(Conference at the French Society of Physics, April 11, 1912.)
When Mr. Abraham came to ask me to close the series of lectures organized by the French Society of Physics, I was at first about to refuse; it seemed to me that every subject had been entirely treated and that I could add nothing to what had been so well said. I could only try to summarize the impression that seems to emerge from this set of works, and this impression is so clear that each of you must have experienced it just as well as I, and that I can not give it any new clarity. by trying to express it in sentences. But Mr. Abraham insisted with so much grace that I ended up resigning myself to the inevitable inconveniences, the greatest of which is to repeat what each of you has long thought, the least of which is to cross a host of various subjects without having time to stop.
A first reflection must have struck all the listeners; the old mechanistic and atomistic hypotheses have in recent times taken enough consistency to almost cease to appear to us as hypotheses; atoms are no longer a convenient fiction; it seems to us, as it were, that we see them, since we know how to count them. An hypothesis takes on weight and becomes more plausible when it explains new facts; but it happens in many ways; most often it must be expanded to account for new facts; but sometimes it loses in precision widening, sometimes it is necessary to graft on it an accessory hypothesis which adapts to it in a plausible way, which does not cuss too much with the rootstock, but which is not no less something foreign, imagined expressly in view of the goal to be attained, which is in a word a kind of help; in this case it can not be said that experience has confirmed the primitive hypothesis, but at the very most that it did not contradict it. Or again, there is between the new facts and the old facts, for which the hypothesis was originally conceived, an intimate connexion of such a nature that any hypothesis which accounts for one must, by that very fact, account for the others, so that the verified facts are new only in appearance.
It is no longer the case when experience reveals a coincidence that could have been foreseen and that can not be due to chance and especially when it comes to a numerical coincidence. Now, coincidences of this kind have lately come to confirm atomistic ideas.
The kinetic theory of gases has received, so to speak, unexpected stresses. Newcomers have exactly copied it; these are on the one hand the theory of solutions, and on the other hand the electronic theory of metals. The molecules of the dissolved bodies, as well as the free electrons to which the metals owe their electrical conductivity, behave like the gaseous molecules in the enclosures, where they are enclosed. Parallelism is perfect and can be pursued to numerical coincidences. By this, what was doubtful becomes probable; each of these three theories, if it were isolated, would only appear to us as an ingenious hypothesis, to which we could substitute other explanations about as probable; but, as in each of the three cases, a different explanation would be necessary, the coincidences observed could no longer be attributed to chance alone, which is inadmissible, while the three kinetic theories make these coincidences necessary. And then the theory of solutions takes us quite naturally to that of Brownian motion where it is impossible to look at the thermal agitation as a fiction of the mind, since it is seen directly under the microscope.
The brilliant determinations of the number of atoms made by Mr. Perrin completed this triumph of atomism. What leads us to believe are the many matches between results obtained by entirely different processes. Not long ago, one would have thought oneself happy provided that the numbers found had the same number of digits; one would not even have demanded that the first significant figure be the same; this first figure is now acquired; and what is remarkable is that we have addressed the most diverse properties of the atom. In processes deriving from Brownian motion, or in those in which the law of radiation is invoked, it is not the atoms which have been counted directly, but the degrees of freedom; in the one where the blue of the sky is used, it is no longer the mechanical properties of the atoms which come into play, they are regarded as causes of optical discontinuity; finally, when we use radium, what we count is the emission of projectiles. So much so that if there had been any discrepancies, we would not have been embarrassed to explain them, but fortunately there was none.
The atom of the chemist is now a reality; but that does not mean that we are close to touching the ultimate elements of things. When Democritus invented atoms, he considered them as absolutely indivisible elements and beyond which there is nothing to look for. That’s what it means in Greek; and that is why he invented them; behind the atom, he did not want any more mystery. The atom of the chemist would not have given him satisfaction, for this atom is by no means indivisible, it is not a true element, it is not without mystery; this atom is a world. Democritus would have thought that after we had gone to such lengths to find him, we are no more advanced than at the beginning; these philosophers are never happy.
Because, and this is the second reflection that imposes on us, each new discovery of physics reveals a new complication of the atom. And first of all, bodies that were thought to be simple, and which, in many ways, behave quite like simple bodies, are capable of decomposing themselves into even simpler bodies. The atom breaks up into smaller atoms. What is called radioactivity is only a perpetual decay of the atom. This is what has sometimes been called the transmutation of elements, which is not altogether exact, since one element does not transform into reality into another, but decomposes into several others. The products of this decomposition are still chemical atoms, analogous in many respects to the complex atoms which gave rise to them by decay, so that the phenomenon could be expressed as the most banal reactions, by a chemical equation, susceptible to be accepted without much suffering by the most conservative chemist.
This is not all, in the atom we find many other things: first we find electrons; each atom then appears to us as a kind of solar system, where small negative electrons acting as planets revolve around a large positive electron that plays the role of the central sun. It is the mutual attraction of these electricities of contrary name which maintains the cohesion of the system and which makes it a whole; it governs the periods of the planets, and it is these periods which determine the wavelength of the light emitted by the atom; it is to the self-induction of the currents of convection produced by the movements of these electrons that the atom which is formed owes its apparent inertia and which we call its mass. Besides these captive electrons, there are free electrons, those which obey the same kinetic laws as the gaseous molecules, those which render the metals conductive. These are comparable to comets that circulate from one star system to another and establish between these distant systems as a free exchange of energy.
But we are not at the end: after the electrons or atoms of electricity, here come the magnetons or atoms of magnetism which arrive today by two different ways, by the study of the magnetic bodies and by the study of the spectrum of simple bodies. I do not have to remind you here of Mr. Weiss’s fine lecture and the astonishing commensurability reports that these experiments have so unexpectedly brought to light. Here too, there are numerical relationships that can not be attributed to chance and whose explanation must be sought.
At the same time it is necessary to explain the curious laws of the distribution of the lines in the spectrum. According to the works of Balmer, Runge, Kaiser, and Rydberg, these lines are divided into series, and in each series obey simple laws. The first thought is to bring these laws closer to those of harmonics. Just as a vibrating string has an infinity of degrees of freedom, which allows it to give an infinity of sounds whose frequencies are multiples of the fundamental frequency; just as a sound body of complex form also gives harmonics, whose laws are analogous, though much less simple, just as a Hertz resonator is capable of an infinite number of different periods, the atom can it to give, for identical reasons, an infinity of different lights? You know that this simple idea has gone bankrupt, because, according to the spectroscopic laws, it is the frequency and not its square whose expression is simple; because the frequency does not become infinite for harmonics of infinitely high rank. The idea must be changed or it must be abandoned. So far she has resisted all attempts, she refused to adapt; this led Mr. Ritz to abandon it. He then represents the vibrating atom as consisting of a rotating electron and several magnets placed end to end. It is no longer the mutual electrostatic attraction of the electrons that regulates the wavelengths, it is the magnetic field created by these magnets.
It is difficult to accept this conception, which has something artificial about it; but we must resign ourselves to it, at least temporarily, since hitherto we have found nothing else, and yet we have sought. Why can hydrogen atoms give more than one line? It is not because each of them could give all the lines of the hydrogen spectrum, and that it actually gives one or the other according to the initial circumstances of the movement; it is because there are several species of hydrogen atoms, differing from each other by the number of magnetons which are aligned therein, and that each of these species of atoms gives a different line; one wonders if these different atoms can be transformed into each other and how. How can an atom lose magnetons (and that seems to happen when we go from one allotropic variety of iron to another)? Can the magneton get out of the atom or can a part of the magnetons leave the alignment to dispose of it irregularly?
This disposition of the magnetons end to end is also a singular feature of Ritz’s hypothesis; Weiss’s ideas, however, must make it seem less strange to us. It is necessary that the magnetons be arranged otherwise end to end, at least parallel, since they are added arithmetically or at least algebraically, and not geometrically.
What is a magneton now? Is it something simple? No, if we do not want to give up the hypothesis of Ampere’s particle currents; a magneton is then a whirlwind of electrons and this is our atom that gets more and more complicated.
However, what better than anything else makes us measure the complexity of the atom is the reflection made by Mr. Debierne at the end of his lecture. This is to explain the law of radioactive transformation; this law is very simple, it is exponential; but if one reflects on its form, one sees that it is a statistical law; we recognize the mark of chance. But chance is not due here to the chance encounter of other atoms and other external agents. It is within the atom itself that the causes of its transformation are found, I mean the occasional cause as well as the root cause. Otherwise we would see the external circumstances, temperature for instance, influence the coefficient of time in the exponent; or this coefficient is remarkably constant, and Curie proposes to use it for the measurement of absolute time.
The chance that presides over these transformations is therefore an internal hazard; that is to say that the atom of the radioactive body is a world, and a world subject to chance; but to beware of it, which says chance, says large numbers; a world formed of few elements will obey laws more or less complicated, but which will not be statistical laws. It is therefore necessary that the atom be a complex world; it is true that it is a closed world (or at least almost closed), it is sheltered from external disturbances that we can provoke; since there is a statistic and consequently an internal thermodynamics of the atom, we can speak of the internal temperature of this atom; well ! it has no tendency to equilibrate with the outside temperature, as if the atom were enclosed in a perfectly adiathermal envelope. And it is precisely because it is closed, because its functions are clearly traced, guarded by strict customs officers, that the atom is an individual.
At first sight, this complexity of the atom is not offensive to the mind; it seems that it should not cause us any embarrassment. But a little thought is not slow to show us the difficulties that escaped us first. What we counted, counting the atoms, are the degrees of freedom; we have implicitly assumed that each atom has only three; this is what accounts for the specific heats observed; but every new complication should introduce a new degree of freedom, and then we are far from it. This difficulty has not escaped the creators of the theory of the equipartition of energy; they were already astonished by the number of lines in the spectrum; but, finding no way out, they had the boldness to override.
What seems to be the natural explanation is precisely that the atom is a complex world, but a closed world; external disturbances have no repercussions on what is going on inside, and what goes on inside does not act on the outside; that could not be quite true, otherwise we would always ignore what is happening within, and the atom would appear to us as a mere material point; what is true is that we can see the inside only through a very small window, that there is practically no exchange of energies between the outside and the inside and therefore not of tendency to equipartition of energy between the two worlds. The internal temperature, as I said earlier, does not tend to equilibrate with the outside temperature, and that is why the specific heat is the same as if all this internal complexity did not exist. Suppose a complex body formed of a hollow sphere whose inner wall would be absolutely impervious to heat, and inside a crowd of diverse bodies; the specific heat observed of this complex body will be that of the sphere, as if all the bodies which are enclosed in it did not exist.
The door that closes the inner world of the atom, however, opens from time to time; this happens when, by the emission of a particle of helium, the atom degrades and descends a rank in the radioactive hierarchy. What happens then? How does this decomposition differ from ordinary chemical decompositions? How does the uranium atom, formed of helium and other things, have more titers in the name of atom than the cyanogen half-molecule, for example, which behaves in so many ways like that of a simple body, and which is formed of carbon and nitrogen? It is doubtless that the atomic heat of uranium would obey (I do not know if it was measured) the law of Dulong and Petit and that it would be that of a simple atom; it should then double at the moment of emission of the helium particle and when the primordial atom decomposes into two secondary atoms. By this decomposition, the atom would acquire new degrees of freedom capable of acting on the outside world, and these new degrees of freedom would result in an increase in specific heat. What would be the consequence of this difference between the total specific heat of the components and that of the compounds? This is because the heat given off by this decomposition should vary rapidly with the temperature; so that the formation of radioactive molecules, very strongly endothermic at ordinary temperature, would become exothermic at high temperature. It would be easier to explain how the radioactive compounds were formed, which did not stop being a little mysterious.
Be that as it may, this conception of these small closed worlds, or only half opened, is not enough to solve the problem. The equipartition of energy should reign without dispute outside these closed worlds, except at the moment when one of the doors should open, and that is not what happens.
The specific heat of the solid bodies diminishes rapidly when the temperature is lowered, as if some of their degrees of freedom successively fit into each other, freeze themselves, so to speak, or, if you like better, lose all contact with the outside and retired in their turn, behind a certain enclosure, in some closed world.
On the other hand, the law of black radiation is not that which would be required by the theory of equipartition.
The law that would adapt to this theory would be that of Rayleigh, and this law, which would, moreover, imply a contradiction, since it would lead to an infinite total radiation, is absolutely contradicted by experience. There is in the emission of black bodies much less light at short wavelength than the hypothesis of equipartition would require.
That’s why Mr. Planck imagined his theory of Quanta, according to which the exchanges of energy between ordinary matter and the small resonators whose vibrations generate the light of the incandescent bodies, could be done only by abrupt jumps; one of these resonators could not acquire energy or lose it in a continuous way; he could not acquire a fraction of quantum, he would acquire an entire quantum or nothing at all.
Why then does the specific heat of a solid diminish at a low temperature, why do some of its degrees of freedom seem not to play? This is because the supply of energy that is offered to them at low temperature is not sufficient to provide a quantum to each; some of them would only be entitled to a fraction of a quantum; but, as they want all or nothing, they have nothing and remain as stiff.
Likewise in the radiation, certain resonators, which can not have the whole quantum, have nothing and remain motionless; so that there is much less light radiated at low temperature than there would be without this circumstance; and since the required quantum is all the greater as the wavelength is smaller, it is especially the short-wavelength resonators which remain silent, so that the proportion of short-wavelength light is much smaller than would be required by Rayleigh’s law.
To declare that such a theory raises many difficulties would be a great naivety; when one emits such a bold idea, one expects to encounter difficulties, one knows that one overturns all the received opinions and one is not surprised by any obstacle any more, one would be astonished on the contrary of none not find in front of you. These difficulties do not seem valid objections.
I will, however, have the courage to point out some of them, and I will not choose the larger ones, the more obvious ones, the ones that are presented to all minds, and indeed it is useless, since everyone thinks of first attempt ; I want to tell you simply by what series of states of souls I have successively passed.
I first wondered what was the value of the proposed demonstrations; I saw that the probability of the various distributions of energy was evaluated by simply listing them, since, thanks to the hypothesis made, they were in finite numbers, but I did not see why they were regarded as equally likely. Then we introduced the known relations between temperature, entropy and probability; this implied the possibility of thermodynamic equilibrium, since these relations are demonstrated by supposing this possible equilibrium. I know that experience teaches us that this equilibrium is realizable, since it is realized; but that was not enough for me, it was necessary to show that this equilibrium is compatible with the hypothesis made and even that it is a necessary consequence of it. I did not have exactly doubts, but I felt the need to see a little more clearly, and for that it was necessary to penetrate a little in the detail of the mechanism.
For there to be a distribution of energy between the resonators of different wavelength whose oscillations are the cause of the radiation, they must be able to exchange their energy; without this, the initial distribution would subsist indefinitely, and since this initial distribution is arbitrary, there can be no question of a law of radiation. Now a resonator can not yield to the ether, and it can only receive light of a perfectly determined wavelength. If, then, the resonators could not react on each other mechanically, that is, without the intermediary of the ether; if, on the other hand, they were fixed and enclosed in a fixed enclosure, each of them could emit or absorb only light of a certain color, it could therefore exchange energy only with the resonators with which it would be in perfect resonance, and the initial distribution would remain unalterable. But we can conceive of two modes of exchange that do not lend themselves to this objection. On the one hand, atoms and free electrons can flow from one resonator to another, collide a resonator, communicate to it and receive energy. On the other hand, the light, reflecting on mobile mirrors, changes its wavelength by virtue of the Döppler-Fizeau principle.
Are we free to choose between these two mechanisms? No, it is certain that both must come into play, and it is necessary that both lead us to the same result, to the same law of radiation. What would happen if the results were contradictory, if the mechanism of collision acting alone tended to achieve a certain law of radiation, that of Planck for example, while the mechanism of Döppler-Fizeau would tend to achieve another? Well ! it would happen that, since these two mechanisms must play one another, but become alternatively preponderant under the influence of fortuitous circumstances, the world would oscillate constantly from one law to another, it would not tend towards an end stable state, towards this thermal death where he will no longer experience change; the second principle of thermodynamics would not be true.
I resolved to examine the two processes successively, and I began with the mechanical action, the collision. You know why ancient theories necessarily lead us to the law of equipartition; it is because they suppose that all the equations of mechanics are of Hamilton’s form and that consequently they admit unity as a last multiplier in the sense of Jacobi. We must then suppose that the laws of the collision between a free electron and a resonator are not of the same form and that the equations governing them admit a last multiplier other than unity. They must have a last multiplier, otherwise the second principle of thermodynamics would not be true, we would find the difficulty of earlier, but this multiplier must not be unity.
It is precisely this last multiplier that measures the probability of a given state of the system (or rather what might be called the density of probability). In the hypothesis of quanta, this multiplier can not be a continuous function, since the probability of a state must be zero, whenever the corresponding energy is not a multiple of the quantum. This is an obvious difficulty, but it is one of those to which we are resigned in advance; I did not stop there; I then pushed the calculation to the end and I found Planck’s law, fully justifying the views of the German physicist.
I then moved on to the Döppler-Fizeau mechanism; suppose an enclosure formed of a pump body and a piston, whose walls are perfectly reflective. In this enclosure is enclosed a certain amount of light energy with any distribution of wavelengths, but no light source; the luminous energy is locked up once and for all.
As long as the piston does not move, this distribution can not vary, because the light will retain its wavelength by reflecting; but when you move the piston, the distribution will vary. If the piston speed is very small, the phenomenon is reversible and the entropy must remain constant; We thus find the analysis of Wien and the law of Wien, but we are no more advanced, since this law is common to old and new theories. If the speed of the piston is not very small, the phenomenon becomes irreversible; so that thermodynamic analysis no longer leads us to equalities, but to simple inequalities from which conclusions can not be drawn.
It seems however that one could reason as follows: suppose that the initial distribution of energy is that of black radiation, it is obviously that which corresponds to the maximum of entropy; if we give some shots of piston, the final distribution will have to remain the same, otherwise the entropy would have diminished; and even whatever the initial distribution, after a very large number of piston shots, the final distribution must be the one that makes the maximum entropy, that of the black radiation. This reasoning would be worthless.
Distribution has a tendency to approach that of black radiation; it can not deviate from it any more than the heat can pass from a cold body to a warm body, that is to say, it can not do it without a counterpart. But here there is a counter-part: by giving shots to the piston, we spend work, which is found by an increase in light energy locked in the pump body, that is to say, which is transformed in heat.
The same difficulty would not be found again if the bodies in motion on which the reflection of the light is made were infinitely small and infinitely numerous, because then their living force would not be mechanical work, but heat; it would therefore be impossible to compensate for the decrease in entropy which corresponds to a change in the distribution of wavelengths by the transformation of this work into heat, and then it will be right to conclude that, if the initial distribution is that of the black radiation, this distribution will have to persist indefinitely.
Suppose, then, an enclosure with fixed and reflecting walls; we will enclose not only light energy, but also a gas; it is the molecules of this gas that will act as mobile mirrors. If the wavelength distribution is that of the black radiation corresponding to the temperature of the gas, this state must be stable, that is to say:
- That the action of light on the molecules should not cause the temperature to vary;
- That the action of molecules on the light should not disturb the distribution.
Mr. Einstein has studied the action of light on molecules; these molecules undergo, indeed, something which resembles the pressure of radiation; Mr. Einstein, however, did not quite put himself in such a simple point of view; he has assimilated his molecules to small mobile resonators, capable of possessing both the translational force and the energy due to electric oscillations. The result would in all cases have been the same, he would have found Rayleigh’s law.
As for me, I will do the opposite, that is to say that I will study the action of molecules on the light. The molecules are too small to give regular reflection; they only produce a diffusion. What is this diffusion, when we do not take into account the motions of molecules, we know it, and by theory and experience; it is she who produces the blue of the sky.
This diffusion does not alter the wavelength, but it is all the more intense as the wavelength is smaller.
It is now necessary to pass from the action of a molecule at rest to the action of a molecule in motion, in order to take into account the thermal agitation; it is easy, we have only to apply the principle of relativity of Lorentz; it follows that various beams of the same real wavelength, arriving on the molecule in different directions, will not have the same apparent wavelength for an observer who would believe the molecule at rest. The apparent wavelength is not altered by diffraction, but this is not the case with the actual wavelength.
We thus arrive at an interesting law; the reflected or scattered light energy is not equal to the incident light energy; it is not energy, it is the product of energy by the wavelength which remains unchanged. I was very happy at first. The result was that an incident quantum gave a scattered quantum, since the quantum is in inverse proportion to the wavelength. Unfortunately it did not give anything.
I was led by this analysis to Rayleigh’s law; that I knew well in advance; but I hoped that by seeing how I would be led to Rayleigh’s law, I would see more clearly what modifications must be made to the hypotheses in order to find Planck’s law. It is this hope that has been disappointed.
My first thought was to look for something that resembled the theory of quanta; it would indeed be surprising that two entirely different explanations give account of the same derogation from the equipartition law, according to the mechanism by which this derogation would occur. Now, how can the discontinuous structure of energy intervene? It might be supposed that this discontinuity belongs to the luminous energy itself, when it circulates in the free ether, that consequently the light does not fall on the molecules in compact mass, but in small separate battalions; it is easy to see that it would not change the result.
Or one could suppose that the discontinuity occurs at the moment of diffusion itself, that the diffusing molecule does not transform the light in a continuous way, but by successive quanta; It does not go well yet, because if the light to be transformed were to go to an antechamber, as if we were dealing with an omnibus waiting to be full to leave, it would inevitably result in a delay. Now Lord Rayleigh’s theory teaches us that diffusion by molecules, when done without deviation in the direction of the incident ray, simply produces ordinary refraction; that is, the scattered light interferes regularly with the incident light, which would not be possible if there was a loss of phase.
If we seek without bias which of our premises it is convenient for us to abandon, we shall not be less embarrassed: we do not see how we could renounce the principle of relativity; Is it then the law of diffusion by the molecules at rest that should be modified? that is very difficult; we can scarcely push fantasy to believe that the sky is not blue.
I will stay on this embarrassment, and I will end with the following reflection. As science progresses, it becomes more and more difficult to make room for something new that does not fit naturally. Ancient theories are based on a large number of numerical coincidences that can not be attributed to chance; we can not, therefore, disjoin what they have gathered together; we can no longer break the frames, we must try to bend them; and they do not always lend themselves to it. The theory of equipartition explained so many facts that it must contain some truth; on the other hand, it is not entirely true, since it does not explain them all. We can neither abandon it nor preserve it without modification, and the modifications that seem to be necessary are so strange that we hesitate to resign ourselves to it. In the present state of science, we can only see these difficulties without solving them.