Rudolf Peierls - Session II

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Interviewed by
Questions dictated by Lillian Hoddeson in Urbana, Illinois
Replies dictated by Peierls in Oxford, England
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Interview of Rudolf Peierls by Lillian Hoddeson on 1981 July 1, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA,

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Solid state physics in its early days in Arnold Sommerfeld’s and Werner Heisenberg’s groups, where Peierls was a student in the late 1920s. Sommerfeld’s group compared to Heisenberg’ s group; interaction between experimentalists and theoreticians in Germany, in England and in the U.S.; Peierls’s thesis work on heat conductivity in non-metallic crystals (suggested by Wolfgang Pauli), done in Zurich in 1920 and defended at Universitat Leipzig; the application of the Heitler-London approach in other works (especially Felix Bloch, Arnold Sommerfeld); attempt to state the first use of names and concepts: energy bands, theory of solid bodies; the periods of optimism in the history of physics; the “accidental” change of fields because of Nazism (Hans Bethe). Significant papers on absorption spectra, diamagnetism, phase changes and statistical foundations, published in the 1930s: reasons for selecting these topics; inspirational contributions in the work; responses; associated circumstances and events. Lev Landau and his work also discussed at length.


Hoddeson's Questions

Hoddeson: The papers I would like your comments on in this interview session fall into three groups:

1) Absorption spectra,
2) Diamagnetism, and
3) Phase changes and statistical foundations.

Since part of our job on the solid-state history project is to place your work into its proper historical context, we would like to know for each of these groups: how you came to work in the area, what you feel are the significant contributions in your papers, and any other connective information that comes readily to mind, for example, responses to the work, specific issues, problems, circumstances, or events associated with it. Some more specific questions are listed below. Their primary purpose is, however, to help tickle your memory and stimulate you to, once again, as in our earlier face-to-face interview, suggest and respond to still better interview questions.

The purpose of your two papers on this subject in 1932 is to generalize studies of the spectra of atoms and molecules to the much more complicated case of solids. As in the case of atoms, there are selection rules for solids which lead to missing lines. But relative to the case of atoms, there are so few missing lines — the spectra are almost continuous — that one learns little by studying these lines. Instead, as you say on page 906 of the longer paper, it is far more interesting to compute the absorption coefficient as a function of frequency, for solids absorb almost everywhere, with varying intensities. Had anyone else attempted to do such a calculation? The shorter of the two papers, which was published in the Physicaliche Zeitschrift der Sowjet Union, volume 1, is based on a talk written for a conference held in Kharkow. We would be very much interested in any recollection you have of this meeting. For example, who organized it? Who were some of the attendees? What were the main subjects of interest at the meeting? The paper addresses the question of the origin of the wide bands, which come both in the visible and the ultraviolet regions, from the lines of single atoms when they are put together in a solid. I gather that the classic paper in this area was Frenkel’s 1931 work [Phys. Rev., 37, (1931)] attributing line broadening in ordered systems primarily to thermal motion, rather than also to the resonant coupling of atoms. You present a more precise argument than Frenkel did, which takes into account the coupling between the atoms and the lattice. You deal both with the case of weak and strong coupling. In the strong case, one gets true absorption, but it is not calculable by available methods — one can’t solve the Schrodinger equation. But one can calculate the location and width of the bands (i.e., proportional to the square root of the temperature for high temperatures, and constant at low temperatures). In the weak case, practically all of the energy goes into fluorescence. Did you discuss this subject with Frenkel? If so, when and where? How did he respond? What were the responses to this work in the Soviet Union? Were experiments being done there on this phenomenon? Your longer paper on this subject of absorption spectra, published in the Annalen der Physik, was your Zuricher Habilitationsschrift. For the benefit of scholars who will be using this interview but are unfamiliar with the German professional system in physics at that time, could you briefly explain what this Habilitations work meant then? You deal with two classes of solids: the absorbers, which are in the majority and transform energy significantly; and the rest, which you call dispersers. (Actually, as you mention, in different temperature ranges, materials act differently.) You treat both classes, solving the Schrodinger equation within certain approximations due especially to symmetries in the crystals. From my very preliminary look at the literature, it appears to me that this work, by providing a mechanism for absorption of light by solids, was the basis for much of the later work done in this area, for example by Mott and Seitz. Could you help us in our research by telling us of some of the key points we will need to examine in order to fill in the historical connections in this development?

Here we are dealing with two papers, both published in 1933, and both in the Zeitschrift fur Physik (volumes 80 and 81). Is it fair to say that this work was stimulated by Landau’s 1930 paper on diamagnetism [Landau, Zeitschrift fur Physik, 64, (1930), p. 629], which was based on a free-electron degenerate gas model as well as the recent experiments by de Haas and van Alphen on the de Haag-van Alphen effect? Landau had shown that the earlier classical arguments for free electrons, which conclude that the orbital magnetic effect is zero because the Hamiltonian is independent of the magnetic field, simply don’t apply. He showed instead that in quantum mechanics, the finite motion of the electrons in the plane perpendicular to H leads to a partial discreteness of the eigenvalues of the system, which in turn leads to non-zero magnetization due to the orbital electron motion induced by the field. Any attempts to determine the three components of the velocity of an electron at any given instant would disturb the system and cause the electron to jump from one quantum state to another, nullifying the classical arguments of Niels Bohr and Miss Van Leeuwen. Do you recall the response to this classic paper by Landau when it appeared? Do you know (or know how we can find out) how Landau came to do this work? Were you at this time aware of the de Haas—van Alphen results? Landau apparently missed this experiment, although in his paper predicts the effect, stating however that “it would hardly be possible to observe this effect experimentally because the in homogeneity of available fields would always lead to an averaging.” Landau calculated that the magnetic susceptibility for diamagnetism is -1/3 that of the Pauli spin paramagnetism susceptibility. In your two papers on diamagnetism, you do a much more complete analysis, putting in the lattice and also considering strong fields. You find in these much more complicated cases that the Landau magnetic susceptibility is still the same order of magnitude as the Pauli susceptibility. Is this what you expected? You find in the first of your two papers, that Landau’s formula applies to metals, provided kT is very, much larger than h/τ, where h/τ is the collision frequency. And only if the inequality is reversed — if h/τ is less than kT — is there a noticeable deviation from the Landau formula. In your strong field paper published a year later, you show a way to work the problem out for the case in which uH is much greater than kT, and you get rather good agreement with the de Haas—van Alphen experiment. Did you discuss this work with Landau? If so, where and when? What were his responses? Any correspondence with de Haas or van Alphen about this work? In our earlier interview, you commented that your work on diamagnetism was of some interest to Pauli because there was a chance that it might be a step towards explaining superconductivity, which many people were challenged by at that time. I neglected to ask you whether your interest in the problem connected with an interest in superconductivity. You wrote both of these papers on diamagnetisms in Rome, in successive years. Did you have any interactions with Fermi about them? Was Fermi generally interested in solid—state problems of this kind?

Before we deal with the five papers in this category, I want to ask you a question about the work of Landau and Ginzberg on the subject of phase changes. We are very much interested in the historical events that connected the early experiments and theories of superconductivity, including the London theory of superconductivity, Landau’s work on phase transitions, the Ginzberg-Landau theory of superconductivity and Landau’s theory of Fermi liquids. But we have very little ability in the West to probe this history — to get at the connections and motivations between the various scientific achievements. Could you give us some guidance by filling in a few of the high points and/or perhaps offering a few suggestions for how we might go about investigating this history? Any anecdotes you may have heard in the Soviet Union about this history would also be helpful. Which paper do you consider to have been the motivating paper in this sequence? Was it Landau’s paper on phase changes, or did the interest go back much earlier, for example, to the work of Ehrenfest? In your 1934 paper on the statistical basis for the electron theory of metals, you consider whether it is legitimate to use statistical methods in dealing with conductivity and the electron theory, because the assumptions seemed to be unfulfilled at high temperatures. You show here that it’s all right to do so for metals and other systems that can be described as fully degenerate Fermi gases, due to the possibility of making other simplifying assumptions. But the argument doesn’t hold for non-degenerate systems like semiconductors or systems in which variations of the free wavelength play a role. You show that most of the cases that seem suspect can be treated by a method due to Landau, which you explain here, based on the fact that at high temperatures, the collisions between electrons can be taken as elastic. Did Landau publish this method? How did you learn about it? You promise a joint note with Landau elsewhere on the matter, but I haven’t found it. Was this note ever written? If so, was it published? Where? Who did you discuss the problem with, either at Manchester or elsewhere? According to your references, E. Kretschmann also worried about this problem. Kretschmann’s name comes up often in papers of this period, and we know very little about him. Anything you could fill in about his role in solid-state physics at that time would be helpful to us. Was your 1934 paper on the transition temperature one that you gave at a society meeting? If so, it would be of interest to us to know the subject of the meeting. Where was it held? Who else was there? Who organized it? As you mention in the first page of the article, the relationships at the transition point depend on the simplifying assumptions one makes. This is a serious problem, because it means that at the transition point the usual methods often fail. Did attempts to analyze a certain set of experiments cause you to look into this problem? The particular issue that you focus on here is whether a transition temperature exists in 1, 2, and 3-dimensional systems. This depends on the consideration, which you say you and Bethe arrived at in discussion, that small deviations from the ideal ordering don’t disturb the coherence of an ordered state. In what connection did you and Bethe take up this discussion? When and where? In the paper, you verify Bethe’s consideration geometrically, starting out with a one-dimensional argument, and then extending it to three dimensions. You show that in one dimension fluctuations in the distance between atoms cause the coherence to disappear, but in three dimensions there is a transition temperature, because the more atoms that are involved and the further away they are, the weaker is any disturbance of the coherence. This then implies two qualitatively distinct phases with the transition temperature between them. You publish the calculations that provide a background for the geometrical arguments you make here for one and three dimensions in your review paper “Quelques proprietes typiques des corps solides” in the Annales de L’Institut Henri Poincare, vol. 5 (1934), on pages 198—201. Did you carry out similar calculations for 2 dimensions? Could you also, while we’re on the Institut Henri Poincare paper, give us some background about it? Besides you, Landau and Bethe, who else was working on this subject of transition temperature at the time? For one and two dimensional lattices, you argue that there is no sharply defined transition temperature, but it seems that here you are excluding superconductivity and ferromagnetism. Why did you exclude them? Your work on the statistical theory of super-lattices seems to have been motivated by a series of papers on the degree of order in an alloy by W.L. Bragg and E.J. Williams. Is this so? Do you know how Bragg and Williams got started on this work? The problem is to determine the function V(s), that is, the dependence of the average ordering force on the existing order, where s is the order at a given temperature. Bragg and Williams assumed that V(s) is equal to V0s. Bethe based his paper on this subject (which I have included in the enclosed package) on the assumption that there is appreciable interaction only between neighboring atoms, as in Felix Bloch’s theory of ferromagnetism, where the interaction is between the spins of neighboring atoms. Bethe distinguishes between two kinds of order: order at high temperatures, where there is a correlation only between near neighbors and the state is very similar to a liquid, and order at low temperatures, order of the whole crystal, which he calls “solid-like” and is restricted at long distances to two or more dimensions. You take off from Bethe, generalizing to include a different kind of super-lattice. Do you recall the specific motivations for this work? Did you and Bethe work closely on this subject? Do you know who coined the idea of ordering in alloys; the idea of a super-lattice? Was Landau in on this work as well? In your 1936 paper on the magnetic transition of superconductors you generalize the work of Gorter and Casimir to bodies which are other than long-shaped. Your analysis depends on the assumption that all currents are microscopic (this is on page 614) and that the peculiar magnetic properties of superconductors are therefore due to the relationship between H and B at each point. You then go on to set up the boundary conditions and solve the problem for an ellipsoid. And you come up with (see page 622) three states: superconducting, the transition state (in which H=Hc and 0 < B < Hc) and finally the normal state (in which B=H > Hc). You say that there is always a range in which the body is in a transition state, which is in agreement with the magnetization experiments of Schoenberg, and the predictions about specific heat measurements. Did you interact much about this with Schoenberg? If so, when and where? What led to it? Your results imply that the experiments can’t be explained with the assumption of alternating superconducting regions (in which the induction B vanishes) and normal regions, and that one must assume the existence of a transition state intermediate between superconducting and normal, with a field-dependent permeability between 0 and 1. This seems not to agree with today’s concept of the intermediate state of superconductors. Could you please explain this point? At the time you wrote your 1936 paper about Ising’s model of ferromagnetism, Ising’s paper was more than 11 years old. How central was it then in discussion of ferromagnetism? When did you come across Ising’s classic paper? You and Bethe, as well as Bragg and Williams, and others, had been working on statistical problems, such as magnetic transitions and the theory for adsorption, which gave you the mathematical machinery for dealing with the Ising problem in more than one dimension. You refer to “a good deal of controversy” stimulated by Ising’s paper. Who else besides you, Bethe, Bragg and Williams were involved in this controversy at this time? What were the main issues? Did you have the feeling in the mid-30s that you (as well as Bethe and Bragg and Williams and any others working in this area then) were getting close to solution of the phase change problem? You aim in this paper to give a rigorous proof for the 2-dimensional case and you do this geometrically, putting boundaries between regions of positively and negatively oriented magnets. Each boundary has energy LU, where L is the length and U the energy per length. Your proof shows that at sufficiently low temperatures the area enclosed by closed boundaries and cut off by open ones is only a small fraction of the total area. The Ising model in two dimensions, therefore, shows ferromagnetism and, as you say, the argument also holds a fortiori for three dimensions. Did you at this time try to calculate the transition temperature using the Ising model in two or three dimensions? You thank Bethe for suggesting part of the proof. Do you remember which part? My last question concerns the Landau-Peierls theorem on the lack of one-dimensional ordering in a 3-dimensional system. Could you please distinguish your contribution from Landau’s? Did you interact with Landau about this? I want to thank you very much once again, for your help with this interview and with our research in general. (Lillian Hoddeson)

Peierls' replies

Peierls: The first question relates to the papers about the absorption spectra of solids. How did I become interested in this? I don’t think I remember that very clearly, but it is natural that having been interested in various physical properties of metals and non-metals one should think about other similar situations. In fact, it is quite possible that the occasion arose in writing my review article in the Ergebnisse der Naturwissenschaften. I tried of course to visualize clearly the general basis of solid-state theory and the relation between electronic and atomic coordinates, but that is only a guess. Of the two papers the one published in the Soviet Union is evidently just an abstract of a talk at a conference in Kharkow about the work I was then doing. I believe my long paper was not written at the time of that conference because in the paper I acknowledge comments by various people including Obreimov, whom I would probably have met at the conference.

Turning to the specific questions: Had anybody else attempted to look at the absorption coefficient as a function of frequency? I think the answer was I was always trying to make rather general statements and therefore the statement describing the difference between atoms and solids — in the atom just a knowledge of the absorption frequency is informative, whereas in the solid, you have normally a continuous function of response as a function of frequency so you really must think about intensities. I don’t know whether anybody had stated this in that form, but it was clearly implicit for example in the work of Frenkel because it was of course realized that in most solids you are dealing with absorption bands rather than lines and obviously if you have a band you are dealing with a continuous function. But otherwise I have no recollection of what the literature was like at the time. As far as I remember nobody had as yet looked at the problem in full detail as I was trying to do in that paper.

Next about the conference in Kharkow. I’m afraid I don’t know who organized it. Presumably the research institute in Kharkow of which Landau then was a member, and certainly the actual invitation to that conference came to me from Landau. Who else was present, I do not recall. I think one could probably find out some of this by looking at the journal, the Physikalische Zeitschrift der Sowjet Union, in which my paper appeared because there would no doubt be other papers about the same time giving other talks at the conference. I haven’t had a chance to look at that as yet. I think the conference was about solid-state theory in general, but then again I can’t recall it in detail.

On the content of my longer paper, the essence is the remark which in fact is outlined in the margin in the Xerox copy I got on page 92, which is the origin of the way of understanding the possibility of radiative transitions. I recall very clearly the difficulty I felt at first about explaining this and I realized that to get rid of the atomic excitation without emission of radiation one would have to create a large number, something like 50, phonons, and if one can use the ordinary kind of perturbation theory this would happen only in the fiftieth order of approximation. And I recall the pleasure I then got from realizing that the analogy with the molecular problem and the Franck—Condon rule meant that the forces acting on an excited atom in a crystal could be appreciably different from those acting on a normal atom, including the difference in the equilibrium configuration, so that therefore the perturbation theory would be likely to fail.

Now, in the paper I say that I acknowledge a comment from Landau on this point. This may have been at the, conference in Kharkow or on, a different occasion because I visited the Soviet Union during several summers and met Landau then, so I do not know when this conversation took place. And I also do not know whether Landau’s comment already pointed out the analogy with the Franck-Condon principle, which really contains the gist of the explanation, but I suspect it did not. I think that probably he was just pointing out that one should think of the analogy with molecules where the situation was simpler and therefore could help one to sort out what was going on.

Then you ask about Frenkel’s reaction and whether I discussed the problem with him. I’m sorry I don’t remember. I do not even remember whether he was at the Kharkow conference. I certainly must have talked with him about these matters on various occasions. My guess would be that he would have been interested and would have immediately acknowledged that the study in his own paper was not terribly detailed or accurate and that I had results which went beyond his findings, though that doesn’t necessarily mean that he accepted every one of my conclusions. I’m afraid my memory doesn’t help on that. Probably by looking at Frenkel’s later papers on the subject, one could see how far he agreed or disagreed, but I haven’t at the moment the time to look that up.

Similarly, I could not comment on the reaction which my talk produced. I think probably it would have been intelligible mainly to theoreticians because there was an awful lot of material and considerations to get into a short conference talk. I don’t think any experimentalists would have been moved to do experiments along those lines. There were of course people studying the optical properties of solids; in any case, some general features were known, for example, the fact that the rare earths had sharp lines and resonance fluorescence but little true absorption, and the other, the square root of T law of the width of bands which I mention, had been found before. These things were known, but I think to make closer contact with experiment, I should have followed these up by looking at specific cases, specific experimental results on specific substances, and trying to interpret these in the light of the theory. But that wasn’t my habit at the time. What I usually tried to do was to understand the general principles and write them down as clearly as I was capable of, which might not have been too clearly, and then leave the subject.

Now, the paper in the Annalen was what is called in Germany a Habilitationschrift and you asked me to explain what that meant. Well, in all the German-speaking universities, the position was that in order to have the permission to give official lectures in a university, the so-called venia legendi, you have to qualify. You have to apply to the faculty board for this permission. Without that, you could give informal lectures or for example is asked by a professor to give a few lectures in his place or things like that. But you did not have the right to announce official lectures on your own initiative and invite students to come to them. Incidentally, though this is getting rather far from the subject, in most universities, one received a small fee for each student registered for a lecture course and paying the statutory small fee for it. That had the result that the most senior professors in each subject always gave the large big introductory lecture course, which was probably a good thing. And if you were a young Privatdozent, that means lecturer who has the permission to lecture, you would get very few students to register. And in fact, it paid to announce a course as being free of charge because then many more students would register. It would, cost them nothing. And then the university paid a small fee, much less than the regular fee per student, but since so many more would register, as it would cost them nothing, it paid on balance. Anyway, to get the venia legendi, the permission to lecture, you had to submit a kind of thesis and you also had to give a kind of model lecture — a lecture in the presence of many of the professors, other lecturers in the subject, and others from other departments of the university, not necessarily on the subject of your thesis. The idea was that this way you would prove that you were capable of giving an intelligible lecture. I think this was mainly a formality, and having written an acceptable thesis meant that you would automatically get the permission. In choosing to submit this paper as a thesis, I had of course to rely on the advice of Pauli, my professor. He thought the paper was suitable and encouraged me to put it in. I imagine his opinion was in fact the determining one in getting me approval. This is perhaps interesting in view of his later complaints about the nature of solid-state physics and the kinds of papers I was writing on it, because although this paper is very largely qualitative and complicated, he must to some degree have approved of it.

Then comes the question, how far did this paper influence work in this field, for example, by Mott? Well, again the answer is I do not know. And probably the only way to find out is by looking at some of the later papers to see whether they quote or use the ideas of my paper, or when they are still around, to ask them. But I do not remember. I have a suspicion that certainly Mott was thinking of optical properties of solids in terms of very concrete substances with very specific ideas about the electronic states involved, and so on. He might not have started from considerations of the kind of generality in my paper, but might have been led to the approximate conclusions by another route just starting from the concrete examples. But that I don’t know.

You ask what are the key points to look for in tracing this development? As far as my paper goes, it is just this question that the radiationless de-excitation, the radiationless transfer of electronic excitation to the vibrational degrees of freedom is not possible if the coupling between the atom and the vibrational states is weak, weak enough to allow perturbation theory, and that this depends on the difference between the natural environment, let’s say the forces or the equilibrium configuration, of an excited atom and an atom in the ground state. This one would expect in general to be a large difference, therefore invalidating perturbation theory, and therefore making radiationless transitions possible. Except in special circumstances, such as the rare earths, for example, in which the electronic excitation is in an inner shell which is fairly well screened off from its environment. Well, that seems to be all about this question of optical properties of solids. I might add that the kind of state my paper is about, an electronic excitation which migrates through the crystal lattice, was given by Frenkel the name of exciton, which I never used, but it’s a very nice name. But later on, the word exciton has also become used for a different concept, a combination in a semiconductor, particularly of an extra electron in the conduction band and a hole in the valence band held together by the electrostatic interaction. So, one now, I think, talks about a Frenkel exciton if one means the kind of excitation discussed in my paper.

Now, the next set of questions is about diamagnetism. As you say, this field was opened up by the classical paper of Landau. I first met Landau when he had just written or was writing, this paper, anyway, when he had just done this work on diamagnetism. It was the sort of thing that when you see the argument you realize immediately it is right, so when he came to Zurich, neither Pauli nor I had any doubt in feeling confident that Landau had got the right argument. The summary you give in your note isn’t quite correct. It is not a question of electrons jumping from one quantum state to another. It is simply that in quantum mechanics, because the electron orbit, at least the part in the plane at right angles to the magnetic field is periodic, it gives rise to a discrete spectrum. And therefore the distribution of electrons over these states, just by Boltzmann factors, is a little different, from what it would be in the continuous spectrum which you have in the absence of the magnetic field. Now, again I cannot recall precisely the steps which led me to thinking about diamagnetism, but it was part of one’s general interest in metals and just as Pauli’s paper on paramagnetism had opened up the quantum approach to metals and to solids, one was interested in magnetism in, general. One obvious puzzle was that if one thinks only of free electrons, then Landau had shown that the diamagnetic susceptibility is just 1/3 of the Pauli paramagnetism, so that the net effect would be still a paramagnetic susceptibility equal to 2/3 of the Pauli value. But in fact it’s known that very many metals are diamagnetic. Now this could be to some extent accounted for by the inner closed shell of the atoms because one knows that closed shells are diamagnetic, and that could give a contribution. But that certainly couldn’t account for as large diamagnetism as that of bismuth. On the other hand it was always clear that Landau’s calculation related only to free electrons since he ignored the periodic field of the lattice. So it was of interest to see how this could generalize. The Pauli paramagnetism one could understand would not be effected drastically because the spin moments are not effected by the periodic potential. The only effect would be that since the Pauli principle requires an increase in the kinetic energy or in the orbital energy of the electrons in order to allow them to turn their spins, if the periodic field of the lattice changes the orbital energy of the electrons, this will somewhat change the magnitude of the Pauli paramagnetism. But the Landau diamagnetism is a kinematic effect connected with the motion of the electrons, and therefore one has to be prepared for much more drastic changes due to the periodic field. So, I set out in that paper to study the diamagnetism of electrons in a periodic potential, of Bloch electrons, if you like and as usual of course, ignored their mutual interaction. Now in thinking about this I encountered two difficulties: one was practical, and the other conceptual. The practical difficulty was that to proceed in the same way as Landau, one would have to find the exact eigenvalues for the motion of an electron in the periodic potential and the magnetic field, and that would be quite a tall order. Secondly, the conceptual difficulty was that the Landau diamagnetism seemed to be due to the existence of discrete levels for the electrons at least discrete energy values for the projected portion in the plane perpendicular to the magnetic field. Now, in a solid, all the electrons undergo collisions with impurities and with phonons and such collisions will lead to a broadening of the otherwise sharp energy levels and you might suspect that if this broadening exceeds the spacing between the levels, then you would in effect get a continuous spectrum and therefore the Landau effect might disappear or at least might be substantially modified. Now, quantitatively, at most temperatures and almost all practical magnetic fields, the ratio between the broadening of levels and the spacing, which can also be expressed as the ratio of the Larmor frequency to the frequency of collisions, is in fact small. That means that — I don’t know whether I’ve put that the right way around — in fact the Larmor frequency is small compared to the collision frequency in practice and therefore the broadening due to the collision is much greater than the spacing of the levels. And therefore, if you realize that, you begin to feel doubts as to whether Landau’s argument is applicable to metals in practical circumstances. Now, in worrying about the second conceptual point, I was very pleased to realize that there was a general theorem in statistical mechanics which said that the effect of the perturbation, in collisions for example, can be small even though the associated energy is greater than the spacing of the unperturbed levels provided that it is small compared to kT. Any effect involving energies small compared to kT, k being the Boltzmann constant, of course, has only a very weak effect on the statistical behavior of the system. That is quite a general and comforting result and it applies only to problems of statistical equilibrium, not to transport problems. But it led me in a much later paper to prove the minimum property of the free energy of the system which again is a general theorem of the same kind. So that meant that therefore for most practical cases, one did not have to worry too much about collisions. And the other question could be solved by realizing that for reasonably weak magnetic fields, one didn’t have to solve the Schrodinger equation in the periodic potential and the magnetic field completely, but one could start from the solution without a magnetic field, and then take the effect of the magnetic field into account by an approximation. Now, as I show in the paper, this could not be done by just treating the magnetic terms in the Hamiltonian as a small perturbation; that wouldn’t do at all. The reason is that the Hamiltonian is normally formulated in terms of coordinates and momenta, and the momentum of a charged particle in a magnetic field is not a physically sensible and not a gauge-invarying quantity. And therefore looking for the magnetic terms in that form doesn’t make sense, and in fact the magnitude of the perturbation depends on the size of the piece of metal under consideration. But I had learned from discussions about electromagnetism in other contexts that what one should do, is to introduce the gauge-invariant combinations, that is to say, momentum minus a multiple of the vector potential. These are physically interesting quantities, but now you get a vector whose components do not commute with each other in the magnetic field. But if the magnetic field is weak enough, they almost commute, the commutators are small, and therefore you’re led to the problem, how to evaluate the free energy or the partition function of a system whose Hamiltonian can be expressed as a function of several quantities which almost commute with each other, so that you want to write an expansion in terms of their commutator. Now I found out how to do this, and then realized that the procedure I had developed was equivalent to one invented by Wigner for treating quantum systems in a temperature region where they were almost classical so that the commutators were small. My derivation had a different mathematical method but was equivalent with what Wigner had done. So now I had the tools ready for looking at the diamagnetism of electrons in periodic potentials, and I found that the main part of the result was expressed in terms of an integral involving the energy of the electron as a function of the wave vector in the absence of the magnetic field. I derived that only in the case of the so-called tight-binding model where the electron wave functions in the solid are very similar to those of the free atom, though later work showed that as far as this particular term in the answer goes, it is much more general than that derivation. I also showed that there were other terms, one essentially like the diamagnetism of the atom, and another, a cross—term, which has to do with matrix elements going from one electron band to another. I expressed the conjecture that the anomalously large susceptibility of bismuth, which everybody knew at the time of course, was due to these extra terms. I think now that is no longer a reasonable view to take. And then I gave a prescription how to calculate at least one important portion of the diamagnetic susceptibility and left it at that; I didn’t follow it up by trying to look at specific cases. Also, like some of my other papers, this one suffers from the fact that too much is heaped together in one paper. If I had published the general statistical mechanics theorems which I was using separately from the work on diamagnetism, it might have made the contents more easily accessible to other people.

Now to the specific questions: You ask, is it fair to say that this work was stimulated by Landau’s paper? That is undoubtedly true, though it took some time before I moved from Landau’s paper to the problem of bound electrons. And then you say as well by the recent experiments by de Haas and van Alphen? No, I don’t think I was influenced by these very much. I did know about them. During a visit to Holland I don’t know if it was 1930 or 1931, I visited de Haas and he talked about this strange effect of the periodic fluctuations in the bismuth susceptibility, which mystified him. And I remember he told me since he didn’t understand what was going on, he was trying to look for the dependence of this effect on everything, including time. So he kept one particular specimen of bismuth in his cupboard, and every few months re-measured the effect to see if it was going to change. You see, I found this phenomenon quite mystifying, but I don’t think I thought about it or attempted to find an explanation at that time.

Now, we come to the second paper. You refer to this as being written a year later, but in fact, the first paper reached the journal in December 1932, and the second paper in the middle of January 1933, so there was only a month between them. Now in the first paper I had used an expansion in terms of the commutators, which meant in fact it turned out that the condition for that was that the spacing of the levels should be small compared to kT. I started wondering what would happen in the opposite limit, if the spacing was large compared to kT, and I did that first of all just for the Landau case for free electrons, and started thinking about what one could do in that case. And then it suddenly dawned on me that you would in that case get a periodic variation in the susceptibility as a function of the field or rather its periodic more or less in the reciprocal of the magnetic field intensity. And this of course immediately reminded me of the funny result of de Haas and van Alphen.

Now, it is quite true as I was recently reminded by Shoenberg, who is probably by now the world’s expert in the experimental study of the de Haas-van Alphen effect, this statement is already in Landau’s original paper. Why didn’t I notice it? I cannot say. Probably it seemed a very abstract problem of such strong magnetic fields, so I must not have paid attention when I read the paper. Also I probably didn’t read the paper very carefully because I had discussed its contents with Landau before the paper was published and knew the main elements very well. It may be that I never looked at the paper in detail, so the discovery of the theoretical oscillations was quite new to me, and it was only a year or two ago now that Shoenberg pointed out to me that this was already in Landau’s paper, and I had not until then noticed it.

Of course, the other point, why did not Landau see the connection to the de Haas-van Alphen, well or why didn’t anybody else? Well, of course, in pointing out this effect, Landau adds that he has discussed this possibility with Kapitza, who assured him that no such effect would in practice be observed. And in a sense Kapitza was right, because for ordinary metals, the effect needs such low temperatures and such carefully controlled homogeneous magnetic fields, that with the techniques then available, it was not observable, bismuth being the exception where the effect was very much larger. Now, Landau had visited Holland, but probably had either not talked about this with de Haas or again had forgotten the implications of this strange phenomenon. Of course, I was very excited about having seen this possibility of accounting for the mysterious effect and I must have talked to lots of people about it. I do not recall talking to Landau about it, and if I did, I imagine he might have said that it was already known to him in his paper. Although as long as he thought that my result was correct, which obviously he must have done because he had obtained it himself, he might well not have bothered to point this out. This was quite within Landau’s nature.

Now you ask whether in Rome Fermi was interested in this. I don’t think he was. I don’t recall whether I told him about the results; I probably did, being in his department when writing the paper. I think the main work on diamagnetism probably had been done already before I came to Rome, I mean the contents of the first paper. I probably just tidied it up and wrote up the paper in my first month or so in Rome — I must have arrived there during October. And the second paper, once you see the point, doesn’t take very long. Fermi might have listened politely to my story about this, but it wasn’t his main field of interest.

I didn’t do any more about this in Rome; many years later in Cambridge, I talked about this problem with Maurice Blackman, and he seemed interested in following up this problem and getting a better fit to the case of bismuth. I had found that I could not get a quantitative fit; I could not find values for the effective mass of the electrons and the effective number of free electrons to fit both the amplitude and the period of the oscillations. Blackman’s work involved, first of all, doing more accurate numerical calculations over a wider range of parameters, but also allowing for the fact that there was not one effective mass, but one could assume a band consisting of very anisotropic branches in which therefore the effective mass was different in different directions, but these several branches were oriented in such a way that overall the symmetry of the crystal was preserved. And in that way, a fit could be obtained. Now, that work was done by computing the actual energy levels and, by means of Fermi statistics, averaging over the thermal distribution, which was quite elaborate. When later David Shoenberg was visiting Moscow and obviously talked to Landau about the phenomenon, Landau showed how to simplify the calculations where the ratio of the level spacing to kT is still less than 1 but not negligible. And then one could get an expression in closed form in terms of the so-called Poisson summation formula, a device which I had learned about from Pauli, and which I liked to use, but it had never occurred to me to use it on this particular problem. There’s an interesting sideline on that. When Shoenberg returned to England, he had written a paper about the experiments, and wanted to add Landau’s theory as an appendix. By that time, Landau was in trouble with the authorities of the Soviet Union. I believe he was already in jail, and therefore his work couldn’t be published there. All that Shoenberg had was a page of notes by Landau which he couldn’t quite interpret, so I helped him to write out the calcu1ation. I recognized that Landau had used Poisson’s summation formula, and so the Shoenberg paper in the Proceedings of the Royal Society now contains Landau’s work as an appendix. There is a Russian version published of this paper, which of course could not refer to Landau’s work, so that now Russians always have to quote the English version of Shoenberg’s paper for the theory.

Now, there is a question about Pauli’s interest in this. Here you refer to what I said in an earlier interview, namely that Pauli was interested in this because of its possible connection with superconductivity [The work that interested Pauli on the grounds that it might explain superconductivity concerned the Umklapp processes. L.H.] I haven’t got the transcript, of this interview in front of me, and because of my move, it would take a rather long time to find. However, I can’t recall that there could be any possible connection between this and superconductivity. I think Pauli’s interest in this problem was because like Landau’s original work on diamagnetism, it was a very clean quantum mechanical problem. If you take the idealization of free electrons, then the problem is exactly soluble, and you can find practically useful approximations to the answer with a range of approximation that is easy to judge and which, incidentally, made use of the Poisson summation formula of which Pauli was very fond. I don’t know if I mentioned that on the earlier occasion. When I wrote my book on the quantum theory of solids, which of course contains a passage on the de Haas-van Alphen effect. I of course sent a copy to Pauli, and after a while, I got a furious letter from Pauli because he had referred to this de Haas-van Alphen effect in some lectures and had found it convenient to quote some results from my book, and discovered that just at that place, there was a misprint, at least in the first printing of my book, so he was very cross. At far as I know, that was the only major misprint that has been discovered. But I don’t think the reason for his interest was to do with superconductivity. What I do recall is that in connection with problems about the validity of the normal approximations to electrical conductivity at low temperatures and the possibility of an exponential rise of conductivity at low temperatures, one might have suspected that that had something to do with superconductivity, and certainly when I discussed that with Pauli, that would have come up.

Next we come to a number of questions on statistical mechanics in general. Now, you ask for some general outline of the development there and ways of going about it. Well, I think in outlining the theoretical development one has first of all to remember the work of Meissner in Germany on the Meissner effect, which showed that, subject to certain limitations, if a sphere is cooled in a magnetic field, the magnetic field lines will be expelled at the transition point, which shows that the field-free state of the superconductor is an equilibrium phenomenon and not merely a result of the infinite conductivity by which the magnetic field is unable to penetrate into the body. Now, that I think had a very great influence on people’s thoughts, and this led Casimir and Gorter to apply thermodynamics to the superconducting transition and derive a relation between the critical magnetic field which can destroy superconductivity and its dependence on temperature. The connection between that and the anomalous specific heat of the superconductor showed that you’re really dealing with an equilibrium situation to which thermodynamics is applicable. Now about that same time, and I’m not sure of my chronology, comes the Londons’ theory of superconductivity, which exploits the idea of the superconductor being basically a diamagnetic substance with the maximum possible diamagnetism and which leads to some interesting concepts, in particular the idea of the penetration depth which has some relation to experiment. But also it was not very satisfactory in its theoretical foundation. Then Landau and Ginzburg took this further in their paper in which they made a model which is still phenomenological, but based on the concept of what is called an order parameter, a model which is really meant in the first place as an expansion valid in the neighborhood of the critical point, but which in fact proves to be a useful model for a much wider range than that. Now, this I think has very little connection with Landau’s theory of Fermi liquids. As I think I have mentioned in the paper at the London conference, Landau had been thinking about electrons in metals, in particular on the question why the electrons could be with such success described as practically free, or rather independent of each other, when in fact their Coulomb interaction was very large. And he had pointed out very early — I remember having discussions with him on that point in the very early 30s, though I can’t say exactly when and where they took place — that this was due to the Pauli principle, that although the matrix elements of the interaction between the electrons were large, they could not cause any change in the state of the electrons because there was essentially nowhere for the electrons to go because of the Pauli principle. Now this Fermi liquid theory is a formalization of that idea and its applicable to electrons in metals, above the critical point if you are in a superconductor, and also to liquid 3H, above any phase transition point. In fact, any phase transition like the super fluidity of liquid 3H or the critical point for superconductivity is just the place where the Landau approximation involved in the Landau’s Fermi liquid theory breaks down, so the connection is rather negative. I’m not aware; I don’t know the literature that well, that Landau and Ginzburg had done anything special about phase transitions as such. They of course had done their classical work about superconductivity. However, the person to ask about that of course is Ginzburg. I know that now he is mainly interested in astrophysics, I’m sure he remembers his work with Landau, and probably remembers work of Landau’s in which he was not involved. And I think if somebody with suitable interest has the occasion to visit Moscow, he might talk with Ginzburg about that. Or one may catch him at some conference; he does come to conferences in the West from time to time. There are no doubt other people who might remember, but on that aspect, the Landau-Ginzburg theory, he is perhaps the best person. About Landau’s work on phase transitions, the person to talk to there would be E.M. Lifschitz. He was here recently and no doubt will appear again.

Now we come to the question of the paper I call the statistical foundations of the electron theory of metals which in fact is the question of the validity of the perturbation theory. The history of that is that Kretschmann, about whom you ask, who was generally known to be somebody who quibbled about the current theory without really understanding it — I don’t know whether that’s a fair assessment, but that was our impression at the time — Kretschmann had written a paper in which he claimed that the whole foundation of the current electron theory of metals, in particular the work of Bloch, also mine, was incorrect because it made use of perturbation theory in circumstances where this wasn’t permissible. Well, I decided to disprove this, and defend the current theory and therefore set out to lay down the conditions for the applicability of perturbation theory and to show that they were satisfied in the normal situation in metals. And to my great surprise, I came to the conclusion that it wasn’t that the conditions were not satisfied. Not precisely for the reasons claimed by Kretschmann, but, never mind, they were not satisfied. And what actually happened was that it looks as if in applying the perturbation theory, one was relying on the fact that h/τ < kT where τ is the collision time; in other words, h/τ is a quantity of the dimension of an energy measuring the intensity of the collisions. One was assuming that this should be small compared to kT. In fact, if you compare these quantities for most metals, at reasonable temperatures you found that they are of the same order of magnitude, sometimes one a little larger, sometimes the other a little larger. Never terribly different. Now, in fact I discussed that with Landau, who pointed out this was natural because if you asked dimensionally what determined these two quantities and found where they came from. In metals at temperatures above the Debye temperature, where things are classical, these two things were dimensionally the same. So now what to do? And then Landau gave an argument showing that in conditions where you could treat the electrons as moving in a static potential including the effect of the collisions, then things are better. Now this is of practical interest because first of all of course impurities are essentially static, and also lattice vibrations can be treated as static at temperatures above the Debye temperature, where then the energy of lattice vibrations is less than kT. Now in that case the condition is not that h/τ should be small compared to kT, but only that it should be small compared to the Fermi energy. Now that is in most situations, about 100 times larger than kT and therefore this condition can easily be met. That’s true for metals; in semiconductors it’s different because there the Fermi energy is of the same order as kT. Also at low temperatures, where the finite speed of propagation of the lattice is important, the argument by Landau fails and whether the result is very different nobody knows. At very low temperatures, when the resistance is almost entirely residual resistance, that is to say due to impurities, everything is all right again because they are static.

You asked whether this argument of Landau was ever published, and/or what happened to a promised joint paper by Landau and me. The answer is this was never written, the only mention in public of Landau’s argument is in my paper, and I enlarged on it a little in my book The Quantum Theory of Solids. I understand from E.M. Lifschitz that in a book about to be published or just published on the statistical mechanics of irreversible phenomena, which includes of course the theory of conductivity, he has again written up this argument, I mean that book is also based on Landau’s lectures. But probably there he would just refer to my paper. I haven’t yet seen what he is writing.

You ask how I found out about Landau’s argument, well this was in conversation. Again, I don’t know when and where, but probably during one of my summer visits to Leningrad or possibly at the Kharkow conference. Actually, I think I visited the Soviet Union in ‘34, and this is quite likely where we might have talked about this.

Now this paper and the paper about transition temperatures were on account of talks given at a conference in Geneva published in the Swiss journal, Helvetica Physika Acta. The conference at the University of Geneva was about solid-state theory, or the electron theory of metals. It was organized by the professor of theoretical physics there. I believe his name was Mercier and I have somewhere a photograph taken at that conference. It was not a very large conference, Hans Bethe was there, Sommerfeld, R. H. Fowler from Cambridge, Nordheim, and if I find the picture, I can probably add a few more names. I had undoubtedly discussed these problems with Bethe in Manchester. I do not think anybody else had been interested.

Well now the other paper about transition temperatures also was a talk at that same conference. The origin of this was that one was inclined in those days to look at the problem of solid state, in particular the vibrations of atoms, by looking at a one-dimensional case as a simple model. And if you think about one dimension, then you notice that the fluctuation in the distance between two atoms increases with the square root of the distance apart, so that in the case of a very long chain there is no coherence; the periodicity in the position of the atoms has been lost as between one end of the chain and the other. Once you see that, you wonder how you could ever get sharp X-ray lines from solids and it was thinking about that problem that I suddenly understood that there was a fundamental difference between the one-dimensional case and the three-dimensional one: that in three dimensions, in fact, for at least reasonably low temperatures, there is a clear correlation in the position remaining over infinitely long distances in a big crystal. That meant that there was now clearly a qualitative difference between a crystal at low temperatures below the melting point, and a liquid and that, therefore, there must be some sharply-defined temperature where you change from one to the other. And it was this argument that I expounded in this lecture and in the paper. The same point is made in rather more detail in my lectures at the Institut Henri Poincare a little later. Of course, this is a fairly naive argument and it’s an early case of what one now calls an order parameter. In these papers, particularly the one at the Swiss conference, I make the point that in two dimensions, you could not have a sharp melting point, that of course has to be taken with a pinch of salt, there are much more complicated arguments relating to that but I shall not go into details.

You ask about the nature of the discussion with Bethe on this subject. I can’t recall. It might well be that this was connected with our discussions about the order/disorder problem in super-lattices, where the Bethe method involves essentially introducing a long-range order which is also an order parameter, and therefore also leads to a sharp critical point. So there is a connection between these things, but whether this connection came out in our discussion, or whether it was the basis for our discussion, I’m afraid I don’t remember. Actually, in formulating your questions, you say that I refer in my paper to Bethe’s considerations which give the impression that it was essentially Bethe’s argument. That, I think is not right. As far as I recall, it was essentially my argument, but in developing it, I had been talking about it with Bethe, and he made some useful comments. That at least was what my acknowledgement in the paper would imply.

Now you ask about the lectures at the Institut Henri Poincare. Well, it was customary in those days for people to be invited to give a series of probably four lectures, four or five, I don’t remember, which at the Institut at that time had to be given in French. My French has always been rather poor, but I struggled through it somehow. There is nothing much else to say about the occasion. The chairman was Langevin, and I saw a lot of Bauer, who was a solid-state physicist interested in magnetism mainly, a very charming person. And I just talked about some of the work I’d been doing on solids, picking out the parts which had a more sort of abstract mathematical background, because of the rather mathematical interests of French theoreticians at that time. You ask who else was working on that subject; I do not know.

You also mention that I excluded the cases of ferromagnetism and superconductivity. Of course, my actual argument related to lattice vibrations, and I commented on other situations only by analogy. Now the reason I make an exception for ferromagnetism is because in a real ferromagnet where the magnetic interactions are taken into account, the ground state is, as one knows, not the one in which all spins are parallel and therefore has the highest degree of order because that would give too high a magnetic energy. And in fact the ground state in the absence of an external magnetic field is one where the substance is divided into layer-shaped domains, with opposite spin orientation, that’s been analyzed particularly by Landau. There’s another reason, which I don’t mention, where I threw some doubt of the possibility of melting in two dimensions. There is certainly no doubt on the possibility of ferromagnetism in two dimensions, because in another paper, which probably we shall come onto, I have in fact given a proof that in the Ising model ferromagnetism does exist in two dimensions.

Now we get on the question of super-lattices, which certainly was motivated by the work of Bragg and Williams. How did they get onto it? I don’t know exactly, because it happened before Bethe and I went to Manchester in 1933. But Bragg of course was interested in crystals; that was his main life’s interest, and it is not unnatural that he should become interested in super-lattices. Perhaps through contact with a metallurgist called Sykes, a very distinguished metallurgist with whom Bragg certainly had many discussions about alloys and super-lattices. Whether these preceded or followed Bragg’s interest in super-lattices, I do not know.

Anyway, Bethe and I came to Manchester, saw his work and realized that the approach of Bragg and Williams was rather crude while instructive. And this obviously was a challenge, to see whether one could do better, and so we talked about this, Bethe and I, and tried to see whether we could develop an approximation in which one would do better. Now, the beginning of it was a model where you essentially looked at a finite array of atoms surrounding a chosen atom which is called the central one and did the statistical mechanics of that, and then tried to find out from that something about the averages and about correlations. And according to Bethe’s paper, I must have started off that approach. I don’t recall that phase in our discussions. But then Bethe made the important further step of describing the effect of atoms not included in a particular array by an order parameter, which expresses the fact that the configuration of the atoms outside favors atoms in their right as opposed to wrong positions. And that was the clue to getting a reasonable approximation, at least an approximation that is reasonable at high and low temperatures. No approximation of this kind would really work satisfactorily in the neighborhood of the transition point, let alone give the mathematical nature of the transition point right. The work in first of all devising and then evaluating this approximation was entirely Bethe’s, though we kept in close touch and were discussing this work regularly.

I then got interested in some further applications of that method. The cases Bethe had treated were those of alloys with a 50-50 composition, but there were cases where the atoms were in a proportion of 3 to 1, for example, the copper-gold alloys with a 3-to-1 composition in a face-centered cubic structure. There you have the complication that two neighbors of a given atom can also be nearest neighbors of each other, and that is not the case, in the systems Bethe had considered, and that requires a modification therefore. That was quite an elaborate theory which I worked out when I had moved to Cambridge to the Mond laboratory, Bethe was already in America. And it was quite interesting; it brought up the possibility of getting first-order phase transitions in such systems, or the decision whether it was a first-order or second-order transition seemed rather marginal, and therefore sensitive to the approximations used. There was also the possibility of this forming a two-phase system in which one part of the metal would have an excess of atoms of one kind with a regular ratio and some of the other would have correspondingly fewer of those atoms. And some work on that was done under my supervision by a student, Colin Easthope, who was then in Cambridge. I also found when I got to Cambridge that R.H. Fowler and his students were working on adsorption on the formation of layers of atoms on solid surfaces where you had the problem in taking the interaction between the adsorbed atoms with each other into account. And there again, Bethe’s method could be applied, and helped to improve the results over those of Fowler and others.

You ask about the origin of the idea about ordering in lattices. Well, this was experimental. The fact that certain alloys formed super-lattices below a certain critical temperature was an empirical finding presumably of metallurgists or X-ray crystal structure experts. And, as far as I know, Bragg and Williams were the first to pick up the theoretical problem posed thereby. And then Bethe and I tried to develop a more accurate molecular theory. Landau I think was not at that time interested in this problem. Later of course, in his general classification of transitions in terms of symmetry, I imagine he would have included that as a special case.

Next, you ask about the paper on magnetic transitions in superconductors. This was written in Cambridge when I was at the Mond laboratory and so was David Shoenberg. So we certainly interacted, and I knew about these experimental results, and thinking about those prompted this idea of an intermediate state which I describe purely phenomenologically. And then later on Landau gave an actual theory of this intermediate state as consisting of alternate layers of superconducting and normal material. You see clearly in this context, the remark in the footnote on page 616 of the paper, that no arrangement of superconducting and normal regions seems to be possible which satisfied all the boundary conditions. Now, this was meant in the context of the attempt by Casimir and Gorter to find such an arrangement with the layers being macroscopic and therefore satisfying the macroscopic field equations. In the Landau model, the layers are microscopic; they are so thin that the surface energy of the interfaces between them is not negligible compared to their volume energy, and then the macroscopic equations and the boundary conditions between regions need no longer be satisfied at the interfaces between the regions. That is why this comment of mine is not in contradiction with the Landau theory, which of course is accepted today as correct.

Now, about the paper on the Ising model of ferromagnetism. It is true that at that time, Ising’s paper was rather old, but interest in ferromagnetism had grown recently because with the advent of quantum theory, Heisenberg had shown the physical basis of ferromagnetism; Bloch and Bethe had given approximations to the theoretical problem of ferromagnetism. And so, while all this was leading to rather complicated mathematical problems, one did look-back to the very simple and attractive model of Thing. And also, Bethe and I realized that the problem as formulated by Ising was identical with the problem of superlattices and, with certain limitations, (i.e., you could not have an external field present) you have a complete equivalence. But, in both cases, one assumed interaction only between nearest neighbors, so there seemed to be a physical application of the Ising model. Now, in fact, if you like, what Bethe and I had done was to study approximations to the three-dimensional case of the Ising model. Well, my particular interest was aroused in Cambridge by listening to a seminar in which some mathematician reported on Ising’s paper. I can’t remember his name. Of course Ising’s paper only treats the one-dimensional case, and concludes that that does not show any ordering, any ferromagnetism, and this in fact can be proved in a few lines. But the man who gave the talk spent the whole seminar hour on proving this, and then in the last few minutes gave reasons why the same argument should also apply in two or three dimensions. Well, I didn’t believe that. First of all, I had shown previously that there was a large qualitative difference between one and three dimensions, and also because that would have meant that things like Bethe’s approximation would have been completely misguided. Of course, if you talk about a rigorous proof, then relying on an approximation isn’t very good, but rather than disproving the arguments of the man whose talk I had listened to, I thought it might be easier to prove the contrary, in other words to prove that in two or three dimensions in fact there is an ordered state at sufficiently low temperatures, and so I sat down to try and do that, and indeed found that it was quite easy for the two dimensional case. I think nobody doubts that if this is true in two dimensions, it must be even more so in three. Now, that result pleased me of course because of its simplicity, and I was interested recently to learn from experts in statistical mechanics that until a few years ago at least, this had been the only paper, the only method giving a convincing proof of the existence of the phase transition in any system. I found this gratifying but surprising because the argument was so simple. And also because it’s my only experience where a proof I had devised was accepted as really a valid proof by people insisting on mathematical rigor. The simplicity of the method however implies that the value it gives as a lower limit for the transition temperature is not very close to the real value.

Now, you ask did I try to calculate the transition temperature with the Ising model. Well, the only manageable approximation to the Ising model I knew was the Bethe method. And in effect Bethe had calculated the transition temperature to that approximation for the Ising model, and there was no point in doing any more about that. I also see that I acknowledged that Bethe had suggested part of the proof, I’m afraid I have no idea what part this was.

Finally, on your last question I’m afraid I have to give up. You refer to a Landau-Peierls theorem on the lack of one-dimensional ordering in a three-dimensional system. I don’t recognize this, and I don’t know what it means, what one-dimensional ordering in a three-dimensional system would be, so perhaps there was some error in the formulation of the question. Anyway, if you will let me know a little more what this refers to I shall try to answer it. I think that about covers all the questions and I cannot think of any other material to add spontaneously, but if I can think of something before I have a chance of mailing these tapes, I shall add it.

Note Added 10/28/81: I think I now see what was meant by the question referred to in the, last paragraph. In my 1934 paper on transition temperatures, which we have already discussed, I point out the fundamental difference between one and three dimensions. No doubt Landau refers to this also in his books and possibly in papers. I cannot remember specific conversations with him about this, but I was quite excited about this finding, and I am pretty sure I must have mentioned it to him. He presumably also saw my paper. Whether he had seen this already himself, I cannot say. But I do not believe I was influenced by him in arriving at my argument.