Po.R,Soo.Lond.A371,39-48(1980) Printed in Great Britain Solid state physics 1925-88 opportunities missed and opportunities seized BY A H WIlsoN, F.R. S. Wilson, Sir Alan H. Born Wallasey 1906. Studied in Cambridge and Germany University lecturer in mathematics at Cambridge until 1945. Research in electrons in solids; author of Theory of Metals(Cambridge)1936, 2nd edition 1953. Career since World War II in industry. Deputy Chairman of Courtaulds, Deputy Chairman of the Electricity Council, Chairman of Gla.o Group THE PERIOD UP TO 1929 The discovery of the electron by J.J. Thomson in 1897 enabled P Drude to produce in 1goo what seemed at first to be a very satisfactory theory of the electrical and thermal conductivities of metals. In Igo4-5, H. A Lorentz gave an improved mathe matical formulation of Drude's theory but without essentially adding anything to its physical content. However, as endeavours were made to embrace more and more of the properties of metals within the theory, it became clear during the next decade or so that it was impossible to encompass all of them without introducing a number of ad hoc and conflicting assumptions. The theory therefore fell into a state of dis repute and disorder which is well portrayed in the Report of the Solvay Conference of1924. It was not until I25 that a theory could have emerged which would have been an advance on that of Drude and lorentz, the occasion being the publication by W. Pauli of his paper'Uber den Zusammenhang des Abschlusses der Elektronen gruppen im Atom mit der Komplexstruktur der Spektren. This formulated the Exclusion Principle in the following words(slightly simplified) Es kann niemals zwei oder mehrere aquivalente Elektronen im Atom geben fur welche die Werte aller Quantenzahlen. . ubereinstimmen. Ist ein Elektron im Atom vorhanden, fur das diese Quantenzahlen., bestimmte Werte haben, so ist dieser zustand‘ besetzt’ Eine nahere Begruindung fur diese Regel konnen wir nicht geben, sie scheint sich jedoch von selbst als sehr naturgemasz darzubieten Pauli did not go on to extend his Exclusion Principle to the conduction electrons in a metal. Neither did E. Fermi, though he is often credited with having done so The paper published in 126 in which the Fermi distribution function is introduced is entitled"Zur Quantelung des idealen einatomigen Gases, and its abstract is as follows Wenn der Nernstsche Warmesatz auch fur das ideale Gas seine Gultigkeit behalten soll, muss man annehmen, dass die gesetze idealer Gase bei niedrigen
例如c. R. Soc, . 4. A 371, 39-48 (1980) Prin!W. in Gt-w.t Bri阳"、 Solid state physics 1925-33: opportunities missed and opportunities seized By A. H. WILSOl霄， F.R.S 1. Sir Alan H. Born JVal 8ey 1906. Sludied Cω ，.; and Ge俨many versity lurer in mathemalic8 at Q， mbridge 也饨til1945. e8 rch in elwr0n8 in80l坤，酬tlw ofTh ry ofMetals (Cambridge) 1936, 2nd edilion 1953 向时 "阳 JVvrld IV ar 11 in indust Deputy Chairman 01 Courta叫白 Deputy irman 01 the El lricity Co cil Chairma ofGlaxo Gr THE PEBIOD UP TO 1929 The di.scovery of heelec ron by J. J. Thomson in 1897 enabled P. Drude to produce 呵。 what seemed at first be a very satisfac阳'y恤回ry of the electrical and hermal conductivities of me als. In 11)04-5 , H. A. Lorentz gave an improved mathematical formulation of Drude's 由四ry but wi hou entially adding anything to its physical co ent. However, as endeavours were made to embrace mo and more of the properties of metals within he heory ，他 became clear during the next decade or 80 that it was impo ibletoen mp all ofthem without introducing a number of ad hoc and confiicting 嗣.s ump ions. The theory therefore fell into a state of dis pute and '0 er which is well portrayed in the Report of he Solv町Conference of 1924 It not untill92S that a th ry could have emerged which would have been an advance on th of Drude and Lo rent毡， the occasion being the publication by W. Pauli of his e，‘ Über den Zusammenhang Abschlus der Elektronengruppen im Atom mit der Komplexstruktur der Spektren.' This formula.ted the Exclusion Prìnciple in he following words (sligh sìmplified) Es kann niemals zwei oder mehrere äquivalente Elektronen im Atom geben, für welche die Nerte al1er Quantenzahlen... übereìnstimmen. ein Elektron im Atom vorhanden, für diese Quan ，由len... bestimmte Nerte haben ，回 ist die zustand 'bese t' Eine nähere Begtündung für diese R.egel können wir nicht geben, sie sche皿也 sich jedoch von selbst als sehr natu em darzubieten Pauli did no go on to extend his Exclusion Principle to the conduction electrons in a metal. Neither did E. Fermi ，也hough he is of credited with having done so Thep er published in 1926 which the Fermi distribution function is introdu is enti Zur Quantelung des idealen einatomigen Gases', and its abstract ìs follows Wenn der Nemstsche Wärmesatz auch für das ideale 跑回ine Gu tigkei behalten soll, mu man annehmen, da die Gesetze idealer Ga bei niedrigen [ 39 ]
A.h. wilson Temperaturen von den klassischen abweichen. Die Ursache dieser Entartung is in einer Quantelung der Molekularbewegungen zu suchen Bei allen Theorien der Entartung werden immer mehr oder weniger willkiirliche Annahmen uber das statistische Verhalten der Molekule, oder uber ihre Quante- ng gemacht. In der vorliegenden Arbeit wird nur die von Pauli zuerst ausges- prochene. Annahme benuzt, dass in einem System nie zwei gleichwertige Elemente vorkommen konnen, deren Quantenzahlen vollstandig ubereinstim men. Mit dieser Hypothese werden die Zustandsgleichung und die innere Energie des idealen Gases abgeleitet; der Entropiewert fur grosse Temperaturen stimmt mit der Stern-Tetrodeschen uiberein In addition to assuming the Pauli principle, Fermi used the old quantum theory to determine the allowed energy levels of the individual atoms by supposing that they behaved like harmonic oscillators with quantum numbers 1, 82, 83(8=0, 1, 2, . )and energies hvs, with 8=81+82+83 Then, if the total number of atoms is N and the total energy is Ehv,∑N=N,∑8N= where N, is the number of atoms with quantum numbers 8. The number of com plexions for given 8 is Q=(8+1)(s+2)N, and the number of arrangements of the N, atoms over the @, levels complying with the Pauli principle is W=QV/IN1(Q,-N)1 Hence, maximizing W, subject to the conditions(1), we have N=Q This is the first appearance of the Fermi function, but, though the derivation is correct according to the assumptions made, Fermi (or Pauli)statistics is inapplic- able to structureless monatomic gases(i.e gases whose atoms haveno uncompensated electronic or nuclear spin). The same error was made by P. A. M. Dirac later in I926 in his paperOn the theory of quantum mechanics, and perhaps with less justifi cation. Fermi based his arguments on the old quantum the eory, whereas Dirac wrote in the context of the new quantum theory. Starting from the consideration that the Hamiltonian of a system of indistinguishable particles is a symmetrical function of the coordinates of the individual particles, Dirac correctly deduced that the wavefunction must be either a symmetrical or an antisymmetrical function of those coordinates To comply with the Pauli principle, symmetrical wavefunctions must be excluded. Dirac wrote, The solution with symmetrical eigenfunctions must be the correct one when applied to light quanta, since it is known that the Einstein Bose statistical mechanics leads to Planck's law of black-body radiation. The solution with antisymmetrical eigenfunctions, though, is probably the correct one for gas molecules, since it is known to be the correct one for electrons in an atom and one would expect molecules to resemble electrons more closely than light
40 A. H. Wilson Temperaturen VQn den klaasischen abweichen. Die Ursache die Entartung is in einer Quantelung der M:olekülarbewegungen zu suchen Heg 80n to Bei allen Theorien der Entartung werden immer mehr oder weniger willkürliche Annahmen über das statistische Verhalten der Moleküle, oder über ihre Quan lung gemach ln der vorJi唔enden Arbeìt wird nur die von Pauli zuers .u :es prochene ... Annahme benuz a.s in einem System nie zwei gleichwertige Elemer vorkommen können, deren Quantenzahlen voIl ndig übereinstim men. Mi di erHypo lese werden die Zustandsgleichung und die innere Energie des idealen Gases abgelei也剖; der Entropiewert für gro Temperaturen stimmt der Stern-Tetrodeschen überein ln addition to assuming he Pauli principle, Fermi the old quantum heory to de rmine the a.llowed energy levels of the ìndivìdual atoms by supposing h.t t.hey behaved like harmonìc oscillators with quantum numbers 81, 82, 83 (8‘= 0, 1, 2, ) and energies hV8, with 8 = 81 +82+83 , Then, if he total number of atoms is N and the total energy is Ehv, :E 飞 = N , :E 8N~ = E, (1) where N, ìs the number of ms with quantum numbers 8. The number of m plexiona for given 8 is Q. ~ ,(8+ 1)(8+ 2)N, (2) and the num ber of a. ngements of the atoms over the ~ levels complying with the Pauli principle is 月~ !/[ !(Q -l飞) !] Hence, maximizing r~ subject to the conditiona (川， we have fJ /(1 +ae (3) (4) This is the fi且也 appearance of he Fermi function, but, though the derivation is correct according to the ump ons made, Fermi (or Pauli) 吼叫自tics is inapplic a.ble to 8tructureleωmona阳皿cgaa (i.e ga whose a.tomsha.ve noun mpensa electronic or nuclear spin). The same error made by P. A. M. Dirac la也erin 1926 in hia paper 'On the heory of quantum mechanics', and perhapa wi less justific. on. Fermi based his arguments on the old quantum heo 巾， where Dirac wrote in he ntext of the new quantum heory. Starting from the conaideration that the Hamiltonian of a. system of indistinguishable pa icles is a. symmetrica.1 function of he ∞。rdinates ofthe individual particles, Dirac correctly deduced h.t the wa.vefunction must be either a symmetrical or an an symmetrica. func创。 of 血。因∞{)rdin也恤s. To comply with he Pa.uli princìple, symmetrical wavefunctions must beexcJ uded. Dira.cwrote, 'Thesolution with symmetrical eigenfunctiona must be he oorrect one when a.pplied to light qua. 恤， sln曲比 is known h.t he EinsteinBose statistical mechanÎcs leads to Planck' l a. of bla.ck-body ra.diation. The lution wi antiaymmetrical eigenfunctions ，也hough Îs probably he rrec one for molecules since is known to be the correct one for electrons in a.n atom, and one would expect molecules 阳回回mble electrons more closely than light
Opportunities missed and opportunities seized quanta. 'Dirac's mistake, like Fermis, is, of course, the omission of the spin of the electron. But whereas the electron was considered to be a structureless mass point in 1925, in 1926 the hypothesis of G. E. Uhlenbeck S Goudsmit, that the electron possessed an intrinsic spin, was generally accepted, and it therefore followed that the eigenfunctions of an ideal spinless gas should be symmetrical functions of the space coordinates, and that Fermi-Dirac statistics is not applicable to such a gas The first practical as distinct from theoretical problem to which the Fermi- Dirac statistics was correctly applied was that of the behaviour of White Dwarf stars. In what is probably his most important paper, published late in 1926 and entitled'On dense matter,R. H. Fowler put forward the hypothesis that matter in a White Dwarf consists of bare nuclei and free electrons, and that the ultimate fate of a White dwarf is to become a Black Dwarf similar to a single gigantic molecule in its lowest quantum state, the specific heat of the condensed electron (and nuclear) gas being effectively zero. This major step forward, though acclaimed by astrophysicists, received scant attention by physicists, and a paper by Pauli, published early in 1927 and entitled Uber Gasentartung und Paramagnetismus', received little more. Pauli wrote Die,. von Fermi herruihrende Quantenstatistik des einatomigen idealen Gases wird auf den Fall von Gasatomen mit Drehimpuls erweitert. Betrachtet man die Leitungselektronen im Metall als entartetes idealen Gas-was gewiss nur als ganz provisioned anzusehen ist-so gelangt man auf Grund der entwickelten Statistik zu einen wenigstens qualitativen theoretischen Verstandnis der Tat- sache, dass trotz des Vorhandenseins des Eigenmomentes des Elektrons viele Metalle in ihrem festen zustand keinen oder nur einen sehr schwachen und annahernd temperaturunabhangigen Paramagnetismus zeigen e. This paper was largely taken up by a discussion of the difference between instein-Bose and Fermi-Dirac statistics, and, except for the derivation of the paramagnetic susceptibility, contained very little of physical interest. It will be seen that in this paper Pauli is far from insisting that the conduction electrons in a metal should definitely be treated as a degenerate gas, but anyone really familiar with the Drude-Lorentz theory of metallic conduction could have seen in a com bination of Fowler's and Pauli's papers a key to the solution of the difficulties that had beset the theory. The most outstanding stumbling block(there were many more)was, on the one hand, the necessity for the number of the conduction electrons to be of the order of one per atom, and, on the other hand, for the number to be negligibly small. The first requirement arose from the magnitude of the Hall coefficient, while the second was the basis for one explanation of the fact that the specific heat per atom was the same for insulators and conductors. These require ments could now be reconciled, but it was not until a year later(1928)that Sommer old published his paper 'Zur Elektronentheorie der Metalle auf grund der Fermi- schen Statistik, which is the real starting point for the major developments that were to follow Why had it taken so long to arrive at this point? So far as Fermi and Dirac are
Oppoγt侃侃ities missed and 'portunitics seized 41 quanta.' Dirac's mistake, like Fermi's, is, of ∞"""' he omi ion of he spin of the electron. But whereas the electron was considered to be a structureless mass point in 1925, ìn 1926 he hypothesis ofG. E. Uhlenbeck & S. Goudsmi hat the electron posse鸣幽 an intrinsic spin, was generally accep and it herefore followed that he eigenfunctions of an ideal spinIess should be symmetrical functions of he space coordinates, and at Fermi Dirac atistic8 is not applicable to such a ga.s The firs practical as distinct from heoreticaI problem to wruch FcrmiDirac statistics was rre tly applied was hat of the behaviour of White Dwarf stars. In what is probab!y his mos importan paper published late in 1926 and enti led ' On dense matter' , R. H. Fowler put forwaro he hypo eSls hat mat in a Whi Dwarf consÎsts of bare nuclei and free electrons, and hat he ultimate fate of a. White Dwarf is to become a Black Dwa.rf similar to a single gigantic molecule in its lowes qu a.ntum state, the specific heat of the condensed electron (a.nd nuclear) gas being effectively zero This 坷。 step forwaI址， though acclaimed by 剧也rophysicists ，目。eived sca.nt attention by physicis恤， and a er by Pauli, published early in 1927 a.nd entitled 'über Ga8entartung und Paramagnetismus', received little more. Pauli wro Die ... von Fermi herrührende Quan nsta.tistik des eina.tomigen idcalen Ga wird auf den Fa.ll von atomen mi Drehimpuls erwe crt ... Betrach阳也 man die Leitungselektronen im Metall als en rtetes idealen Ga8 - was gewi nm ganz provisioncl anzusehen ist - 80 gelangt man a.uf Grund der entwickelten 阳岛istik zu cincn wenigstens qua. litativen heoretischen Ve tändnis der Tat. sache, das.s trotz Vorhanden ins des Eigenmomen des Elektron8 viele Meta.lle in ihrem fcs zustand keinen oder nur einen hr schwa.chen und annähcrnd tempera.turunabhãngigen Paramagnetismus zcigen This 叩时 was largely taken up by a discussion of the differen between Einstein-Bose a.nd Fermi- Dirac atisti曲， and, except for the derivation of he paramagnetic suscep bili conta.ined very little of physical in erest. It will be en at in thi8 paper Pauli i8 far from insisting that the conduction electrons in a meta.1 should del1nitely be 也， 也阻 as a degcnerate gas, but anyone really fami Jia.r wi仙也he Drude-Lorentz ~ory of metallic conduction could have 8een in a com. bination of Fowler's and Pauli's papers a key to the solution of the di culties hat had beset the theory. The mo outstanding umbling block (there were many more)w剧， on the one hand, the necessity for he numberofthe conduc nelectrons b
A. H. Wilson concerned the answer probably is that they were much more interested in genera theory than in specific applications, But Fowler and Pauli were interested in appli cations but missed the main one. I knew Fowler well, but I only met Pauli once in Copenhagen in April 1931. When I brought the subject up with Fowler, he said I had the thing right under my nose but I couldn't see it was there. I kick myself whenever I think of it. Pauli was more explicit. He had been engaged over many years in dealing with various magnetic problems by means of the old quantum theory, with varying success. Some problems could be solved satisfactorily, others yielded to a mixture of sound theory and currently unfounded conjectures, while others were quite intractable. One of the problems of the third kind was the weak paramagnetism of the alkali metals. But he had left this somewhat narrow field behind him for the more exciting developments which led to the birth of the new quantum mechanics. However, when the papers of Fermi and Dirac appeared, it occurred to Pauli in a flash that here was the solution to a minor problem which had long been troubling him. But once he had written his paper, solid state magnetic problems were to him a completed chapter, and it never occurred to him that there might be another more exciting chapter on a related theme. His main interest was to establish his theory of the spinning electron To revert to Sommerfeld, he took over Lorentz's theory in its entirety, but with the free electrons obeying the Fermi-Dirac statistics instead of the classical, Maxwellian, statistics. It was therefore essentially a phenomenological theory depending upon two parameters, n, the number of free electrons per unit volume, and 4, the mean free path of the electrons. Since the specific heat of the electrons temperatures, n and I could be deduced purely from the conduction phenomen 4 was negligible compared with that of the lattice vibrations, except at very lo e. The theory of the Hall effect showed that n must be of the same order of magnitude the number of atoms per unit volume, and, to obtain the correct value of the conductivity, for example for silver at room temperature, the mean free path l had to be of the order of 100 interatomic distances and be proportional to 1/T, This behaviour of the mean free path was inexplicable on any classical collision theory, and the correct explanation was given by F. Bloch in 1928 by a thoroughgoing application of quantum theory, on the assumption that a single-electron theory was adequate for this purpose It was shown by G. Floquet in 1883 that the fundamental solutions of any linear differential equation L[f]=0, with one independent variable a, whose coefficients are periodic functions of a with period 2T, are of the form f(a)=e/tu(a), where the exponent u is either complex or purely imaginary and where a(er)=u(a + 2T). Now the potential energy of an electron moving in a crystal lattice must have the same periodicity characteristics as those of the lattice, and Bloch generalized Floquet's theorem to show that the wavefunction of such an electron must be of the form y(r)=e ru (r), where u(r) has the periodicity of the lattice In other words, provided that k is real, the wave function of an electron in a crystal lattice s a modulated plane wave spread over the whole crystal, and a conduction electron
42 A. H. Wilson concerned the answer probably is that they were much more interested in genera.l 白白'y由an in specific applica.t.ions. But Fowler and Pauli were inte sted in appli cations bu missed the main one. 1 knew Fowler well, but 1 only met Pauli once - in Copenhagen in April t931. When 1 brough the subject up with Fowler, he said '1 ha.d the hing right under my no bu也 I uldn' see there. 1 kick myself wheneve I 也hink of 'Pau li more exp1icit . He ha.d been engaged Qver ma.ny years in dealing wi various magnetic problems by means of the old quantum eory Wl varying succ s. Some problems could be solved satisfactorily, others yielded to a mixture of sou nd eory and currently unfounded conjectures, while 。也hers were qui ractable. One of the problems of the third kind WM the weak paramagnetism of he a.lkali metals. Bu he had left this somewha narrow field behind him for the more exciting developments which 时也 the bir也h of the new quantum mechanics. However, when he papers of Fermi and Dirac a. ppear时，她 occurred to Pauli in a. flash hat here was the solution to a minor problem which had long been roubling him. Butonce he had wri ten his paper, solid state a.gn ie problems were him a completed chap and never occurred him that there might be ano her more exciting chapter on a. re ated heme. His main interest was to establish his th ryof 'p in electron To revert to Sommerfeld, he took over Lorentz's theory in its entire bu with the free electrons obeying the Fermi- Dirac sta.tistics in.stead of the sical Ma.xwel1ian, statistics. It 础也herefore entially a phenomenological theory depending upon two para. mete阻，饨，也he number of free ec rons per um volume a.nd 1, the mea.n free pa.th of electrons. Since the specific he of the electrons was negligible mpa. red w比 that of the la.ttice vibrations, except very low temperatures, n a.nd 1 could be deduced purely from the conduction phenomena The theory of Hal! effec showed atnm stbeof he sa.me oroer of magnitude as the number of atoms per unit volume, and ，也 obta.in the correct value of the conductivity, for ex nple for silver at room temperature ，也 he mea.n free pa.th l ha.d to be of the order of 100 interatomic distances and be proportional to 1fT. This beha.viour of the mean fr path inexplicable on any cl sica ∞IIi sion theory, and heωη阳也 exp anation was given by F. Bloch in 1928 by a thoroughgoing application of quantum 怕回ry on the assumption that a single ee也，on heory was adequa.te for 也hi purpo was shown by G. Floquet in 883 a. the fund a.mental solutions of a.ny linea.r differential equation L[f] = 0, wi one independen a. riable x , wh
Opportunities missed and opportunities seized can be conveniently described as a quasi-free electron. Houston(1928)had also arrived at a similar conclusion n accordance with the concepts outlined above, a non-zero electrical resistance can only arise in a metal if the atomic lattice is imperfect, the major source of the imperfections being the thermal vibrations of the metal ions. Once this was realized it was possible to give a physically plausible explanation of the magnitude and temperature variation of the mean free path, and Bloch gave a detailed mathe- matical derivation of the appropriate formulae. The most difficult part of the calculation was the determination of the eigen- ralues of the quasi-free electrons. For, whereas Floquet s theorem gives precise information about the form of the eigenfunctions, it only gives qualitative and not quantitative information about the eigenvalues. Bloch therefore had recourse to the following approximate method If, for simplicity, we consider a perfect simple cubic lattice with lattice constant a, a conduction electron moves in a field in which its potential energy is of the form v(r)=∑U(-g,g=n,9 where the g' s are integers. Bloch assumed that the wave functions were of the form yk(r)=∑C(r-ga) and he made the further assumption that the integral J(g, h)=(V(r)-U(r-ga)o(r-ga)(r-ka)dr is only non-zero for g=h or when one of g,, ga and ga differs from h,, h 2 and ha by unity. That is, when the electron can, in the zero approximation, be considered to be tightly bound to the atom g, and in the first approximation to have a small probability of moving to the vicinity of the six neighbouring atoms. With these approximations, Bloch deduced that the ground state energy level Eo of an isolated atom gave rise to G energy levels in a metal containing Ga atoms, and that these energy levels were given by the formula Ek=Eo-a-2B(cos ak,+cos alg+ cos ak3). where =J(g,g)andB=J(1,y293i1+1,92,93) C C=exp(iak·g) (10) Bloch further showed that the velocity v of an electron with the wavefunction ya(r) is given by hu= grad Ek. For tightly bound electrons with the energy spectrum(8), the current is given by v=(2Ba/m)sin ak
Opporl nit四阴阳edand Iport ni sei: 43 can be conveniently described aa a. qu 础卜free elec衍。 Houswn (1928) had arrived at a. similar conclusion ln accor wi th the on ts outlined above, a non-zero electrical resis急剧ce can on1y ar in a metal if the atomic Jattice is im perfect ，也 he a.jo urce of he imperfections being the ermal vibrations of the metal iOll8. 00 this was realized, it was possible to give a. physically plausibJe ex plana.tion f 也 magnitude and mperature variation of he mean free path, a.nd Bloch gave a detailed mathe rnatical derivation of the appropria forrnul a.e The mo difficult par也。 the calculation was he rmination of t he eigenvalues of the quasi-free electronB. For, whereas Floquet's heorem iv ec se information abou the form of the eigenfunctiolls, it on1y gives qualita.tive and not qu a.ntit a.tive i1 or io about the eigenva.lues. Bloch the fore ha.d recourse he following a.pproximate hod If. for simplicity, we consider a perfect simple cubic Ia.ttice with lattice nsta.nt G, a. conduction electron moves in a. field in which its potenti energy is of he form • V( 叶~ ~ U( , - ga), g ~ (0,,0,, 0,), (5) ,.-... where the g's are intege Bloc 创酬med that the wa.ve functions were orm ifF,( ' ) ~ ~ G, Ø('- ga), (6) ,..-... a.nd he made t he further ass umption .t integral 峭的 f(V )-U( )}Ø(叫)制 a)d (7) is only non-zero for g = h or when one of gl, g2 and (13 differs from h1, h2 and h3 by unity. That is, when the electron can, in the 肘。岛pprox im tiol\ be considered to be tightly bound to the atom g, and in the first pproxim at on av a small probability of moving to the vicini of the six neighbouring atoms. With hese proximations Bloch dedllced that the round ate energy level Eo of an isolated gave rise to 0 3 energy levels in a me con nin 0 3 atoms, and that these energy levels were given by formula where E. = Eo-a-2p(cosaι+cosak a = J (g, g) and p = J(gl , g2,g3;gl + l , g2, g3), C, being by C, = exp (iak. g) (8) (9) (10) Bloch further showed th8.t the velocity V of an electron with t he w8.vefunction (r) is given by /iv = gra.d"Ek. 时也 htl bound electrons wi t he energy spectrum (时，也 he current is given by 叫~ (2 a/苑) sin ak, ( 1 1 )
A.H. wilson The next major contribution was made by Peierls in 1929 in a paper entitled Zur Theorie der galvanomagnetische Effekte. In order to calculate the electrical conductivity of his model, Bloch had shown that the mean wavevector k of a wave packet was connected with the applied electric field 8 by the relation dk/dt=(-6/), where -e is the electronic charge, and hence that, with v given by(11), the accele ration of an electron due to a field (6, 0, 0)is du,a du, dk,_2pa-edco Commenting on the physical significance of this last equation Peierls stated, Diese Gleichung hat folgende merkwuirdige Konsequenz: Fur k>Ira nimmt mit wachsendem k, der Strom ab, das heisst im Felde wird ein solches Elektron ver- zogert, statt beschleunigt zu werden. Diese Tatsache is so unanschaulich dass es notwendig erscheint ihre Richtigkeit moglichst ohne Vernachlassigungen und Annahmen zu beweisen. Peierls succeeded in this, and he went on to use(13)to explain the existence of anomalous (that is, positive)Hall coefficients. If the conduction electrons were such that the wave numbers k,= k g= ha- ko of the highest filled energy level were less than Ir/a, the Hall coefficient would be negative. If, on the other hand, ko lay in the range(lr/a, m/a), the Hall coefficient would b This discovery gave a rational explanation of the existence of both negative and positive Hall coefficients, and completely cleared up a major mystery, to account for which more than twenty implausible theories had been advanced since 1879 THE PERIOD 1929-33 By the middle of 1929 the state of knowledge was as follows. By making the eroic assumption that the valency electrons in a perfect crystalline solid were not firmly bound, each to a single atom, but had a non-zero chance of jumping to a neighbouring atom, the electrical properties of metals had for the first time been given a rational explanation. The valency electrons could be considered to be quasi-free, and, surprisingly, the energy spectra of quasi-free electrons were such that they could result either in negative(normal) Hall effects or positive(anomalous) Hall effects. However, a number of substantial criticisms could be levelled at the theory. It is sufficient to give two examples In the first place, the problem had only been made tractable by neglecting the electrostatic forces between the valency electrons, except in so far as they could be deemed to give rise to a smeared field having the same symmetry as that due to the atomic nuclei and the core electrons. This meant neglecting the exchange forces between the valency electrons, the dominant effect of which was the basis for the theory of ferromagnetism, first put forward by Heisenberg in 1928 In the second place, while Bloch's theory, as supplemented by Peierls, gave
44 A. H. Wilson The next major contribution made by Peierls in 1929 in a. a.阳 entitled 'Zur Theorie der galvanomagnetische tTek 恤'. In order to a. lcul a. te he electrical conductivity of his model, Bloch had shown that he mean wavevecoor k of a. wave packet was nnect‘时 wo the applied electric field 6 by he relation dk/dt - (-./~) (f, (12) where -e he electronic charge, and hence 出盹 wi th v given by (11)，也heac le '"饥 00 of an electroJ1 due a field (1,0, 0) i8 守=技等=-~于eßcosak ( 13) Co mmen川$创】 ng on 也"恤、随 phys刨>0'叫 ifì a. c<。回 of rusl 盹也 equa时也ωion Pe创'"'叶 st ι' D wachs ndem羽、 kι der Strom ab, da.s ei困也 im Felde wird ein solches Elektron ver. zoger毛， sta.tt beschleunig zu werden. Diese Ta.t8ache is 80 una.nscha.uJich a.ss回 notwendig erscheint ihre Rich igk创也 möglichst ohne Vemachlässigungen und Anna.hmen zu bewei n.' Peierls succeeded in this, and he wen也。 se 3) explain the exiatence of anomaloua (that 恼， poaitive) HaU coeffiωents. If出 cond uction elec ons were such h. the wave numbers k1 = k2 - k3 ... ko of highest fill energy level were less than /α，也he HaU coefficient WQuld be negative Iιon the 。由er hand, ko lay in the range (i n: /α π/叫， the Hall coefficient would be positive This dis ery gave a rational explanation of the existence of both negative and positive HaU coeffic ien and completely cle町时 up a major mystery, 10 account for which more h.n wenty implausible ries had been advanced since 1879 THE PERIOD 1929--33 By the middle of 1929 the state of knowledge 翩翩 follows. By making the heroic umption tha.t the valency ec衍。ns in a perfect crysta.lline solid were not firmly bound, each to gle atom, but had a non-zero chance of jumping to a neighbouring 1o ，由 elec~rical propertie8 of metals had for the firs time been given a rational explanation. The valency electrons could be con8idered to be quasi-free, and, surprisingly, the energy spectra of qu i-free electrons were 8uch ha也也 heycould resu either in nega ive(norn Hall effects or positive (anomalous) Hall effects. However, a number of 8ubstantial criticisms could be Jevelled at the theory.1 is uffi ien to give wo example8 1n the fir8t place, t he problem had only been made ractable by neglec ing he electrostatic for 四. be ween the valency electron6, except in 60 far M they could be deemed to give ri to a smeared field havin he 8ame symmetry aa hat due he atomic nuclei and he core electrons. Th.is mean neglecting the exchange forces be he valency electrona ，也 he dominant effect of which was he ba: for he 怕回ry of ferromagnetisffi ，耻，也 put forward by Hei nberg in 1928 Io 回∞nd place, while Bloch's theory. a.s supplemented by Pcie巾， gave
Opportunities missed and apportunities seized satisfying explanations of the electrical and thermal conductivities and the hall effect both at room temperatures and at low temperatures, it had not had the same success in dealing with the magneto-resistance effect the increase in the electrical resistance produced by a uniform magnetic field. Various theories had indeed been put forward, but they had not stood up to detailed criticism. Simplifying assump tions had to be made to permit a rudimentary start on these and other problems nd it was not easy to light upon assumptions that were sufficiently plausible It was at this stage that I first became interested in the theory of metals, largely because P. Kapitza, who was then working in Cambridge, published a number of papers on the electrical resistance of bismuth crystals in strong magnetic fields (1929). In these pioneering experiments Kapitza used magnetic fields of up to 200000G(20T)and worked at temperatures down to liquid air, and he came to the conclusion that the change in resistance was proportional to the magnetic field but that this linear behaviour was masked in weak fields by disturbances in the metal that are equivalent to that produced by an inner magnetic field. I had not got very far in tackling this problem when I was extremely fortunate in receiving the offer of a Rockefeller Travelling Fellowship, and I decided to spend the best part of a year in Leipzig, interspersed with shorter stays in Copenhagen I arrived in Leipzig in the first week in January 1931, and Heisenberg immediately pressed me into giving a colloquium on magnetic effects in metals, remarking Peierls work is undoubtedly important, but the mathematics is complex and the physical ideas not easy to disentangle. We ought to have a thorough discussion of the whole subject, and, as you have more free time than the rest of us, would you agree to give the first talk? 'I had studied Peierls's papers and had not found them too easy, and though I thought I had understood them well enough for my own purposes, to give a colloquium on them, and in German, was a formidable task. I had to find ways of shortening and simplifying the arguments. I was unsuccessful for a couple of weeks, but just at the end of January it suddenly occurred to me that the Bloch-Peierls theories could be enormously simplified and made intuitively more plausible if one assumed that quasi-free electrons, like valency electrons in single atoms, could form either open or closed shells Bloch's theory had in fact proved too much. Before his paper appeared it was difficult to understand the existence of metals. Afterwards it was the existence of insulators that required explanation. This problem had been by-passed because it had been taken for granted that insulators were merely bad conductors rather than non-conductors, the difference between metals and insulators being a quantitative and not a qualitative one. This concept I now challenged I went to see Heisenberg the next day and told him that I wanted to change and broaden the subject for the colloquium which was scheduled to take place two weeks later. I expounded my thoughts to Heisenberg who grasped the significance of them at once and fetched in Bloch from the adjoining room to take part in the discussion. Bloch was highly sceptical and stuck to the view that insulators were substances for which the overlap integral(g, h)(defined in(7)above)was negligibly
Oppor tniti甜刑臼sed apportuniti seized 45 satisfying explanations of the electrical and hermal conductivities and the Hall effect both at room temperatures and at low temperatur咽， it not had the same success in dealing wi the neto -resista effect he incre in the electrical resistance produced by a uniform magnetic field. Various 仙。ories had indeed been pu forward but they not stood up to detailed criticism mplifying sump tions had to be made to pefffi rudimentary 的时也 on these a. nd 。也her problems, and no to li gh upon su mptions that were 8ufficiently plausible It was 抽出is stage hat 曲曲 became interested in the theory of metals, largely because P. Kapitza, who was hen working in Cambridge, published a number of paper on the electrical resistance of bismu crys也叫 in strong magnetic elds (1929). In these pioneering experiments Kapitza used magnetic fields of up to 200000 G (20 T) and worked "'mp atures down to uid air，创ld he came he conclu on that the change in re糊的ance was proportional to magnetic field, but that is Iinear behaviour masked in weak fields by disturbances in metal that are 呵旧valent to 也hat produced by an inner magnetic field 1 had not got very far in tackling his problem when 1 extremely fortunate in receiving the offer of a Rockefeller Travelling Fellowship, and 1 decided to spend the best of a year in Leipz 毡， 阳回persecl with shorter stays in Copenh en 1 arrived in Leipzig in he firs week in January 1931 , and Heisenberg immedia.tely pce ed me in giving a colloquium on magnetic effects in metals, remarking 'Peierls' work is undoubtedly importa. 毛， but t he mathema.tics is complex and he physical ideas not to disentangle Ne ough也 to have a thorough discu ion of whole subject, and, as you have mo free tiroe than the rest of us, would you agree to give the firs talk?' 1 had studied Peierls's papers and had no found hem oe :y and though hough ha.d understood them well enough for my own purp田田， to give a. colloquium on hem and in German, formida.ble task. 1 had to find ways of shortening a.nd simplifying the arguments. 1 unsuccessful for a. couple of weeks, bu乞 )u，也 he end of a. nu时'Y'也 suddenly occurred to me that Bloch-Peierls theories could be enormously simplified and made intuitively more plausible if one assumed that quasi-free electrons, like valency electrons in single a.toms, couJd form ei er open or closed sheJls Bloch's theory ha.d in fa.ct proved too much. Before his paper appe ed it was d; cult to understand exÎstence of metals. Af rwa. rds the existence of insulators tha requu唱d expla.nation. This problem had been by-passed because i
A. h. wilson small. As a refutation of my hypothesis, he pointed out that it would entail the monovalent elements being metals while the divalent alkaline earth elements would be insulators The discussion ended there for that day, but it was resumed on the following morning, by which time I had marshalled arguments against Bloch's views, which had depended upon the energy spectra of one-dimensional and three-dimensional lattices being qualitatively similar. I pointed out that, whereas in one dimension the number of states in each energy band is equal to the number of atoms, a similar result does not necessarily obtain in three dimensions; the energy levels split up into bands, but the bands can overlap. It therefore followed that an elemental solid(that is, composed of atoms, not molecules)with an odd valency had to be a metal, whereas elements with an even valency might produce either a metal or an insulator Bloch was eventually convinced, but the arrangements for the colloquium were hanged, the reason being that the experimental physicists in Leipzig were far from certain what the real differences were between metals, semi-metals and insu lators. In the end, two colloquia were held, some 3 months apart, the second of which was attended by a large contingent from Erlangen, headed by B Gudden The most recent survey of the conductivity of solids had been written by E. Gruneisen(1928)for the Handbuch der Physil, in which solids were classified as metals, semi-metals and insulators according to the shapes of their resistance- temperature curves. He defined a semiconductor as a metallic conductor, the resistance of which was high at low temperatures but which diminished with increasing temperature until a minimum was reached, whereafter it increased. As examples of such substances he cited boron, silicon, titanium, zirconium, germanium selenium, graphite and various alloys. However, more recent work had tended to show that the peculiar shapes of the resistance curves of the above substances might well be due to the presence of poorly conducting oxide layers, and, if they could be purified sufficiently, they would be found to be metals The first colloquium failed to clarify the situation, and I therefore wrote a paper for the Royal Society giving details of my theory of the difference between metals nd insulators, but leaving open whether true semiconductors existed, that is whether there were insulators with so small an energy gap that, at room temperature, electrons could be thermally excited from a full band to the next higher one The next colloquium was much more fruitful, largely because Heisenberg had drawn my attention to Gudden,s view (193o)that no pure substance was ever a semiconductor, and that the conductivity of such conductors is due to the presence of impurities. These impurities could be either electropositive or electronegative according as they gave rise to negative or positive Hall effects. In other words, the impurities could act either as donors or acceptors of electrons. The mechanism by which free electrons could be produced in an otherwise empty energy band, or free holes in an otherwise full energy band, could readily be explained by a simple expansion of my theory, and this was given in a second paper
46 A. H. Wilson small. As a. refutation of my hypotheaÎs, he pointed ou hat it would entail the monovalent elements being meta.ls while the diva.Jent alkaline ea.rth element8 would be insulawrs The discu酬。 ended there for a. 町. but &S r. umed on the folJowing mom by which time 1 had marshal1ed arguments ainst Bloch's view8, which had depended upon the energy spectra of one.dimensional a. nd hree.dimensÎonal lattices being qua.Htatively similar. 1 pointed Qut 由旧， where in one dimen on the number of state in each energy band is equa the number of a.toms, a simiJar resul does no necessarily obtain in th dimcnsions; the energy levels sp li up into bands, but the bands can overl叩It therefore followed tha.t a.n elementaJ solid (也 a. is, compos of atoms, no molecules) with an odd valency ha.d to be me也时， wher~ ements with an even va1ency might produce 拙。 8. meta.1 or a.n insulator Bloch eventu a. lly conv皿曲d.bu也也he arrangements for the lI oquium were changed. he re捕。 being t hat the experimental physicists in Leipzig were far from certain what the real differences were be ween me ，.肥mi-met Is 8.nd insu- 1ators. In the end ，也wo colloquia were held, some 3 months apa鸭， the second of which attended by a large contingent from Erlangen, headed by B. Gudden The most recen survey of the conductivi也.y of solids had been written by E Grüneisen (1928) for the Handbuch der Ph归曲， in which solids were ified as meta.ls, semi-metals 8.nd insulators &Ccording sh8.pes of their 回跑回ncepe ature curves. He defined 8. semi nductor as a met&llic conductor, the resist&nce of which was high at low mpera. tures but which ruminished wi increasillg temperature until a minimum was reached, whereafter it increased. As exampl ofsuchsubst町、。es he cited boron. 副Ii con ，也 itanium zìrconium, germanium, se1enium, gra.phite and various alloys. However, more recen work had tend时也 show th8. he peculiar shapes of the resistan curves of the above substances might well be due to 也he pr. ence of poorJy conducting oxide 1ayers, and, if .Y could be purifi sufficient1y they wouJd be found to be .1 The firs也∞110quium failed to CI ify the situation, and 1 therefore wrote a 叩.， fo，也he Royal Society giving 9.etails of my heory of the difference be ween metals and insu1ators, bu leaving open whe her true m. nduc阳"回国础，也hat is, whe h.，也here were insulators wi so smo.lJ an energy gap hat atroom temperature, electrons could be hermallyex归国 from a. full ba.nd to the nex higher one The next colloquium was much more fruitf1时， largely becau Hei mberg ha.d drawn my a. ention to Gudder view (1930) that no pure 8ubstance w
Opportunities missed and opportunities seized Subsequent developments are either well known or would be better described by others. I shall therefore conclude my personal reminiscences by posing two questions, namely why were my discoveries not made earlier, and why was so little done in the 1930s to follow them up? As regards the first question, both Bloch and especially Peierls might well have forestalled me, and they may be able to remember why they did not do so. The only clue that I can offer is a remark made by Peierls in his paper of 193o on electrical and thermal conductivity in which he states: ' Die Eigenfunktionen sind in zwei Fallen bekannt: erstens in dem Grenzfall freier Elektronen... und zweitens in dem Grenzfall stark gebundener Elektronen, wo man sie nach Bloch durch ein Storungs- verfahren aus den Eigenfunktionen des einzelnen Atoms verhalten kann... Von dem zweiten Grenzfall(stark gebundenen Elektronen) weiss man wegen seinen Konse- quenzen fur die magnetische Suszeptibilitat dass er praktisch nie realisiert ist With the concept of nearly free electrons uppermost in his mind, it is understandable that Peierls did not give much attention to insulator As regards opportunities missed or seized during the remainder of the 1930s I can speak only about Cambridge with first-hand knowledge. On my return there n October 1931 considerable interest was shown in my discoveries, but, when I suggested that germanium might be a very interesting substance to study in detail the silence was deafening. However, I eventually got J. D. Bernal to consider the project, though he saw little hope in it. In a paper published in 1927, H. J Seemann had investigated the temperature coefficient of the resistance of ten single crystals of silicon between-80C and room temperature. He found that not only was it positive but that it was of the same order of magnitude as for normal metals. (This was confirmed by Schulze in 193I ) In view of this fact, that silicon, in its pure state and freed from oxide films, was a metal, germanium would also be a metal and of no more interest than grey tin. But, since Bernal possessed a large single crystal of silicon of outstanding purity, he agreed to examine its properties. After some six months Bernal told me that he thought that Seemann was probably right, though he did not reveal his grounds for saying so. It is likely that he was merely being polite, and that the subject held no interest for him The canard that silicon is a good metal continued to be believed in many circles for a long time. For example, in the Handbuch der Metallphysik, published in 1935 both U. Dehlinger and G. Borelius stressed that all the elements which had at one time or another been classed as semiconductors were, in the pure state, true metals Borelius ends his review ruling out the existence of semiconductors with the follow ing statement(p. 354): 'Man kann qualitativ nur zwischen metallischen Leitern (Elektronenleiter)und elektrolytischen Leitern(Ionenleiter)unterscheiden. Whether anything interesting would have emerged from a proper study of germanium in the 1930s is extremely doubtful. Methods of purification would almost certainly not have been established because there were no incentives, either acade- mic or industrial, to do so, and impure germanium is one of the least interesting of
Oppo "8 issed αnd opportunities seized 47 Subsequent developments well known Qr would be be er described by others. 1 shall therefore conclude my personal reminiscences by posing tWQ que ons namely why were my discoveries no de earlier, and why was 80 Ii灿le don in 1930s to follow them up ? As rcgards fìrst question, both Bloch and especially Peierls might well have re lled me, and they be ble to remember why they did not do 80. The only c1 ue tha也 1 can offer is a. remark made by Peierls in his paperof 1930 on ec ri ca and thermal conductivity in which he states: ' Die Eigenfunk onen sind in zw Fällen bekannt: in dem Grenzfal! freier Elektronen, ... und zweitens in dem Grenzfal! stark gebundener Elektronen, wo man sie nach Bloch durch ein rungsverfahren aus den Eigenfunktionen des einzelnen Atoms verhalten kann ... Von dem zweiten Grenzfall (stark gebundenen Elektronen) weiss man wegen nen Konse quenzen für die magnetische Suszeptibîlîtät dass er praktisch nie eali ier .' Ni怕也 concept of nearly free electrons uppermos也 in rns mind, itis understanda.ble at Peierls did not give much attention to insuJators As regards opportunities missed or seîzed during le remainder of 1930s 1 can speak onJy about Cambridge with first-hand knowledge. On my re um here in October 193 1 considerable inter.曲也 shown in my d>scoveries, but, when 1 suggeste 出岛 germ üum a. very nteres ng subs 也a.nce study in detail, the silence was deafening eve 1 even ual1 go J. D. Bernal consider the project ，也 hough he saw li le hope in it. In a er published in 1927, H. J. Seemann had in ve ig ated he mp atur coefficient of he resis nceof single crystals of silicon between - 80 oC ld room mperature. He found that nly was positive bu at asof he same order of nitude as for normal 回国 s. (This was confirmed by Schulze in 1931.) In view of this fact, that silicon, in ur state and freed from oxide films, was a metal , germanium would al50 be a metal and of no more in忧而睛也han grey tin. But, since Bcrnal po回国回 a large singlc crystal of silicon of outstanding puri 'y hc ag ed to examine its proper cs. After some six months Bernal told me 创陆也 he thought that Soomann was probably right, ough he did not reveal his grounds for saying so. It is likely tha也 he was merely being poli阳， and t hat the suçject held no inter困也 fo him The canard 出叫 ili con is a good metal continued to be believed in many circles for a ong im e. For example, in the Randbuch der Metallphysik, published in 1935, both U. Dehlinger and G. Borelius stressed that lJ乞 element8 which had at one u
A. H. wilson To conclude, one further gaffe is perhaps worth mentioning. For many years cuprous oxide was the most interesting semiconductor as an object both of academic study and for commercial applications. When I first discussed it in 1931(a, b)it was deemed to be anexcess conductor in which the current is carried by electrons derived from excess donor copper atoms. In 1935 it was discovered by Schottky Waibel that in all previous measurements the Hall coefficient had been given the wrong sign, and that the current is in fact carried by holes produced by excess acceptor oxygen atoms REFERENCES Bloch,F.1928Z.Phy8.52,555 Borelius, G. 1935 Handbuch der Metallphysik, vol 1, p. 353 Dehlinger, U. 1935 Handbuch der Metallphysil, vol 1, p. 79 Dirac, P. A M. 1926 Proc. R, Soc. Lond. A. 112, 161 Drude, P, Igoo Annin Phys.(4)1, 566 Fermi, E. 1926 Z. Phys. 36, 902 Floquet, G. 1883 Annls Ec norm. sup.(2)12, 47 Fowler, R. H. I926 Mon. Not, R. astr. Soc. 87, 114 Gruneisen, E. I928 Handb. Phys. 13, 61-64 Gudden, B. 1930 Sber. phys med. Soz. Erlangen 62, 289 Houston, W. V. 1928 Z. Phys. 48, 449 Kapitza, P. 1929 Proc. R Soc. Lond. A 123, 292. Lorentz, H. A, 1904-5 Proc. Acad. Sci. Amst. 7, 438, 585, 684. Pauli, W. 1925 Z. Phys, 31,765 Peierls, R. 1930 AnnIn Phys.(5)4, 121 Schottky, w.& Waibel, F, 1935 Phys. Z. 36, 912. Schulze, A. 1931 Z. Metall, 23, 261 n.H. J. Sommerfeld, A. 1928 Z, Phys. 47, 1 wilson, A H. 193Ia Proe. R Soc. Lond. A 133, 458. Wilson, A. H. 1931b Proc. R Soc. Lond. A 134, 277 Conductibility electrique des meaux et problemes connects Conseil de Physique, tenu a Bruxelles du 24 au I'Institut International de Physique Solvay(Paris ga pport Villars, 1927)
48 A. H. Wilson To conclude, one fur er ga.ffe Îs pc worth mentioning. For many ycars cuprOU8 oxide was thc most intere tin co ndu cto as an object both of academic study and fOf commcrcial applications. When 1 firs disc ssed it in 19 忡，的 was deemed to be an • exccss ndu r' in hi currcnt is carried byelcctrons derivcd from ex 盹s8 pper atoms. In 1935 it was d iscovered by Schottky & Wa ibel that in aU previous mCa.<!urements the H a.U cocffi cient had becn given the wrong sign, nd mt thc currcn in fact carried by holes produ byex accep oxygen atoma REFEREN CES Ji'rom 1m ~. 1928 Z. Phy,. 52, 555 Borolius, G. 1935 H (J'‘dbuch d~r Maallphy8ik . vol. 1, p. 353 Dohlinger. U. 1935 nd.buch MttaU内时， vol. 1, p. 79 Dirac, P. A. M. 1926 Proo. R. SQc. Lond. A 111, 161 Drudo. P. 1900 A 饰州，、 ν8 (4) 1, 566 Formi, E. 1926 Z. Phν.，. 36, 902 F1oquet, G. 1883 Annlø Gc. norm 'p. (2) 47 Fowler, R. H. 1926 Mot~. Not. R. aøtr. Soc. 87, 114 ，也 ae E. 1928 Handb. Phy" 13, 61- 64 Gudden, B. 1930 SlMr. 'P ，.-med. So:. 也呻剧 89 OU8 W. :;: Z. Phy 咽， 44 a. pi a.， P. 1929 肉:JC. R. Soc. J.nn4. A 123, 292 Lorontz, H. A 吨。 PrOC. Awd. Sci. f7 438, 585, 684 a. uli ，飞，V. 1925 Z. Phy', 31, 765 a. uli ，飞 1927 Z. Phν.，. 41 , 81 Peierlø, R. 1929 Z. Phy,. 53, 255 PeierJs, R. 1930 An Phy'. (5) 4, 121 S< 。‘吐:y ，飞鸟，. l< 飞， a. ibel F. 1935 Phy'. Z. 36, 912 Sehulw, A. 1931 Z. Mttallk. 23, 261 Seema.nn , H . J. 1927 Phy,. Z. 28, 765 Sommerfeld, A. 1928 Z . Phy', 47, I Wil A. H. 1931 a Proo. R. Soc. Lmd. A 133, 458 Wil80n, A. H. 1931 b Proe. R. Lond. A 134, 277 Oenual 。衍Klua liu éledriq~ m/阳回 et problbnt 阳出:u. a. ppor也矶 diøcu 8II Îonø du qua.trième Conøeil de Phyøique, tenu à. Bruxellee du 24 a.u 29 A vri1 t 924 ØOU8 les a.uøpices de Inøt阳也 Inωma. ti on a. de Phyøique Solva.y (Pa.riø: Ga.utier Villa.rø, 19
