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On the Question of Superconductivity*
11. ON THE QUESTION OF SUPERCONDUCTIVITY ZUR FRAGE DER SUPRALEITUNG Proofs, 1932 TEXT AND TRANSLATION
See Introduction to Part I, p. [7].
X I , 7. 32 (Art. 492. Bohr ) Syamurvune Ruchdruckerci in Leipzig.
Fatrzrwisscnschaften 4, Springer.
492 Zur Frage der Supraleitung. Bekanntlich bietet die Deutung der metallischen Elektrizitatskitung vgm fjtandpunkt der k l w i sch en E k k t r o n e n t h w j e prinzipielle Schwierigkeiten, Zwar dieferten unter dec Annahme der freien fkveglichkeit der Elektronen im Metall sehpn die. klassischen Metho: d e n ,, eine Rrklarung des charakteristischen Verhaltpisses zwischen Elekttizit&tsr und WWneleitungsverfgogen der &letal)e ,und dessen Abhbngigkeit von d e r Tamperatur. v e d e r der normale Wert der spezifischen Warme der jMetaIle ppch die Kieinheit des Thamson-Effektes waren aber von dieser Auffassung a u s verstCnd1,ich. v i e zuerst ~ O M M E R F E L D zeigte, verschwinden jedoch diese Whwierigkeiten, wenn anstatt der klassischen Statistik eine dem PauLischen AUST gchlieDungsprinzip entsprechende Quantenstatistik zur Behandlung des Elektronengases in vetallen 'herangezogen wird. f m AnschIuO zu diesem entscheidenderi Fortschritt erwiei es sjch dann auch moglich, mit Hilfe der quantenmechanischen Methpdep die weiteren augmscheinlichen Schwierigkeiten, die der Vprstellung der Bewegungsfreiheit der Metallelektronen vom klassisth'en Standpunkt anhaften, zu iiberwinden und ins: besondere, wie von BLOCHnachgewiesen wurde, die rasche gunahme der LeTtfOhigkeit mit abnehmender Temperatur eu erklaren. Auf Grund der Vorstellung, daB die Flektronen sich im 'Metal1 unabhiingig voneinander bewegen, konnte aber keine Deutung gegeben ONNES entdeckte, werden fur das von KAMERLINGH platzliche Verschwinden des Widerstandes einiger M e t a l k EU e h e r gewissen Temperatur. In dieser Ver: bindung dorite eq vqn Interesse sein, darauf hinzuweisen, daB auf Grudd der Quantenmechanik der Strom sich unter SJmstbnden auch auffas4en &Bt als eine gemeinsame Tranelationsbewegung deq ganaen zuwmmengekoppelten Elektronengystems durch das vom Gitter der Metalliongn gebildete Gerilst. Dies fikhrt in der T a t z u einer Varstellung der Supraleitung, die eiae gewipie phnlichkeit hat mi$ einer Auffassung der metallischen Leitung, die vor meh'reren Jahren hesonders von L~NDEMANW vertrefen wurde um ?ie aben erwahnten Schwierigkeitep der FJektronengasvorstellung umaugehen, die aber vom damaligen StandF n k t njcht durchqefiihft werdeq konnhe, EWrachten wir 'ein Metallstlick vom Volurnen v , welehes N-Elektrorien q n t h g t , w o b i wir Z U ~ L C ~die L~C Gitterianan d s unbeweglicl' a p e h m e n . p e r normale .stromlose Eustand des l$ektronensystems wird dar,$estellt durch eine Wellenfunktion '
~
3
.
i .- Eot 7 T("1,. , a,; Q, , * as)en wo a(,. . ., zN die Raumkoordinaten, o,,, . ., up die
v(0)
..
.. ,
I
.
Spinkoordinaten der 8-Elektronen, n.-&e--dwd--it-% d&y&krtrt..B.&xetrscirer X m g @ - m n d E, die Gesamtenergib des Elektropensysteivs bedeuten; ferirer besitzt r d & n & e n . e i ~ d e m8 w a g i t t e r t .und ist unter BerWksich%igung der Spinkoordinaten in iibllcher Weise antiPymmetrisch. in .bezug auf jedes Elektronenpaar. Man iiberzeugt qich nun leicht, daB jeder solchen Losung sich eipe unendjiche ljchar von LSsungen anschlieBt, d i durch folgenden approximativen Ausdruck darg J t e l l t werden konnen:
h e r 1st 6 ein kleiner Parameter, a eine Lange von der GrO~enOrdniiUgdei Gitterkonstanten, madieE*QaeLcl
241
c -
h
2 masse, y,eine von der Ausgangsliisung y ( 0 ) abhangige, pericdische und antisymmetrische Funktion und A , eine ebenfalls von dieser Losung abhangige Konstante. WBhrend ly ( 0 ) stromlos war, entspricht .yr (a) ein Strom in der z-Richtung von der mittleren Dichte
wobei s die Ladung eines Elektrons bezeichnet.
Strom hauptsachlich Dirac-Fermi-Statistik Werten von 8 von der
den einzelnen Gitterionen gebunden kijnnen, wird asymptotisch A, = 0. Unter den Urnsttinden, die in den Metallen realisiert S W . wedie Abstande zwischen den Gitterionen mit den
der g r o k n Dichte der Metallelektronen bekom'mt man daher einen betrachtlichen Strom bereits f u r sehr kleine Werte des Parameters 8 . So wird, ffir A, = I O - ~und NIV = o v a = ofa as, die Stromdichte S, bereits von ' eit ddr GrbDenordnung kiner elektrom
a
'
Aa, I0
d h A
.
3 Wellenhagea vergleichbar sind mit a/S auch der Befund von D nicht nur reine Metalle, sondern auch Zegiermgen und sogar Ralbleiter supraleitend werden konnen. Det unter gewtrhnlichen Umstanden auftfetende Restwiderstand der Legierungen la5t sich, nac'hgewiesen, durch die Streuungwie vop NORDHEIM der Elektronen am unregelmai3igen Gitter erkllren.. Auch diese Ursashe des Widesstandes diirfte aber verschwinden, sobald die DarsteUung des Stromes vom Typus. Yr (&jzul&sig ist. Be] welter skigender Temperatur miissen wir annehmen,. da@ die Lbsung; Y (8 instabil wird, in dern Sinne, daS auger de-m tiefiten Energiezustand des ms- auch die Mannigraltigkeit der d e m r Geltung kommt. Die Statistik .m-rd schlie5lich mit einer MAXWELLschens Vwteilang der als f?ei zu betrachtenden Elektronen zusammenfallen, aber range vorher wird sie mit g.roSer Kmlherun durch die FERmrsche Verteilung gegeben. Den &ergan$ zwischen diesen verschiedenen Stu&n im einzelnen zu verfblgen, dtirfte eine sehr schwierige A u f s b e sein. Der empirische Befund des sprungha6ten be1 einm gewissen Yunehmenx, daD es einem Schmelzvorgang l h n wie von k d a n g an KAwEzxLrivmi ONNW betont hat. Ib der Tat durften auch einfache theoretische tfberahme stiitzen, daB im betreffenden.
wBhrend bei ghichteitigem Vorhandensein in anSchlieBenden Raumgebieten des Metalls sich je. nach der Temperatur der erste bzw. der zweite Zustand auf Kosten des anderen ansbreitet. jDiese Betrashhngen sind aber nur qualitativer Art und, bemr sich eine mehr quantitative Beschreibung entwickeln llBt, ist es schwierig, die Stlchhaltigkeit der hier vorgesehlagenen Auffassung der Supraleitungserscheinungenund deren Anwendbarkeit auf die bemerltenswerte, van KAMERLINcB QNNES und von MCLENNANentdeckte Beeinflussung des dprengpunktes durch Magnetfeider und elektrische Wechselfelder zu beurteilen. Bei dieser Gelegenheit mochte ich noch F. BLOCK uild L. RQSENPELD fiir erlauternde Diskussio herzlichen Dank aussprechen. Kopenhagen, Institut €or teoretisk Fysik,
M..BOIIB,
PART I: OVERVIEW AND POPULARIZATION
TRANSLATION [Editor’s comment: Essential handwritten corrections in the proofs are reproduced in square brackets, whereas text that Bohr asked to be removed is overwritten with a horizontal line. Formulae and symbols are taken from a manuscript for the proofs in the BMSS, since the proofs are not fully consistent in this regard. Two handwritten comments in the margin, which are difficult to read and the latter of which is in Bohr’s handwriting, have been included. The footnotes have been added by the editor with the kind assistance of Gordon Baym.]
On the Question of Superconductivity As is well known, the interpretation of metallic conduction of electricity in terms of the classical electron theory meets with difficulties of principle. It is true that, assuming free mobility of the electrons in the metal, the classical methods could provide an explanation of the characteristic ratio between the electric and thermal conductivity of metals and its temperature dependence. Neither the normal value of the specific heat of metals nor the smallness of the Thomson effect’ could, however, be understood from this point of view. As was first shown by SOMMERFELD,~ these difficulties disappear if instead of classical statistics a quantum statistics corresponding to PAULI’Sexclusion principle is invoked for treating the electron gas in the metal. Following this decisive step, it also then proved possible, with the aid of quantum mechanical methods, to resolve the other apparent difficulties to which the idea of freely mobile metal electrons give rise from the classical point of view, and, in particular, as was shown by BLoCH,3 to explain the rapid increase of conductivity with decreasing temperature. The conception of electrons moving in the metal independently of
’
The production or absorption of heat when an electric current passes through a circuit of a single material with a temperature difference along its length. It was discovered by the British physicist William Thomson (Lord Kelvin, 1824-1907) in 1854. A. Sommerfeld, Zur Elektronentheorie der Metalle, Natunviss. 15 (1927) 824-832. E Bloch, Uber die Quantenmechanik der Elektronen in Kristallgittern, Z . Phys. 52 (1928) 554600.
P A R T 1: O V E R V I E W A N D P O P U L A R I Z A T I O N
each other did not, however, permit any explanation of the sudden disappearance, discovered by KAMERLINGH ONNES?of the resistance of some metals at a certain temperature. In this connection it could be of interest to point out that in quantum mechanics the current may also in some circumstances be regarded as a common translatory motion of the whole coupled system of electrons through the framework formed by the lattice of metal ions. This actually leads to a conception of superconductivity which has a certain resemblance with a view of metallic conduction proposed several years ago, particularly by LINDEMANN,’ in order to avoid the difficulties of the electron gas conception mentioned above, but which could not be carried through on the basis of the viewpoint held at the time. Consider a piece of metal of volume V containing N electrons, assuming at first the lattice ions to be fixed. The normal current-free state of the electron system will be represented by the wave function
where X I , ... ,ZN denotes the space coordinates, 01, . .., ON the spin coordinates of the N electrons, h PLANCK’S constant divided by 2rr, and EO the total energy of the electron system; moreover q~ has in each electron coordinate a periodicity corresponding to the space lattice, and, taking into account the spin coordinates, is antisymmetric with respect to any two electrons in the usual way. One can now easily see that each such solution corresponds to an infinite family of solutions, which can be represented by the following approximate expression:
here 6 is a small parameter, a a length of the order of the lattice constant, m the electron mass, I),a periodic and antisymmetric function depending on the initial solution W(O), and A, a constant which also depends on this solution. Whereas W(0) carried no current, W(6) corresponds to a current in the x-direction with the average density EhN ma V
S, = 6-A,,
where
E
denotes the electron charge.
H. Kamerlingh Onnes, Further Experiments with Liquid Helium. D. On the Change of the The Disappearance of Electrical Resistance of Pure Metals at very low temperatures. ... the Resistance of Mercury, Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam, Proceedings of the Section of Sciences 14 (191 1) 113-1 15. E Lindemann, Note on the Theory of the Metallic State, Phil. Mag. 29 (1915) 127-141.
PART I: OVERVIEW AND POPULARIZATION
[With respect to the coordinates of any single electron, the solution Q(6) is of the same kind as the so-called modulated waves used by BLOCHto describe the states of motion of the “free” electrons. Apart from the fact, essential for our considerations, that the solution Q(6) represents the state of the combined system of all electrons, another significant difference is that the electron waves, which in the representation in question contribute the most to the resulting current, correspond, because of the use of Fermi-Dirac statistics, to values of 6 of the order of unity, and that therefore the expressions for energy and current in their dependence on 6 are much more complicated than suggested by the formulae given above, in which 6 is assumed to be a very small quantity. The constant A,, which for a given 6 is decisive for the current density S,, is determined by the forces to which the electrons in the lattice are subject. In the limit in which these forces are negligible, A, = 1; in the opposite limit in which the forces are so strong that the electrons may be regarded as being bound to individual lattice ions, A, approaches zer0.1~In the circumstances realized in metals, in which the distances between the lattice ions are comparable to the atomic diameters, A, should not decrease from unity by more than one or two powers of ten, at least if we restrict the consideration to the valence electrons. Because of the high density of electrons in a metal, we then find a considerable current already for very small values of 6. Thus, for A, = and N / V = = the current density S, reaches the order of an electromagnetic unit akeaciy [even] for 6 its d its [of the order of] lo-*. We therefore arrive at the result, paradoxical from the classical point of view, that a current of the strength relevant to the experiments on superconductivity can be conceived as a [slow] “displacement” of the electron system through the lattice without an [appreciable] disturbance of their mutual coupling. The preceding considerations are valid in the first place only at the absolute zero of temperature, where also the assumption of an independent mobility of each electron leads to an infinite conductivity. According to this latter view the resistance is of course due to the irregularities in the ion lattice caused by the thermal vibrations and can be understood as the reflection of the electron waves by the longitudinal elastic waves in the lattice. Because of the current-carrying electrons’ short wavelengths of the order a, the reflection plays a substantial role even at low temperatures and gives rise to a resistance proportional to a low power of the temperature. For the current represented by the solution q ( 6 ) such a transfer of momentum between the e - k t w ~sys-taa [individual electrons] and the lattice does not, however, arise. Instead, the lattice vibrations
This section in square brackets is crossed out by hand in the proofs.
It 1s not the case that Q(x,)
=e~kaa(x,
The Fermi distnbution dealt with by Bioch
+a)l
PART I: OVERVIEW AND POPULARIZATION
can be included in the description of the normal current-free electron state (at least when one disregards lattice waves of a wavelength comparable to a/J). k €aww ef W view is [This view is also in accordance with] the finding of DE HAAS and MEISSNER7 that not only pure metals, but also alloys and even semiconductors, can become superconducting. The residual resistance of alloys occurring under ordinary circumstances can be explained, as NORDHEIM has shown,’ by the scattering of electrons from lattice irregularities. Also this source of the resistance should disappear as soon as the representation of the current by q ( 6 ) is applicable. On a further rise of temperature we must assume that the solution 9(6) becomes unstable, in the sense that besides the lowest energy state of the electron system also the manifold of higher states comes into effect. The statistics of these states will eventually coincide with a MAXWELL distribution of electrons, considered as free, but long before that, it will be given in good approximation by the FERMIdistribution. To follow the passage between these various steps might present a very difficult task. The empirical finding of the discontinuous disappearance of superconductivity makes it natural to assume that the transition point represents a phenomenon similar to the process of melting, as was emphasized from the start by KAMERLINGH ONNES.~ Indeed, also simple theoretical considerations support the assumption that in the relevant temperature region both the “solidly” coupled state symbolized by q ( 0 ) and the “liquid” state symbolized by the Fermi distribution each form separately a stable solution, whereas in the simultaneous presence of both in neighbouring space regions of the metal, the former or the latter of these states will spread at the expense of the other according to the temperature. However, these considerations are only qualitative in nature, and before a more quantitative treatment can be developed, it is difficult to judge the reliability of the view of superconductivity phenomena proposed here and their applicability to explain the remarkable influence on the transition temperature of magnetic and alternating electrical fields discovered by KAMERLINGH ONNESand MCLENNAN. I would like to take the opportunity to express my sincere thanks to F. BLOCH and L. ROSENFELD for instructive discussions. Copenhagen, Institute for Theoretical Physics, [August] 1932. N. BOHR
’ See R E Dahl, Superconductivity:
Its Historical Roots and Development from Mercury to the Ceramic Oxides, American Institute of Physics, New York, 1992, p. 137, and the work cited there. L. Nordheim, Zur Elektronentheorie der Metalle. I , Ann. d. Phys. 9 (1931) 607440; Zur Elektronentheorie der Metalle. 11, ibid., 641478. Ref. 4.
See Introduction to Part I, p. [7].
X I , 7. 32 (Art. 492. Bohr ) Syamurvune Ruchdruckerci in Leipzig.
Fatrzrwisscnschaften 4, Springer.
492 Zur Frage der Supraleitung. Bekanntlich bietet die Deutung der metallischen Elektrizitatskitung vgm fjtandpunkt der k l w i sch en E k k t r o n e n t h w j e prinzipielle Schwierigkeiten, Zwar dieferten unter dec Annahme der freien fkveglichkeit der Elektronen im Metall sehpn die. klassischen Metho: d e n ,, eine Rrklarung des charakteristischen Verhaltpisses zwischen Elekttizit&tsr und WWneleitungsverfgogen der &letal)e ,und dessen Abhbngigkeit von d e r Tamperatur. v e d e r der normale Wert der spezifischen Warme der jMetaIle ppch die Kieinheit des Thamson-Effektes waren aber von dieser Auffassung a u s verstCnd1,ich. v i e zuerst ~ O M M E R F E L D zeigte, verschwinden jedoch diese Whwierigkeiten, wenn anstatt der klassischen Statistik eine dem PauLischen AUST gchlieDungsprinzip entsprechende Quantenstatistik zur Behandlung des Elektronengases in vetallen 'herangezogen wird. f m AnschIuO zu diesem entscheidenderi Fortschritt erwiei es sjch dann auch moglich, mit Hilfe der quantenmechanischen Methpdep die weiteren augmscheinlichen Schwierigkeiten, die der Vprstellung der Bewegungsfreiheit der Metallelektronen vom klassisth'en Standpunkt anhaften, zu iiberwinden und ins: besondere, wie von BLOCHnachgewiesen wurde, die rasche gunahme der LeTtfOhigkeit mit abnehmender Temperatur eu erklaren. Auf Grund der Vorstellung, daB die Flektronen sich im 'Metal1 unabhiingig voneinander bewegen, konnte aber keine Deutung gegeben ONNES entdeckte, werden fur das von KAMERLINGH platzliche Verschwinden des Widerstandes einiger M e t a l k EU e h e r gewissen Temperatur. In dieser Ver: bindung dorite eq vqn Interesse sein, darauf hinzuweisen, daB auf Grudd der Quantenmechanik der Strom sich unter SJmstbnden auch auffas4en &Bt als eine gemeinsame Tranelationsbewegung deq ganaen zuwmmengekoppelten Elektronengystems durch das vom Gitter der Metalliongn gebildete Gerilst. Dies fikhrt in der T a t z u einer Varstellung der Supraleitung, die eiae gewipie phnlichkeit hat mi$ einer Auffassung der metallischen Leitung, die vor meh'reren Jahren hesonders von L~NDEMANW vertrefen wurde um ?ie aben erwahnten Schwierigkeitep der FJektronengasvorstellung umaugehen, die aber vom damaligen StandF n k t njcht durchqefiihft werdeq konnhe, EWrachten wir 'ein Metallstlick vom Volurnen v , welehes N-Elektrorien q n t h g t , w o b i wir Z U ~ L C ~die L~C Gitterianan d s unbeweglicl' a p e h m e n . p e r normale .stromlose Eustand des l$ektronensystems wird dar,$estellt durch eine Wellenfunktion '
~
3
.
i .- Eot 7 T("1,. , a,; Q, , * as)en wo a(,. . ., zN die Raumkoordinaten, o,,, . ., up die
v(0)
..
.. ,
I
.
Spinkoordinaten der 8-Elektronen, n.-&e--dwd--it-% d&y&krtrt..B.&xetrscirer X m g @ - m n d E, die Gesamtenergib des Elektropensysteivs bedeuten; ferirer besitzt r d & n & e n . e i ~ d e m8 w a g i t t e r t .und ist unter BerWksich%igung der Spinkoordinaten in iibllcher Weise antiPymmetrisch. in .bezug auf jedes Elektronenpaar. Man iiberzeugt qich nun leicht, daB jeder solchen Losung sich eipe unendjiche ljchar von LSsungen anschlieBt, d i durch folgenden approximativen Ausdruck darg J t e l l t werden konnen:
h e r 1st 6 ein kleiner Parameter, a eine Lange von der GrO~enOrdniiUgdei Gitterkonstanten, madieE*QaeLcl
241
c -
h
2 masse, y,eine von der Ausgangsliisung y ( 0 ) abhangige, pericdische und antisymmetrische Funktion und A , eine ebenfalls von dieser Losung abhangige Konstante. WBhrend ly ( 0 ) stromlos war, entspricht .yr (a) ein Strom in der z-Richtung von der mittleren Dichte
wobei s die Ladung eines Elektrons bezeichnet.
Strom hauptsachlich Dirac-Fermi-Statistik Werten von 8 von der
den einzelnen Gitterionen gebunden kijnnen, wird asymptotisch A, = 0. Unter den Urnsttinden, die in den Metallen realisiert S W . wedie Abstande zwischen den Gitterionen mit den
der g r o k n Dichte der Metallelektronen bekom'mt man daher einen betrachtlichen Strom bereits f u r sehr kleine Werte des Parameters 8 . So wird, ffir A, = I O - ~und NIV = o v a = ofa as, die Stromdichte S, bereits von ' eit ddr GrbDenordnung kiner elektrom
a
'
Aa, I0
d h A
.
3 Wellenhagea vergleichbar sind mit a/S auch der Befund von D nicht nur reine Metalle, sondern auch Zegiermgen und sogar Ralbleiter supraleitend werden konnen. Det unter gewtrhnlichen Umstanden auftfetende Restwiderstand der Legierungen la5t sich, nac'hgewiesen, durch die Streuungwie vop NORDHEIM der Elektronen am unregelmai3igen Gitter erkllren.. Auch diese Ursashe des Widesstandes diirfte aber verschwinden, sobald die DarsteUung des Stromes vom Typus. Yr (&jzul&sig ist. Be] welter skigender Temperatur miissen wir annehmen,. da@ die Lbsung; Y (8 instabil wird, in dern Sinne, daS auger de-m tiefiten Energiezustand des ms- auch die Mannigraltigkeit der d e m r Geltung kommt. Die Statistik .m-rd schlie5lich mit einer MAXWELLschens Vwteilang der als f?ei zu betrachtenden Elektronen zusammenfallen, aber range vorher wird sie mit g.roSer Kmlherun durch die FERmrsche Verteilung gegeben. Den &ergan$ zwischen diesen verschiedenen Stu&n im einzelnen zu verfblgen, dtirfte eine sehr schwierige A u f s b e sein. Der empirische Befund des sprungha6ten be1 einm gewissen Yunehmenx, daD es einem Schmelzvorgang l h n wie von k d a n g an KAwEzxLrivmi ONNW betont hat. Ib der Tat durften auch einfache theoretische tfberahme stiitzen, daB im betreffenden.
wBhrend bei ghichteitigem Vorhandensein in anSchlieBenden Raumgebieten des Metalls sich je. nach der Temperatur der erste bzw. der zweite Zustand auf Kosten des anderen ansbreitet. jDiese Betrashhngen sind aber nur qualitativer Art und, bemr sich eine mehr quantitative Beschreibung entwickeln llBt, ist es schwierig, die Stlchhaltigkeit der hier vorgesehlagenen Auffassung der Supraleitungserscheinungenund deren Anwendbarkeit auf die bemerltenswerte, van KAMERLINcB QNNES und von MCLENNANentdeckte Beeinflussung des dprengpunktes durch Magnetfeider und elektrische Wechselfelder zu beurteilen. Bei dieser Gelegenheit mochte ich noch F. BLOCK uild L. RQSENPELD fiir erlauternde Diskussio herzlichen Dank aussprechen. Kopenhagen, Institut €or teoretisk Fysik,
M..BOIIB,
PART I: OVERVIEW AND POPULARIZATION
TRANSLATION [Editor’s comment: Essential handwritten corrections in the proofs are reproduced in square brackets, whereas text that Bohr asked to be removed is overwritten with a horizontal line. Formulae and symbols are taken from a manuscript for the proofs in the BMSS, since the proofs are not fully consistent in this regard. Two handwritten comments in the margin, which are difficult to read and the latter of which is in Bohr’s handwriting, have been included. The footnotes have been added by the editor with the kind assistance of Gordon Baym.]
On the Question of Superconductivity As is well known, the interpretation of metallic conduction of electricity in terms of the classical electron theory meets with difficulties of principle. It is true that, assuming free mobility of the electrons in the metal, the classical methods could provide an explanation of the characteristic ratio between the electric and thermal conductivity of metals and its temperature dependence. Neither the normal value of the specific heat of metals nor the smallness of the Thomson effect’ could, however, be understood from this point of view. As was first shown by SOMMERFELD,~ these difficulties disappear if instead of classical statistics a quantum statistics corresponding to PAULI’Sexclusion principle is invoked for treating the electron gas in the metal. Following this decisive step, it also then proved possible, with the aid of quantum mechanical methods, to resolve the other apparent difficulties to which the idea of freely mobile metal electrons give rise from the classical point of view, and, in particular, as was shown by BLoCH,3 to explain the rapid increase of conductivity with decreasing temperature. The conception of electrons moving in the metal independently of
’
The production or absorption of heat when an electric current passes through a circuit of a single material with a temperature difference along its length. It was discovered by the British physicist William Thomson (Lord Kelvin, 1824-1907) in 1854. A. Sommerfeld, Zur Elektronentheorie der Metalle, Natunviss. 15 (1927) 824-832. E Bloch, Uber die Quantenmechanik der Elektronen in Kristallgittern, Z . Phys. 52 (1928) 554600.
P A R T 1: O V E R V I E W A N D P O P U L A R I Z A T I O N
each other did not, however, permit any explanation of the sudden disappearance, discovered by KAMERLINGH ONNES?of the resistance of some metals at a certain temperature. In this connection it could be of interest to point out that in quantum mechanics the current may also in some circumstances be regarded as a common translatory motion of the whole coupled system of electrons through the framework formed by the lattice of metal ions. This actually leads to a conception of superconductivity which has a certain resemblance with a view of metallic conduction proposed several years ago, particularly by LINDEMANN,’ in order to avoid the difficulties of the electron gas conception mentioned above, but which could not be carried through on the basis of the viewpoint held at the time. Consider a piece of metal of volume V containing N electrons, assuming at first the lattice ions to be fixed. The normal current-free state of the electron system will be represented by the wave function
where X I , ... ,ZN denotes the space coordinates, 01, . .., ON the spin coordinates of the N electrons, h PLANCK’S constant divided by 2rr, and EO the total energy of the electron system; moreover q~ has in each electron coordinate a periodicity corresponding to the space lattice, and, taking into account the spin coordinates, is antisymmetric with respect to any two electrons in the usual way. One can now easily see that each such solution corresponds to an infinite family of solutions, which can be represented by the following approximate expression:
here 6 is a small parameter, a a length of the order of the lattice constant, m the electron mass, I),a periodic and antisymmetric function depending on the initial solution W(O), and A, a constant which also depends on this solution. Whereas W(0) carried no current, W(6) corresponds to a current in the x-direction with the average density EhN ma V
S, = 6-A,,
where
E
denotes the electron charge.
H. Kamerlingh Onnes, Further Experiments with Liquid Helium. D. On the Change of the The Disappearance of Electrical Resistance of Pure Metals at very low temperatures. ... the Resistance of Mercury, Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam, Proceedings of the Section of Sciences 14 (191 1) 113-1 15. E Lindemann, Note on the Theory of the Metallic State, Phil. Mag. 29 (1915) 127-141.
PART I: OVERVIEW AND POPULARIZATION
[With respect to the coordinates of any single electron, the solution Q(6) is of the same kind as the so-called modulated waves used by BLOCHto describe the states of motion of the “free” electrons. Apart from the fact, essential for our considerations, that the solution Q(6) represents the state of the combined system of all electrons, another significant difference is that the electron waves, which in the representation in question contribute the most to the resulting current, correspond, because of the use of Fermi-Dirac statistics, to values of 6 of the order of unity, and that therefore the expressions for energy and current in their dependence on 6 are much more complicated than suggested by the formulae given above, in which 6 is assumed to be a very small quantity. The constant A,, which for a given 6 is decisive for the current density S,, is determined by the forces to which the electrons in the lattice are subject. In the limit in which these forces are negligible, A, = 1; in the opposite limit in which the forces are so strong that the electrons may be regarded as being bound to individual lattice ions, A, approaches zer0.1~In the circumstances realized in metals, in which the distances between the lattice ions are comparable to the atomic diameters, A, should not decrease from unity by more than one or two powers of ten, at least if we restrict the consideration to the valence electrons. Because of the high density of electrons in a metal, we then find a considerable current already for very small values of 6. Thus, for A, = and N / V = = the current density S, reaches the order of an electromagnetic unit akeaciy [even] for 6 its d its [of the order of] lo-*. We therefore arrive at the result, paradoxical from the classical point of view, that a current of the strength relevant to the experiments on superconductivity can be conceived as a [slow] “displacement” of the electron system through the lattice without an [appreciable] disturbance of their mutual coupling. The preceding considerations are valid in the first place only at the absolute zero of temperature, where also the assumption of an independent mobility of each electron leads to an infinite conductivity. According to this latter view the resistance is of course due to the irregularities in the ion lattice caused by the thermal vibrations and can be understood as the reflection of the electron waves by the longitudinal elastic waves in the lattice. Because of the current-carrying electrons’ short wavelengths of the order a, the reflection plays a substantial role even at low temperatures and gives rise to a resistance proportional to a low power of the temperature. For the current represented by the solution q ( 6 ) such a transfer of momentum between the e - k t w ~sys-taa [individual electrons] and the lattice does not, however, arise. Instead, the lattice vibrations
This section in square brackets is crossed out by hand in the proofs.
It 1s not the case that Q(x,)
=e~kaa(x,
The Fermi distnbution dealt with by Bioch
+a)l
PART I: OVERVIEW AND POPULARIZATION
can be included in the description of the normal current-free electron state (at least when one disregards lattice waves of a wavelength comparable to a/J). k €aww ef W view is [This view is also in accordance with] the finding of DE HAAS and MEISSNER7 that not only pure metals, but also alloys and even semiconductors, can become superconducting. The residual resistance of alloys occurring under ordinary circumstances can be explained, as NORDHEIM has shown,’ by the scattering of electrons from lattice irregularities. Also this source of the resistance should disappear as soon as the representation of the current by q ( 6 ) is applicable. On a further rise of temperature we must assume that the solution 9(6) becomes unstable, in the sense that besides the lowest energy state of the electron system also the manifold of higher states comes into effect. The statistics of these states will eventually coincide with a MAXWELL distribution of electrons, considered as free, but long before that, it will be given in good approximation by the FERMIdistribution. To follow the passage between these various steps might present a very difficult task. The empirical finding of the discontinuous disappearance of superconductivity makes it natural to assume that the transition point represents a phenomenon similar to the process of melting, as was emphasized from the start by KAMERLINGH ONNES.~ Indeed, also simple theoretical considerations support the assumption that in the relevant temperature region both the “solidly” coupled state symbolized by q ( 0 ) and the “liquid” state symbolized by the Fermi distribution each form separately a stable solution, whereas in the simultaneous presence of both in neighbouring space regions of the metal, the former or the latter of these states will spread at the expense of the other according to the temperature. However, these considerations are only qualitative in nature, and before a more quantitative treatment can be developed, it is difficult to judge the reliability of the view of superconductivity phenomena proposed here and their applicability to explain the remarkable influence on the transition temperature of magnetic and alternating electrical fields discovered by KAMERLINGH ONNESand MCLENNAN. I would like to take the opportunity to express my sincere thanks to F. BLOCH and L. ROSENFELD for instructive discussions. Copenhagen, Institute for Theoretical Physics, [August] 1932. N. BOHR
’ See R E Dahl, Superconductivity:
Its Historical Roots and Development from Mercury to the Ceramic Oxides, American Institute of Physics, New York, 1992, p. 137, and the work cited there. L. Nordheim, Zur Elektronentheorie der Metalle. I , Ann. d. Phys. 9 (1931) 607440; Zur Elektronentheorie der Metalle. 11, ibid., 641478. Ref. 4.