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 akOpporl 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 eigen￾values 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 )