vol II chap 5 sect 3
Previous: 5.2. The photon as a quantum of energy.
5.3. Components of the explanation of scientific theories.¶
In this section we apply the components of the explanation of scientific theories to describe the content of the Nobel Lectures by Wien and by Planck (see Figure 5.7).
Figure 5.7. Components of the explanation of scientific theories.
In the following, for explanatory purposes, we arbitrarily divide the content of the Nobel lectures by Wien and by Planck into the previous six components of the explanation of a scientific theory. Next, the main ideas contained in each component are described. The full contents of both Lectures are included in Appendices 5.1 and 5.2.
Wien´s Nobel Lecture On the Laws of Thermal Radiation.¶
Description of relevant facts.¶
• Using known physical laws, a general law of radiation theory (the displacement law) was derived by considering ideal processes as in a mental experiment. This procedure is like the one followed by Kirchhoff: he assumed an ideal perfectly reflecting body in his derivation of the theorem about the constancy of the ratio of emission and absorption power.
Analysis of critical concepts.¶
• The Kirchhoff theorem is not limited to radiation caused by thermal processes. Furthermore, when a radiation is in thermal equilibrium with a hot body, every radiation has a certain temperature for every color. According to Stefan-Boltzmann Law, that has been derived from thermodynamics, the radiation of a black body is proportional to the fourth power of the absolute temperature.
Proposition of significant relationships.¶
• The colors (frequencies) of a black body radiation change with temperature. Wien based his calculation on an ideal process: the black body was a cylindric oven at a certain temperature, containing in one end a movable piston and in the other end a perfect reflecting wall with a very small hole for the entrance of incident radiation. If an external electromagnetic radiation exercises a pressure on the walls of the piston, it produces a work that moves the piston at a certain velocity. By Doppler effect, the frequency of the radiation reflected by the movable piston is different from the incident radiation already contained inside the cylinder. Afterwards, the change with temperature undergone by the spectral composition of the total black-body radiation in the cavity can be calculated. From this result the displacement law is derived: there is a maximum value of the black body radiation curve for each temperature such that the peak frequency is directly proportional to temperature. This is the same as saying that the peak wavelength is inversely proportional to temperature, meaning that the value of this peak is displaced towards shorter wavelengths.
Solution of challenging questions.¶
• According to Lorentz, if in Maxwell electromagnetic equations all spatial dimensions are displaced in time in the same ratio as the changes in the wavelength due to changes in temperature, the electromagnetic energy must decrease in proportion to the fourth power of displacement. Also, according to the Stefan-Boltzmann law, energy increases with the fourth power of absolute temperature, so the linear dimensions must vary inversely proportionately to the absolute temperature. Therefore, “the radiation energy of a certain wavelength varies with changing temperature so that the product of temperature and wavelength remains constant.”
• As a consequence of Wien displacement law, the color of a star is determined by its temperature. This means that the maximum of solar radiation is in the visible range of wavelengths.
• Calculating of the wavelength of X-rays, which are produced by the impact of electrons on solid bodies. The electric energy of these rays would be a measure of their temperature. By applying the displacement law the wavelength of the maximum of the intensity will indicate a wavelength range of X-rays.
• Thermodynamics is not enough to determine the distribution of the intensity of radiation in terms of individual wavelengths; therefore, the mechanism of the radiation process requires a statistical approach. One possibility is to consider that radiations result from gas molecules moving according to laws of probability. Another possibility is to assume that when electrons generate radiation on striking molecules, they follow a velocity distribution satisfying Maxwell’s law. Anyhow the resulting equation deviates from experimental results for large wavelengths.
• Rayleigh proposed a different approach by applying to the radiation problem the equipartition theorem according to which all the degrees of freedom in a system in equilibrium have the same contribution to the kinetic energy. He calculated the number of free electromagnetic vibration modes per unit volume in the cavity and per unit wavelength. The resulting law indicates that the emission of radiation of a given wavelength is directly proportional to the absolute temperature, and inversely proportional to the fourth power of the wavelength. Measurements of the spectral emission of black bodies revealed that the emission agreed with Rayleigh's calculation at low frequencies but diverged at high frequencies.
Explanation of new results.¶
• Planck´s hypothesis about the quantum of energy assumes that energy is not infinitely divisible but can only be distributed in discrete quantities which cannot be divided further. Planck´s radiation equation accommodates all observed data; it includes the Rayleigh and Wien laws as limiting cases; Rayleigh’s law is satisfied for very long wavelengths and Wien´s law for short wavelengths.
Possibilities for further understanding.¶
• Experimental values of the specific heats in solids do not obey the Dulong-Petit law which assumes that the solid is set of classical oscillators. Even if the energy of these oscillators is quantized, as Einstein proposed, the model is not valid at very low temperatures. Although the theory of specific heats is not derived from the radiation formula, Planck radiation theory provides the first step to the theory of specific heats.
• Einstein considered the fluctuations to which radiation is continuously subjected due to irregularities of the thermal processes. If both sides of a small plate placed inside a cavity filled with radiation are subjected to the radiation pressure, the irregularities of the radiation will produce irregular movements in the plate that can be calculated in probabilistic terms. Two terms contribute to the fluctuations depending on the density of radiation energy: for higher densities one term corresponds to radiation that obeys Rayleigh’s law and when the density is low the second term corresponds to radiation that obeys Wien´s law.
• Sommerfeld attempted to invest the universal constant h of the Planck radiation theory with physical significance: the constant h expresses a universal characteristic of the atoms. By determining how the period in which an electron entering the atom comes to a stop is a function of its velocity, Sommerfeld calculated the wavelength of X-rays with the aid of electromagnetic theory.
• The problem of thermal radiation still requires a complete solution. …. “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.”
Planck´s Nobel Lecture The Genesis and Present State of Development of the Quantum Theory.¶
Description of relevant facts.¶
• Purpose of the Lecture: “to describe the story of the origin and the development of the quantum theory and consider its present-day significance for physics”.
• The study of the physical quantum of action began twenty years ago when Planck tried to solve the problem of the distribution of energy in the normal spectrum of radiating heat.
• According to Kirchhoff, the state of the heat radiation which takes place in a cavity bounded by any emitting and absorbing substances of uniform temperature is entirely independent upon the nature of the substances.
Analysis of critical concepts.¶
• Planck considered the laws of emission and absorption of linear resonators as those proposed by Hertz. When such oscillators are inside a cavity surrounded by a sphere of reflecting walls, they exchange their energy by emitting and absorbing electromagnetic waves. The black-body radiation is the produced stationary radiation corresponding to Kirchhoff’s Law.
Proposition of significant relationships.¶
• There is a general connection, independent on the nature of the resonator, between the energy of a resonator and the energy radiation of the corresponding spectral region in the surrounding field under conditions of stationary energy exchange. The resonators exercise an irreversible effect upon the energy in the surrounding radiation field.
Solution of challenging questions.¶
• Planck considered the entropy of the resonator and its energy, not the temperature. He used existing experimental results: the proof of the energy distribution law established by Wien that was effective for small values of the energy and for short waves.
• Discrepancies with experimental results: for small values of the energy and for short waves the Wien’s law was satisfactorily confirmed but not the case for larger wavelengths.
• A new proposed interpolation radiation formula contained two terms, one of the first power of the energy predominant for small energies and one of the second power of the energy, predominant for larger energies.
• According to Boltzmann the entropy is a measure for physical probability. In Nature entropy and energy are not measured as absolute quantities but as differences; however, for a description in terms of probabilities, quantities defined as absolute values must be considered and two additive constants are required. In connection with the energy distribution of the resonator, the first constant is related to the definition of temperature which corresponds to the average kinetic energy of a molecule in an ideal gas. Corresponding to entropy, the second constant represents action, the product of energy and time.
Explanation of new results.¶
• The quantum of action appeared suitable for explaining experiments involving the action of light. Planck refers to the following phenomena that Einstein explained by considering the quantum of action as a general characteristic of electromagnetic radiation not only an assumption describing the black body radiation: the diffusion of spherical particles through a liquid as described by Stokes´Law, the electron emission in the photoelectric effect, the gas ionization produced by ultraviolet light, and the specific heat of solids where the energy of the solid was identified with the energy of a system of resonators.
Possibilities for further understanding.¶
• Calculation of the specific heat of gases by considering that, in addition to the quantized vibrational motion of the molecules, their rotational energy is also quantized. Furthermore, as affinity properties of a substance are determined by its entropy, the quantum-theoretical calculation of the entropy opens up the way to study chemical relationships.
• When an electron impacts a neutral atom, it causes emission of a light quantum or photon. This is a reverse process to that of electron emission through irradiation by quantized light as in Röntgen or gamma rays.
• The quantum hypothesis is fundamental in the interpretation of spectra and in the derivation of formula describing spectroscopic series. It has been applied in atomic systems with several degrees of freedom and when the variability of the inertial mass is considered according with relativity theory. It is also critical in the calculations of the fine structure of spectra and in the splitting of spectral lines produced by external perturbations.
• The same value of the quantum of action (\(6.52 \times 10^{-27}\) erg sec) has been obtained from different process, meaning that it is a universal physical constant. Nevertheless, the introduction of the quantum of action into classical theories makes difficult to understand that the electron only moves in quantized circular orbits without radiating energy. The old framework for understanding must change.
• The energy of a photon after complete emission could serve for its propagation both in a form described as a boundless field capable of producing wave like interferences or like a punctual projectile being able to concentrate energy in a small space region.
REFERENCES¶
KUHN, T. The Structure of Scientific Revolutions. University of Chicago Press. (1962).