduate stu dent Karl Steinberg, who measured the Hall effect as a function of the iodide concentration, have a modern appearance, they veiled the prob lem of the charge carrier involved in the metallic conduction of cuprous iodide. Steinberg's data showed that with increasing iodine concentration (and decreasing resistance) the absolute value of the Hall coefficient decreases. This tendency alone fit the picture of the electron gas theory very well. The alleged increase in the electron con centration (due to the doping with iodine) would have led to a decrease in the absolute value of the Hall coefficient. But Baedeker and Steinberg measured only below the degeneracy limit, where the

mobility of the carriers decreases with increases in their concentra tion.94

As late as 1912 Baedeker held to the possibility of explaining the variation of the absolute value of the Hall coefficient by varying elec tron concentration.95 The electron vapor theory of thermoelectric

phenomena, the emission of electrons from hot metals, and the simpli city of a single charge carrier were then still very strong arguments for Lorentz' approach.96 It was not possible, however, to ignore the prob lem of the changing signs of the Hall coefficients. Steinberg was very clear about that: "With negative particles alone, the elementary theory can only derive an effect in one direction, namely that opposite to the effect in cuprous iodide." In fact, at that time, 1911, many conductors with positive Hall coefficients were known. Their total eventually brought Baedeker to castigate the electron gas theory of metals as "insufficient."97

Eduard Riecke shared Baedeker's reservations. At the general meeting of the Deutsche Bunsengesellschaft in Aachen in 1909, he

lumped the Hall effect and certain "strange phenomena" discovered

by Nernst and von Ettingshausen together as evidence of deep trouble: "With respect to these phenomena, the unitary theory fails_The

building of electron gas theory has a crack all the way down to its foundation."98 And in 1913, Riecke wrote an article for the Handbuch der Radiologie that recommended against ignoring the Hall effect and

93. Baedeker, "Uber eine eigentumliche Form elektrishcen Leitvermogens bei festen

Korpern," AP, 29 (1909), 566-584, on 581-583.

94., K. Steinberg, "Uber den Halleffekt bei jodhaltigem Kupferjodur," AP, 35 (1911), 1009-1033, 1026-1030.

95. K. Baedeker, "Kunstliche metallische Leiter," PZ, 13 (1912), 1080-1082, on 1081.

96. Baedeker, "Erscheinunge" (ref. 40), 10-14, 77-94, 123. 97. Steinberg (ref. 94); Baedeker, "Erscheinungen" (ref. 40), 122-123. 98. Riecke (ref. 8), 517.



292 KAISER

related phenomena in order to continue with Lorentz' unitary theory. "As long as it excludes these phenomena, the electron [gas] theory will remain a fragment."99 Where physicists followed Riecke's advice they necessarily disturbed the picture of the free electron gas or they stressed the interaction of electrons and metal atoms. The so-called "directive field theories" of Johann Koenigsberger and Niels Bohr, which assumed an interplay of macroscopic and microscopic fields to account for the Hall coefficients, are good examples.100 Hall divided the electron theory of metals into thermionics and thermoelectricity, which he thought amenable to the free electron gas theory, and metal lic conductivity and the Hall effect, which he thought involved a

direct transfer of electrons from one atom to another. He assigned mobility to the transient positive metal ions, which might be deflected in magnetic fields in an amount sufficient to account for the positive

Hall effect. Still, neither he nor others supposed that the Hall effect was thereby explained.101

4. THE BEGINNINGS OF A SOLUTION

The way that released theorists of the electron gas from their difficulties was opened in 1913-14. The decisive stimulus came from Einstein's and Debye's quantum theories of the specific heat of solid bodies published in 1907 and 1912, respectively.102 In these theories, the quantum statistics implicit in Planck's radiation law became a

standard method in electron gas theory as well. Explanation of electri cal resistance even at low temperatures was referred to the vibration of the metal atoms or of the lattice. Owing to a variety of interaction

mechanisms, these vibrating metal atoms limited the mean free path of the electrons. Thus they forced the colliding electrons to behave

according to the statistics of the lattice. Kamerlingh Onnes and Lin demann each proposed theories of this type.103

99. E. Riecke, "Elekronentheorie galvanischer Eigenschaften der Metalle [1913], in

Erich Marx, ed., Handbuch der Radiologie, 6, (Leipzig, 1925), 281-494, on 431.

100. J. Koenigsberger and G. Gottstein, "Uber den Halleffekt," PZ, 14 (1913), 232-, 237; Bohr (ref. 52), 383-395. See also G.H. Livens, "The electron theory of the Hall ; effect and allied phenomena," PM, 30 (1915), 526-548, on 545.

101. E.H. Hall, "Illustrations of the dual theory of metallic conduction," PR, 28

(1926), 393-417; L.L. Campbell, Galvanomagnetic and thermomagnetic effects. The

Hall and allied phenomena (New York, 1923), 89-91; W. Hume-Rothery, The metallic

state. Electrical properties and theories (Oxford, 1931), 223-229; Hall to Lorentz, 13

Nov 1922 (LTZ). 102. A. Einstein, "Die Planck'sche Theorie der Strahlung und die Theorie der

spezifischen Warme," AP, 22 (1907), 180-190, 800; P. Debye, "Zur Theorie der

spezifischen Warme," AP, 39 (1912), 789-839.

103. Seeliger (ref. 10), 864-865; Dahl (ref. 86), 12-13; W. Nernst, "Untersuchungen




The direct transfer of Debye's statistical model to the theory of

gases was particularly important for the electron gas theory. In 1912

Debye had calculated the energy distribution among the vibration modes of a solid body according to Planck's radiation law. H. Tetrode and Willem Hendrik Keesom then adopted Debye's idea to calculate the energy distribution among the acoustic vibrations in a

volume of a gas or liquid. In this way they were able to justify Nernst's earlier statement that specific heats vanish at T = 0.104 At the same time the theory represented the behavior of an ideal gas that is "degenerate" at very low temperatures. This degenerate behavior seemed to occur in the electron gas at considerably higher tempera tures owing to the small mass of electrons.105 Despite its tentative

character, this "most radical" application of quantum theory was the first glimmer of hope for solving the notorious problem of the small contribution of the electron gas to the specific heats of metals at nor

mal temperatures.106

There followed a long period with no real progress. Theory stag nated, attending to criticism and details: "thermionics,"107 the behavior of conductors under high pressure,108 the conductivity of

alloys,109 mechanisms to explain the Hall effect. And it was war time.

liber die spezifische Warme bei tiefen Temperaturen, III," SB (1911), 306-315, on

311-315 (Nernst used Planck's radiation formula only ad hoc to calculate the electric

resistance for low temperatures); F.A. Lindemann, "Untersuchungen uber die

spezifische Warme bei tiefen Temperaturen, IV," SB (1911), 316-321, on 318-319. Cf.

Karl F. Herzfeld, "Zur Elektronentheorie der Metalle," AP, 41 (1913), 27-52; Wien (ref.

79); P. Bridgman, "The electrical resistance of metals," PR, 77 (1921), 161-194, on 163.

104. H. Tetrode, "Bemerkungen liber den Energiegehalt einatomiger Gase und uber

die Quantentheorie fur Flussigkeiten," PZ, 14 (1913), 212-215; W.H. Keesom, "Uber

die Zustandsgleichung eines idealen einatomigen Gase und uber die Quantentheorie," PZ, 14 (1913), 665-670; W. Nernst, "Der Energieinhalt fester Stoffe," AP, 36 (1911), 395-439, on 435.

105. Tetrode (ref. 104), 214. Tetrode did not allow for a complete analogy between an ideal gas and conduction electrons.

106. W.H. Keesom, "Zur Theorie der freien Elektronen in Metallen," PZ, 14 (1913),

670-675, on 671, following Keesom's remarks at the Wolfskehl conference in

Gottingen, in Hilbert, ed. (ref. 46), 193-196. After the Wolfskehl conference, Keesom

discussed his degenerate gas theory with Lorentz and Sommerfeld, who had discussed

the quantum theory of an ideal gas of his coworker Wilhelm Lenz at the conference.

Keesom to Lorentz, 29 Apr 1913 (LTZ); Keesom to Sommerfeld, 29 Apr 1913 (AHQP 31:11).

107. Cf. the lengthy review by articles Richardson, "Thermionic phenomena" (ref.

41); Schottky (ref. 40); and Irving Langmuir, "The pure electron discharge and its appli cations in radio telegraphy and telephony 1915, in Langmuir, The collected works, 3

(Oxford, 1961), 38-58.

108. Bridgman (ref. 103); Erich Kretschmann, "Kritischer Bericht iiber neue Elek

tronentheorien der Elektrizitats-und Warmeleitung in Metallen," 28 (1927), 555-592, on 567.

109. W. Guertler, "Beitrage zur Kenntnis der Elektrizitatsleitung in Metallen und Le

gierungen," JRE, 17(1920), 276-292.



294 KAISER

The war stimulated development of tubes and amplifiers for wireless

telegraphy, which depended at least partially on electron gas theory;110 but it also killed off some leading or promising practitioners and

poisoned relations among the survivors. Much of the wrok that did

go forward was in the character of summaries and overviews.111 And no doubt it was a time of hunting in blind alleys. Haber's, Stark's, and Lindemann's idea of an electron space-lattice that moves through the atomic lattice under an electric field did not work out.112 But all this is food and even fortune for historians because it furnishes a slow

motion picture of a scientific dialogue that was not terminated by an

overwhelming success or by an outstanding contribution of a single person.

In the mid-1920s leading physicists, for example, Einstein and

Schrodinger, again dealt with the degenerate gas and with its possible consequences for the electron gas theory of metals. A breakthrough occurred with the new quantum statistics of Fermi and Dirac, which

obeyed Pauli's exclusion principle.113 R.H. Fowler applied Fermi statistics to stellar matter.114 Closer to home, Pauli used them to

explain the paramagnetism of the alkali metals, which was surprisingly small and to some extent independent of the temperature.115 Stimu lated by galley proofs of Pauli's paper on this subject, Sommerfeld resumed his own research on the electron gas theory of metals. Sommerfeld's revision, based on Fermi statistics, of Lorentz' transport theory led to a much improved version of the free electron gas theory. Theoretical results came into better agreement with experimental data; not only in the vexed matter of specific heat but also in the expression for the Wiedemann-Franz law and for thermionic phenomena.116

110. For the interaction of electron theory and technology during, or as a conse

quence of World War I, see Langmuir (ref. 107). During the war Max von Laue worked with Wien on electronic amplifying tubes for the army; A. Hermann, "Max von Laue," DSB, 8 (New York, 1973), 50-53, and AP, 59 (1919), 465-492. Richardson worked with the Signal School of the Admiralty at Portsmouth; E. Watson to Richardson, 14 June 1918 (RDN).

111. Seeliger (ref. 10); Richardson, "Thermionic phenomena" (ref. 41); Schottky (ref. 40).

112. Lindemann calculated the (small) specific heat of the electron space-lattice in order to remove the "chief stumbling-block of the old theory;" F.A. Lindemann, "Note on the theory of the metallic state," PM, 29 (1915), 127-140.

113. L.H. Hoddeson and G. Baym, "The development of the quantum mechanical electron theory of metals: 1900-28," Royal Society of London, Proceedings, 371A

(1980), 8-23, on 11-20.

114. R.H. Fowler, "On dense matter," Royal Astronomical Society, Monthly Notices,

87:2(1926), 114-122. 115. W. Pauli, "Uber Gasentartung und Paramagnetismus," ZSP, 41 (1927), 81-102. 116. A. Sommerfeld, "Zur Elektronentheorie der Metalle," Die Naturwissenschaften,




Sommerfeld's theory seemed for a short time to be the correct elec tron theory of metals.117 Severe problems soon obtruded, however, for

example, the long free path of an electron in a crowded metal lattice. And contrary to the expression for the Wiedmann-Franz law, where the mean free path does not appear, the temperature dependence of the electric conductivity alone remained a notorious stumbling block.118 Nor could Sommerfeld's theory of a free electron gas deliver the signs of the Hall coefficients.119 The problem of the mean free path or of metallic conduction in general was solved by Felix Bloch, who demonstrated that the wave function of the conduction electrons is

only periodically modified by the lattice. An ideal lattice does not cause diffraction. Only perturbations in the lattice?chemical and structural anomalies, and heat vibrations?lead to diffraction and

eventually to electrical resistance.120

Progress on galvanomagnetic effects came more slowly. The start

ing point here was Pauli's exclusion principle of 1925, which indicated that not only present but also missing electrons in a shell might have a

physical meaning. Atoms with n electrons in a shell resemble atoms that lack n electrons to complete a shell. Referring to the quantum numbers of the missing electrons, Pauli had a notion of "hole values" or Luckenwerte.121 In 1929 Rudolf Peierls, who, like Bloch, was work

ing with Heisenberg, adopted Pauli's idea. He showed the occupation of the shells determines whether the Hall coefficients are positive or

negative.122 In 1931 Heisenberg invented the concept of an electrical conduction carried by "holes" contrary, or in addition to conduction

by negative electrons. He demonstrated that the solution of a

75 (1927), 825-832; "Zur Elektronentheorie der Metalle auf Grund der Fermischen Sta

tistik," ZSP, 47 (1928), 1-32, 43-46; "Zur Elektronentheorie der Metalle," Die Na

turwissenschaften, 76 (1928), 374-381. 117. Einstein to Sommerfeld, 9 Nov 1927, in A. Hermann, ed., Albert

Einstein /Arnold Sommerfeld. Briefwechsel (Basel and Stuttgart, 1968), 111-112; Richard Gans to Sommerfeld, 25 Oct 1927 (AHQP 31:1); A. Rubinowicz to Sommerfeld, 28 Jan 1936 (AHQP 33:6).

118. Sommerfeld (ref. 117), 60, 832; Hume-Rothery (ref. 101), 292-293. 119. Sommerfeld (ref. 117), 55.

120. F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," ZSP, 52 (1929), 555-600. A reformulation of Bloch's theory that includes electric resistance due to imperfections and admixtures is L. Nordheim, "Zur Elektronentheorie der

iMetalle," AP, 9 (1931), 641-678.

121. W. Pauli, "Uber den Zusammenhang des Abschlusses der Elektronengruppen im Atom mit der Komplexstruktur der Spektren," ZSP, 31 (1925), 765-783, on 778-779.

122. R. Peierls, "Zur Theorie der galvanomagnetischen Effekte," ZSP, 53 (1929), 255-266. Peierls recalled that this, his "first paper of any importance," was inspired by Heisenberg's qualitative understanding of the positive Hall effect in terms of "holes." Interview with R.E. Peierls, 17 June 1963, by J.L. Heilbron (AHQP).



296 KAISER

Schrodinger equation for n electrons is approximately that of the solu tion of a Schrodinger equation for N-n holes, where N is the number of electrons of a fully occupied shell:123

If one compares.. .[the wave equation for holes] with the wave

equation.. .[for electrons], one recognizes that under the influence of perturbative external fields the holes behave exactly like electrons with positive charge. The holes also contribute to the current and to the charge density just as electrons with positive charge. Electrical conduc tion in metals that is carried by a small number of holes can therefore be described in every respect like conduction in metals carried by a small number of positive conduction electrons. From this immediately follows the anomalous Hall effect for those metals.

Heisenberg's result, which refers to the energy levels of a single atom, was translated into the language of band theory, which refers to the

energy levels of a whole lattice. This translation can be found in Hans Bethe and Arnold Sommerfeld's influential article in the Handbuch der Physik, which became the basic text in solid state physics for decades to follow.124

Like the kinetic theory of gases in the field of viscosity and

diffusion, the electron gas theory celebrated an early success in

explaining the Wiedemann-Franz law. Moreover, the explanation of

thermoelectricity in terms of a thermodynamic cycle performed by an electron vapor of varying density, and, above all, the almost visible appearance of an electron vapor in the emission of electrons by hot

metals, helped to make the electron gas model almost palpable. The electron inertia experiments of Tolman and Stewart also contributed, although not so effectively, to this concensus.

Owing to the inherent weakness of the electron gas model, how

ever, and to the complexity of solid matter, serious problems emerged that defied the resourcefulness of theorists. The transfer of the

equipartition theorem, the small contribution of free electrons to the

specific heat of metals, the radiation puzzle, and the different signs of

123. W. Heisenberg, "Zum Paulischen Ausschliessungsprinzip," AP, 10 (1931) 888 904.

124. Sommerfeld and Bethe, "Elektronentheorie der Metalle," in H. Geiger and K. Scheel, eds., Handbuch der physik, 2nd ed., vol. 24 (Berline, 1933), 333-622, on 334.




the Hall effect became recognized as severe obstacles to the develop ment of the free electron gas theory. Although Lorentz was an author

ity in the early electron gas theory, he did not dominate the field as Sommerfeld and his school did later.125 Consequently discussion and debate persisted, drawing upon arguments of many kinds, so that the electron gas theory served as a market and a test for the most impor tant ideas of theoretical physics at the beginning of our century.

125. Eckert(ref. 88).



Documents

Early Theories of the Electron Gas