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 )