R.E. peierls required only a few lines of calculation, both because it satisfied me that the nature of the Bloch bands was now qualitatively the same all the way from tight binding to almost free electrons, and because of the neat method of approximation I had invented. This was, of course, not new, being the standard technique of dealing ith the anomalous Zeeman effect, but i was quite ignorant of this. I was satisfied with solving the one-dimensional case, but Brillouin took over the idea and dis- cussed the general three-dimensional problem in depth, which became the theory of brillouin zones din At the same time I started worrying about the conservation of wavevector (pseudomomentum as we say today)in metals. I had found that in non-metallic crystals at low temperatures, when Umklapp processes were rare, it was difficult to get rid of the phonon drift caused by a temperature gradient In a metal the electric field caused an electron drift, which in electron-phonon collisions tended to be passed on to the phonons. What mechanism restores the phonons to equili brium Bloch had by-passed this problem by assuming that the phonon distribution is always in thermal equilibrium. How far was this justified? I set up a Boltzmann equation for electrons and phonons, allowing for their interaction. For an electron band which is about half full. so that neither the num ber of electrons nor that of holes is small, or in the presence of several partly filled bands, there is no difficulty generating Umklapp processes, even at low temperature, and one still finds Bloch's 15 law for the low-temperature resistivity. I first found 14, 2) because of an invalid approximation, and had to correct this in a later paper. 3)The factor of the 15 law could be similar to Bloch's or different, according to whether phonon phonon interactions or electron-phonon interactions were dominant in keeping the phonons in equilibrium However, in metals in which the Fermi surface did not touch the zone boundary, the theory predicted an exponential rise of the electric conductivity at low tempera ture. I was worried by the fact that such a behaviour had never been seen. I was aware of the fact that electron-electron collisions could also cause Umklapp processes, and the criterion for this was somewhat less restrictive than for the electron-phonon interaction. Roughly speaking, it would be sufficient if the dia- meter of the Fermi surface was at least one-half of that of the brillouin zone But it was known that the contribution of electron-electron collusions to the resistivity was proportional to 12; if these collisions were vital to give a finite resistivity it was hard to believe that one would still find a 5 law I was not then aware of the correct explanation: all measurements at these low temperatures were, until recently, made on specimens whose residual (impurity) resistance was much higher than the" resistivity, and had to be subtracted from the measured value, assuming additivity according to Matthiessen's rule This rule lacks any rigorous theoretical foundation, and, while it is empirically quite well satisfied in many circumstances, it is not applicable if the impurities are sufficient to dispose of the excess pseudomomentum, but matter less for the actual32 R. E. Peierls required only a few lines of calculation, both because it satisfied me that the nature of the Bloch bands was now qualitatively the same all the way from tight binding to almost free electrons, and because of the neat method of approximation I had invented. This was, of course, not new, being the standard technique of dealing with the anomalous Zeeman effect, but I was quite ignorant of this. I was satisfied with solving the one-dimensional case, but Brillouin took over the idea and dis￾cussed the general three-dimensional problem in depth, which became the theory of Brillouin zones.(ll) At the same time I started worrying about the conservation of wave vector (pseudomomentum as we say today) in metals. I had found that in non-metallic crystals at low temperatures, when Umklapp processes were rare, it was difficult to get rid of the phonon drift caused by a temperature gradient. In a metal the electric field caused an electron drift, which in electron-phonon collisions tended to be passed on to the phonons. What mechanism restores the phonons to equili￾brium? Bloch had by-passed this problem by assuming that the phonon distribution is always in thermal equilibrium. How far was this justified? I set up a Boltzmann equation for electrons and phonons, allowing for their interaction. For an electron band which is about half full, so that neither the number of electrons nor that of holes is small, or in the presence of several partly filled bands, there is no difficulty in generating Umklapp processes, even at low temperature, and one still finds Bloch's T5 law for the low-temperature resistivity. I first found T4,(12) because of an invalid approximation, and had to correct this in a later paper.(13) The factor of the T 5 law could be similar to Bloch's or different, according to whether phonon￾phonon interactions or electron-phonon interactions were dominant in keeping the phonons in equilibrium. However, in metals in which the Fermi surface did not touch the zone boundary, the theory predicted an exponential rise of the electric conductivity at low tempera￾ture. I was worried by the fact that such a behaviour had never been seen. I was aware of the fact that electron-electron collisions could also cause Umklapp processes, and the criterion for this was somewhat less restrictive than for the electron-phonon interaction. Roughly speaking, it would be sufficient if the dia￾meter of the Fermi surface was at least one-half of that of the Brillouin zone. But it was known that the contribution of electron-electron collusions to the resistivity was proportional to T2; if these collisions were vital to give a finite resistivity it was hard to believe that one would still find a T5 law. I was not then aware of the correct explanation: all measurements at these low temperatures were, until recently, made on specimens whose residual (impurity) resistance was much higher than the 'ideal' resistivity, and had to be subtracted from the measured value, assuming additivity according to Matthiessen's rule. This rule lacks any rigorous theoretical foundation, and, while it is empirically quite well satisfied in many circumstances, it is not applicable if the impurities are sufficient to dispose of the excess pseudomomentum, but matter less for the actual