# Wilhelm Wien

## Nobel Lecture

Nobel Lecture, December 11, 1911

## On the Laws of Thermal Radiation

The kind recognition which my work on thermal radiation has received in the views of your ancient and famous Academy of Sciences gives me particular pleasure to speak to you about this subject which is again attracting the attention of all physicists because of the difficulty of the problems involved. As soon as we step beyond the established boundaries of pure thermodynamic theory, we enter a trackless region confronting us with obstacles which even the most astute of us are almost at a loss to tackle.

The Kirchhoff theorem is not limited to radiation caused by thermal processes. It seems to be valid for most, if not all luminous processes. That the temperature concept can be applied to all luminous processes is beyond doubt. Since we can produce all types of light by means of hot bodies, we can ascribe, to the radiation in thermal equilibrium with hot bodies, the temperature of these bodies, and thus every radiation, even that issuing from a phosphorescent body, has a certain temperature for every colour. This temperature has however no connection whatever with that of the body, nor is it possible as yet to state how e.g. a phosphorescent body comes into equilibrium with radiation. These conditions are bound to be very complicated, in particular in the case of bodies which convert the absorbed radiation and emit it after a long interval of time.

Again using ideal processes and assuming radiation pressure, which at that time had been deduced from the electromagnetic theory of light, Boltzmann derived from thermodynamics the law, previously empirically formulated by Stefan, that the radiation of a black body is proportional to the fourth power of the absolute temperature.

Using this displacement law it is easy to calculate the distribution of the intensity of thermal radiation over the various wavelengths for any temperature, as soon as it is known for one temperature.

The shift of the maximum of intensity in particular is directly accessible to observation. Since the wavelength at which the maximum intensity lies also defines the principal area of the wavelength which is most intense at this temperature, we can, by changing the temperature, shift the principal area of radiation in the direction of short or long wavelengths of any desired magnitude. Of the other derivations of the displacement law, I shall only mention that by H.A. Lorentz. If, in the electromagnetic equations of Maxwell, we imagine all spatial dimensions as being displaced in time in the same ratio, these equations show that the electromagnetic energy must decrease in proportion to the fourth power of displacement. Since, according to the Stefan-Boltzmann law, energy increases with the fourth power of absolute temperature, the linear dimensions must vary inversely proportionately to the absolute temperature. Each characteristic length must vary in this ratio, from which the displacement law follows.

From the displacement law, we can calculate the temperature of the sun if we are entitled to assume that the radiation of the sun must be ascribed to heat, and if we know the position of the maximum of the energy of solar radiation. Different figures are given for this position by different observers, i.e. 0.532 m according to Very, and 0.433 m according to Abbot and Fowle. Depending on the figure used, the temperature of the sun works out at 5,530° and 6,790°. However much the observers may differ, there can be no doubt that the maximum of solar radiation is situated in the visible range of wavelengths. This is to say that the temperature of the sun is the most favourable utilization of the radiant energy of a black body for our illumination and that, in our artificial light sources which utilize thermal radiation we must aim at achieving this temperature, from which we are admittedly far removed as yet.

I wish to discuss yet another application of the displacement law, i.e. the possibility of calculating the wavelength of X-rays. As we know, X-rays are produced by the impact of electrons on solid bodies, and their wavelength must be a function of the velocity of the electrons. According to the kinetic theory of gases, the mean kinetic energy of a molecule is a measure of absolute temperature. If, as is done in the theory of electrons, we assume that this is also valid for the kinetic energy of the electrons, the electric energy of the cathode rays would be a measure of their temperature. If we substitute the temperature thus calculated in the displacement law, we find that the wavelength of the maximum of the intensity indicates a wavelength range of X-rays which agrees well with the wavelengths found by other arguments. It might be objected that we must not ascribe a temperature to the electrons. The permissibility of our procedure can however be justified by an inversion of the above argument. Radiation in an enclosed space must necessarily release electrons whose velocity according to Einstein’s law is proportional to the oscillation frequency. The energy maximum of radiation generates electrons whose velocity is so great that their kinetic energy comes very close to the temperature associated with the maximum of energy.

I myself made the first attempt in this direction. I endeavoured to bypass the problem of applying probability calculation to radiation theory by imagining radiation as resulting from gas molecules moving according to laws of probability. Instead of these we could also imagine electrons generating radiation on striking molecules. What is essential is the further assumption that such a particle will only emit radiation of a certain wavelength dictated by velocity, and that the velocity distribution of the particles obeys Maxwell’s law. With the assistance of the radiation laws derived from thermodynamics we obtain a radiation law which shows good agreement with experience for a wide range of wavelengths, i.e. for the range in which the product of temperature and wavelength is not unduly large.

Imperfect as this first attempt was, a formula had been obtained which considerably deviates from reality for large wavelengths only. Since observations however establish these deviations beyond doubt, it was clear that the formula had to be modified.

Lord Rayleigh was the first to approach the problem from an entirely different angle. He made the attempt to apply to the radiation problem a very general theorem of statistical mechanics, i.e. the theorem of the uniform distribution of energy over the degrees of freedom of the system in the state of statistical equilibrium. The meaning of this theorem is as follows:

In the state of thermal equilibrium, all movements of the molecules are so completely irregular that there exists no movement which would be preferred over any other. The position of the moving parts can be established by geometrical parameters which are mutually independent and in the direction of which the movement falls. These are called the degrees of freedom of the system. As regards the kinetic energy of movement, no degree of freedom is preferred over another, so that each contains the same amount of the total energy.

Radiation present in an empty space can be represented so that a given number of degrees of freedom is allocated to it. If the waves are reflected back and forth by the walls, systems of standing waves are established which adapt themselves to the distances between two walls. This is most easily understood if we consider a vibrating string which can execute an arbitrary number of individual vibrations, but whose half wavelengths must be equal to the length of the string divided by an integer.

There are however further difficulties. The energy elements increase with decreasing wavelength, and an oscillator exposed to incident radiation will, at low intensity, need a very long time before it absorbs a full energy element. What happens if the incident radiation ceases, before an entire energy element has been absorbed? The difficulties implicit in answering this question have recently induced Planck to reformulate his original theory. He now assumes of emission only that it can occur exclusively by whole energy elements. Absorption is assumed to occur continuously according to the electromagnetic laws, and the energy content of an oscillator is assumed to have energy values capable of continuous change. The difficulty of the long absorption time is indeed avoided in this manner. On the other hand the close relationship between emission and absorption for the elementary process is relinquished, and this relationship now becomes valid statistically only. Every atom which only emits whole energy elements and absorbs continuously would therefore in the event of emission suddenly expend energy from its own reserve and would supplement this but little in the event of short irradiation. The special hypothesis must be made that, taken as a whole for many atoms in the stationary state, the absorbed energy after all becomes equal to that emitted. Whereas in the original form of the Planck theory the introduction of the hypothesis of energy elements was sufficient to permit the radiation laws to be derived, the new theories include uncertainties which can only be removed by further hypotheses. On the other hand the new fundamental hypothesis provides the possibility of further application, e.g. to electron emission.

It will be seen from the few observations I am able to offer in this context how great are the difficulties that remain in radiation theory. But the reference to these difficulties which it is the duty of the scientific approach to emphasize must not prevent us from paying tribute to the great positive achievements which the Planck theory has already accomplished.

It has produced a law of radiation which accommodates all observed data and includes the Rayleigh formula and my own formula as limiting cases. In addition, it has thrown unexpected light on an entirely different subject, i.e. the theory of specific heats.

It has long been known that the specific heats do not strictly obey the Dulong-Petit law and that they decrease at low temperatures. Diamonds do not obey the Dulong-Petit law even at normal temperatures. This law can be derived from the theorem of the distribution of kinetic energy over the degrees of freedom and states that, in solid bodies, every atom possesses, in accordance with its three degrees of freedom, three times the amount of energy, and, because of the potential energy, altogether six times the amount of energy of a degree of freedom. If however we apply Planck’s distribution of energy by energy elements, we obtain, according to Einstein, a formula for the specific heat which does in fact show the drop of temperature. This result is characteristic of the Planck theory. This theory of specific heats is not derived from the radiation formula, but from the formula for the mean energy of an oscillator which is based directly on the hypothesis of the energy elements. Unfortunately, difficulties are beginning to appear. The exact measurements of specific heats at low temperatures made in Nernst’s laboratory have shown that the Einstein formula does not agree with observation. The formula which satisfies the experimental data contains half energy elements in addition to the whole energy elements, which cannot be interpreted satisfactorily as yet. There can however be no doubt that the Planck radiation theory provides the first step to the theory of specific heats.

That the theory should remain in many respects incomplete and provisional, is in the nature of the problem which is perhaps the most difficult which has ever confronted theoretical physics. What is involved is to leave behind the laws of theoretical physics confirmed by direct observation, which alone have been applied in the past, and to enter areas which are beyond the reach of direct observation.

The difficulties which beset radiation theory also emerge in an entirely different approach. Einstein investigated the fluctuations to which radiation is continuously subjected even in the state of equilibrium as a result of the irregularities of the thermal processes. If we imagine a small plate in a cavity filled with radiation, this plate will be subjected to a radiation pressure which is the same on average on both sides of the plate. Since the radiation must contain irregularities, the pressure will alternately be greater on one or the other side so that the plate will execute small irregular movements, similar to the Brownian movement of a dust particle suspended in a liquid. These fluctuations can be derived from probability calculations. According to the Boltzmann theorem there is a simple relationship between entropy and probability. The entropy of radiation is known from the radiation law, so that the probability of state is also known, from which the fluctuations can be calculated. The mathematical expression for these fluctuations consists, in a peculiar manner, of two members. The first is readily understandable: it is due to irregularities which arise as a result of the mutual interference of the many independent beams which meet in one point. Where the density of radiation energy is high, this term alone predominates; it corresponds to the radiation range that obeys Rayleigh’s law.

The other term, which cannot be directly explained by the undulation theory, predominates at low density of radiation energy, where the radiation obeys the law formulated by me. It would be understandable if the radiation consisted of the Planck energy elements which would be localized even in an empty space. We cannot however pursue this line of thought. We cannot shake the undulation theory of light, which is one of the most firmly established constructions in the whole of physics. Moreover, the term to be explained by localized energy elements, is not present by itself, and it is a priori impossible to introduce a dualistic approach into optics, e.g. to assume simultaneously Huyghens’ wave theory and Newton’s emanation theory. All we can do is to relinquish the Boltzmann method of applying probability calculations to this type of fluctuations, or to assume that a new irregularity is introduced into radiation with the process of reflection.

In view of the magnitude of the difficulties it is natural that opinions about the path to be pursued should differ greatly. Some are of the opinion that the fundamental principles of electrodynamics must be changed. And yet, previous theory embraces a vast range of facts, it accounts for events even in the most rapid movements of the b-rays, it has proved itself in the most precise optical measurements. In my view, all the signs suggest that the deviations from current theory are due to events within the atom. None of the processes in which the interior of the atom participates are amenable to current theory.

Sommerfeld made an attempt in this direction: He would ascribe to the constant h of the radiation law, which together with the oscillation frequency dictates the magnitude of the energy element, a simple significance for the interior of the atom. It is alleged to determine the period in which an electron entering the atom comes to a stop, as a function of its velocity. On this view, the constant h expresses a universal characteristic of the atoms. This theory permits calculation of the wavelength of X-rays, and it connects two previously independent attempts made by me to carry out this calculation. One method is based on the Planck theory of energy elements in that it assumes that the energy of the so-called secondary electrons released by Xrays is dictated by the energy element. The second method is based on the theory of electrons by means of which it calculates the energy radiated in the X-rays by sudden braking of an electron. From the determination of the energy of the cathode rays and the X-rays we can then calculate the brake path of the electrons and consequently the wavelength of the X-rays. The Sommerfeld theory connects these two theories. It has the great advantage of explaining the generation of X-rays with the aid of electromagnetic theory. A number of conclusions can then be drawn from this which are in complete agreement with observation, e.g. the polarisation of X-rays, the diversity of emission and of hardness in various directions.

The Sommerfeld theory has the great advantage that it attempts to invest the universal constant h of the Planck radiation theory with physical significance. It has the disadvantage that it has been applied so far only to electron emission and absorption, but cannot yet solve the problem of thermal radiation.

We must admit that the result of radiation theory todate is not a very good one for theoretical physics. As we have seen, only the general thermodynami theories have proved satisfactory as yet. The theory of electrons has come to grief over the radiation problem, the Planck theory has not yet been brought into a definite form. Research is faced with exceptional difficulties and we cannot discern when and how they can be overcome. In science, the redeeming idea often comes from an entirely different direction, investigations in an entirely different field often throw unexpected light on the dark aspects of unresolved problems. We must base our hope in the future in the expectation that the present era which has proved so fruitful for physics may not pass without a complete solution being found for the problem of thermal radiation. Far-reaching and new thoughts will have to set to work, but the result will be great, because we shall obtain a profound insight into the world of the atom and the elementary processes within it.

From Nobel Lectures, Physics 1901-1921, Elsevier Publishing Company, Amsterdam, 1967

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