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'En attendant Debye...' ('Until Debye': the specific heat of solids)

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uniquecharacteristic frequency a+, failed to ac- count quantitatively for the observed behaviour; it predicted a temperature dependence of the specific heat ’En attendant I C, X@(+) Debye. l (where the ‘Einsteinfunction’ @(x) behaves like x’ e-x for x +m), leading to an exaggerated de- M Hulin crease of C, when T goes to zero, which does not correspond to the observed T3 behaviour.It was Equipe de Recherche sur la Diffusion et only with the Debye model (Debye 1912) that the 1’Enseignement de la Physique, Universite Pierre T3law was finally understood. et Marie Curie, Tour 32, 4 place Jussieu, 75231 This leaves one with a somewhat puzzling ques- Paris Cedex 05, France tion which we can put in the following way: why Received 9 December 1980 did Einstein not propose the ‘Debye model’? Ein- stein was of course quite familiar with the black body problem, and it is nowadays a very conven- Abstract The problem of the specific heat of solids and its behaviour at low temperatures played an impor- tional remark that a T3specific heat means a T4 tant role in the evolution of basic ideas in physics at the internal energy, that is, via Stefan’s law, a charac- turn of the century. It was finally solved by Debye, but teristic of the black body, which it is easy to trace why did Einstein, who showed a keen interest in that back to the linear dispersion law of electromagnetic problem for several years, fail to propose the Debye waves. A T3behaviour of C, at low temperatures model? This article recalls a few facts and presents a thusseems reasonably direct indication that the few suggestions. solid state thermal properties ought to be explained through the intervention of excitations with, at least RCd Le problkme de la chaleur spCcifique des sol- at low energies, a linear dispersion law, and acous- ides et de son comportement & basse temperature a tic waves (or ‘acoustic phonons’ if we adoptthe un rble important dans l’bvolution des idees fon- jouC modern solid state terminology) appear as the most damentales de la physique au dbbut du sikcle. Ce fut Debye qui en apporta la solution, mais on peut se de- natural candidates. mander pourquoi Einstein, qui manifesta pendant With this question in mind, looking through afew plusieurs anntes un interst soutenu pour ce problkme, original papers? of that period tirstly gives one some hCsitaproposer ce qui estdevenu ‘le modkle de information and secondly raises somequestions Debye’. Cette article rappelle quelques faits et suggkre related to the ‘C, puzzle’; both aspects are consi- quelques 61bments de rCponse. dered briefly below.

2 A point of information:the presentation of 1 Einstein versus Debye: a pdewithin the C, experimental results dilemma One fact must first be stressed: there was no ‘ex- The specific heat of solids was a major problem for perimental T3law’ in 1911. Modern textbooks, for physicists at the turn of the century and played an the sake of brevity, oftenpresent this law at the important role in the introduction of quantum very beginning of their discussions of the C, prob- ideas.This fact is somewhatobscured in physics lem, which may leave the impression that the teaching by the stress put on atomic physics prob- analytical form of the C,(T) function at low temp- lems, which generally appear in curricula before eratures had already been recognised at the time of solid state physics (e.g. through a discussion of the Einstein or, a fortiori, Debye. This was not the case. Bohr model). However, the recollection that one of Actually, the historical development may be the earliestrecognitions of Albert Einstein as a summed up in the following way. Deviations from major ‘star’ of physics was the invitation extended the law of Dulongand Petit, which had been to him to report on the specific heat of solids at the known even at room temperature for some solids 1st Solvay Congress of 1911 should’suffice to show (e.g. diamond) since the middle of the19th cen- the importance which was attached to this problem tury, became more and more frequent when lower by theleaders of the physics community atthat temperatures could be reached (that is, when liquid time. nitrogen,then liquid hydrogen were increasingly In 1911, Einstein had already made an important used). It was clear that Dulong and Petit had only contribution to the C, problem, and the ‘Einstein given high-temperaturea asymptotic law. The solid’ (Einstein 1907) hasremained a classical model. It is now emphasised that, while it explained t The papers by Einstein are described by Lanczos for the first time the low-temperature deviation of (19741, whose book is a valuable, readily accessible C, from the Dulong-Petit law, this model, with its source of great interest. 0143-0807/80/040222+03$01.500 The Institute of Phpics & the EPS ‘En attendant Debye.. .’ 223

Einstein model (Einstein 1907) removed the con- 3 Further, more speculative, remarks straint of having C, fixed at its classical (Dulong Although he did not have the benefit of the very and Petit) constant value obvious reasons, connected with the T3law, which areset out in modern textbooks, to suggest the C, = 3R; introduction of acoustic vibrations as the ‘oscil- lators’ of his 1907 model,Einstein felt-and, we but everybody, including Einstein,admitted from could say, from the very beginning-that this model the very beginning that there was no quantitative had to be improved by taking into account a variety agreement with experimentalobservations, and of excitations, with various frequencies. However, that this was to be expected since the unique vibra- his four-years’ research in this field is marked by tion frequency characterising the Einstein solid was whatappears to bea curious reticence. Thus, al- clearly a crude, unrealistic, oversimplification. though he writes down (Einstein 1911b) the ‘mod- However, the impact of the Einstein theory was ern’ formula for U and C,, involving the density of such that most authors, including experimentalists, vibration states, g(o), and what is now the Bose- were careful not to go too far from it, and made Einstein distribution function, together with a sum- efforts to present theirresults in the language it had mationover frequencies, he remainsreluctant to introduced. take completeadvantage of this formula, and A typical example is provided by the experimen- makes no clear attempt to use effectively a whole tal ‘Nernst-Lindemann formula’(Nernst and Lin- spectrum of vibrations. demann 1911), which played a major role in the What may havebeen the reasons for this at subsequent reflections of Einstein(1911b), Born titude? We would like to make the few following and von Karman (1912) and Debye (1912). Nernst proposals. and Lindemann, at the time the leading experimen- Firstly, Einstein was still quite close to the origi- talists in the field, showed that their measurements nal derivation of Planck’s black bodyformula, could be accounted for by the sum of two Einstein which introduced a ‘resonator’ in equilibrium with functions, one with an ‘Einstein frequency’ one-half the electromagnetic field. The fundamental reasons of the other. Several remarks are in order. for the success of Planck’s theory were not at the The Nernst-Lindemann measurements could not time fully understood, and Einstein may have be carried out below 20 K, and it would have been wished to be careful not to venture too far from very difficult to find a T3 law. In modern terms, this the original theoretical conditions examined in this T3behaviour appears for T at most one-tenth or so theory. of the Debye temperature 0, and if @-which is ‘a It also seems clear that, like other physicists of function of T’(actually just a way of saying that the that period, such asNernst, Einstein needed to Debye model is not to be taken too seriously on persuade himself that mechanical (acoustical) lat- quantitative grounds)-is reasonably constant. This tice vibrations effectively played a role in the opti- leaves one with quite a restricted domain of varia- cal and thermal properties of solids. In several tion of T,say between 15 and 35 K or so, where a papers(Einstein 1911a,b) heseeks aconnection T3 law could have been valid, but was very difficult between the ‘Einstein frequency’ of his model, and to ‘invent’ with no a priori guess prompted by other crystal properties, e.g. infraredabsorption, theoreticians. compressibility, etc. . . , and he is a keen observer In fact, Nernst and his school were content with of experimentalresults. This search involves di- opening the ‘modern’range of temperatures be- mensional analysis considerations and the repeated tween 100 and 20 K or so; they could not pay too use of a ‘local model’, where the vibrations of an much attention to what was happening at the lower atom on a cubic lattice, elastically interacting with end of this range;and they were very normally 26 nearestneighbours, are studied. He remarks satisfied with aclean, and seemingly promising, (Einstein1911b) that, within the 26-neighbours formulation of their measurements that was closely local model, taking into account these neighbours’ related to recent theoretical proposals. It remained own vibrations will profoundly modify the vibration to thsoreticians, as Einstein puts it, to find out why spectrum of the centralatom. But hedoes not the solid state oscillators spend half their time make the next step, which is to Fourier-analyse the vibrating at half their normal frequency. vibrations of the crystal as a whole, and thus intro- All this provides a nice example of the complex duce the acoustical waves. interaction which so frequently occurs between Maybe some mathematical uneasiness has played theory and experiment, and a deeper look into the arole, but it is probable that extending Planck’s history of that part of solid state physics would not treatment to such an abstract thing as an acoustical be without interest. wave, instead of a clearly defined, localised, Wehave not yet tackled what is probably the mechanical-like ‘resonator’ seemed to Einstein too most puzzling point: the fact that Einstein did not risky a step to takewithout many previous cautious consider acoustic vibrations, which should have ap- considerations. peared as rather naturalprotagonists in this C, Finally, one cannotescape the feeling that the dilemma. We lack here explicit facts; yet, we may problem of the electroniccontribution tothe put forward some suggestions. specific heat, which we have not mentioned up to 224 M Hulin

now, always remained present in Einstein’s mind. soon as Debye hadproduced his model. (For 15 By 1907, atoms had ceased for several years to be years or so, solid state physics will almost disap- regarded as indestructible entities, and, in metals at pear from the field of pre-quantum physics, and least, electrons were known to wander through the atomic physics will strut in the foreground of the ionic lattice. They seemed not to contribute to C,, stage. This should not lead us, however, to let our and this was another puzzling problem (only to be students ignore theimportant part previously solved with the Pauli principle and Sommerfeld’s played by solid state considerations.) Perhaps we theory). When proposing his model Einstein (1907) might also recall that some features of the Debye insists on the fact that the ‘Einstein frequency’, for theory have remained not entirely clear even up to a vibrating mass m, may be expected to increase as now (Wannier 1966): the extent of its success was l/m; thus the ‘Einstein temperature’ 0, for light partly fortuitous, and it could be worthwhile to try particles like electrons will be much higher than and understand what helped luck in this affair. room temperature, and the electronic contribution to C, will be reduced by a factor = ,-@Eir -x 1 from its classical value. One of the advantages of Ein- References stein’s model is thus that it provides a way out of the Born M and von Karman T 1912 Phys. Z. 13 297 electronic specific heat dilemma.Einstein seem- Casimir H B G 1977 in Storia della fisica del XX secolo ingly remained very attentive to this point; it was (Roc. Int. School Enrico Fermi) (New York: thus very important for him not to abandon the Academic) p 159 Debye P P 1912 Ann. Phys., Lpz 39 789 simple mechanical picture of an oscillator with a Einstein A 1907 Ann Phys., Lpz 22 180 definite mass, and this may have led him to refrain - 1911a Ann. Phys., Lpz 34 170 from the necessary generalisations. - 1911b Ann. Phys., Lpz 35 679 By contrast, Born and von Karman (1912) and Lanczos C 1974 The Einstein Decade (London: Paul Debye (1912) were much bolder. The formula put Elek) forward by Einstein one year earlier, which yields Nernst W and Lindemann F 1911 Sitzungsber. Preuss. U(T)and C,(T) starting with g(w), is used without Akad. Wiss. 22 494 undue scruples. The Fourier analysis of lattice vib- Wannier G H 1966 Statistical Physics (New York: rations is developed along lines that have remained Wiley) p 278 standard in textbooks since that time; it produces a spectrum thatBorn and von Karman keep in its original form, while Debye simplifies it to obtain finally the conventional ‘Debye model’, thus mak- ing the decisive final step. All these authors make no mention of the electroniccontribution tothe specific heat. It is worth noting that Debye notices the analogy between the C, a T3behaviour at low temperatures and Stefan’s law for black body radiation. But he fits his theoretical calculations tothe ‘Nernst- Lindemann’formula, and his majorargument is that the fit he gets is much better than with Ein- stein’s model: the relative deviation does not exceed 15% down to T=@/l0 (while such adeviation alreadyappears for T=012 in Einstein’s treat- ment). As compared with Einstein’s cautious ponder- ings, Debye thus rushes towards the solution of the specific heat dilemma, deliberately ignoring the, not yet ripe, problem of the electroniccontribution, and accepting withoutquestion the legitimacy of the generalisation of Planck’s black body treatment that hadbeen tentatively proposed by Einstein himself. Is it this undue lack of regard to a funda- mental problem that motivated the extremely harsh welcome Nernst gave to Debye’s theory (Casimir 1977)? Whatever the answer, it is clear thatthe specific heat question remained, until its end, a fine example of the complexity of scientific progress, and would seem to merit closer historical and epis- temological studies. A final remark: the C, problem ceased to be a major component of physicists’ speculations as