Planck's Law Research Articles

Planck's Radiation Formula: a Correction

SINCE passing the proofs of my letter which appeared in Nature of April 20* I have found, as a result of a conversation with Dr. G. J. Whitrow, that the conditions prescribed for the radiation formula admit of a more general solution than that of Planck. Any function of I¾ = I½In-3/4 which satisfies the equation meets those conditions. It may be that dimensional or other considerations will be found to impose somewhat severe limits on the form of the function, and it would be possible to invoke the (purely macroscopic and continuous) Wien's law to limit n to the value 1/2, but at the present stage the possibilities appear to be too numerous to justify the statement that Planck's law has been deduced on macroscopic grounds. This statement must therefore be withdrawn.

Freezing temperature of irons and steels

This chapter starts with the historical study of blackbody radiometry. It includes the investigation of the Stefan–Boltzmann constant and the verification of Planck's law. The features of early blackbodies are described. In recent years, blackbodies covering a higher temperature range have come into use, primarily to calibrate secondary energy sources spectrally, over the range 0.25–2.6 μm, and spectroscopic or radiometric instruments over a wavelength range produced by the higher temperature. Arising from the design and technique in operating a blackbody as an absolute calibration source are several possible sources of error. The first four errors associated with the blackbody are as follows: (1) differences in temperature scales, (2) value of the Stefan–Boltzmann constant (σ), (3) nonuniform temperature within the radiating cavity, and (4) apparent emissivity of the blackbody. Originating in the source-detector calibration setup are errors from stray radiation, geometry of source-detector system, and atmospheric attenuation of radiation.

The Interpretation of ε Aurigae

view Abstract Citations (53) References (5) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Interpretation of ɛ Aurigae Kuiper, G. P. ; Struve, O. ; Strömgren, B. Abstract The light-curve of this eclipsing binary is rediscussed with the help of the spectro- graphic elements determined at the Yerkes Observatory. The ratio of the radii and the mass ratio are determined with the assumption that the visible component falls upon the empirical mass-luminosity-curve. The orbital inclination is found to be about 70°. The spectrum is formed by the smaller, F2 star. The larger component has a temperature of about i 2000_14000 and gives no appreciable light in the region covered by the spectroscopic observations. The spectroscopic and photometric observations duriiig the eclipse show that the infrared star is semitransparent and that its nonselective opacity for visual and photo- graphic light is concentrated in an outer shell. It is suggested that this effect is produced by photo-electric ionization from the F2 star. The spectral lines of the F2 star are visible throughout the total phase of the eclipse. In addition, there are lines produced in the ionized shell of the infrared star. The latter are displaced by the rotation of this component, and the displacements and intensities are in accord with the hypothesis. The close proximity, before mid-eclipse, of the ro- tationally displaced lines of the infrared star to the normal lines of the F2 star results in a relatively small increase of the equivalent breadth. After mid-eclipse, when the two lines are better separated, the increase in the equivalent breadth is more pronounced. The equatorial velocity of rotation of the infrared star is approximately ~o km/sec. The problem of the formation of an absorbing layer in the atmosphere of the infrared component by the radiation of the other component is investigated. Relations giving the extent of the region of the infrared star ionized by the outside radiation are derived. The optical thickness of the ionized region in visual and photographic light is calcu- lated as a function of the maximum density along the ray traversing the atmosphere. An approximately constant optical thickness is found for a r~nge of maximum density sufficiently wide to account for the observed constant minimum. A considerable excess of the amount of ionizing radiation over that calculated from Planck's law is required in order to give an optical thickness equal to that observed. The opacity of the non- ionized region of the infrared component between the ionized region considered and the observer is calculated. The effect of this opacity is found to be suficiently small, pro- vided the relative hydrogen content of the atmosphere of the infrared component is below a certain limit. Some aspects of the problem of line absorption in the atmosphere of the cool companion are considered. The results obtained are summarized, and the possible importance of a source of opacity other than the electron scattering and photo- electric transition opacity is briefly discussed Publication: The Astrophysical Journal Pub Date: December 1937 DOI: 10.1086/143888 Bibcode: 1937ApJ....86..570K full text sources ADS | data products SIMBAD (1)

CVI.On the relation of electronic waves to light quanta and to Planck's law

CVI.<i>On the relation of electronic waves to light quanta and to Planck's law</i>

A Generalized Form of the New Statistics

Instead of assuming the probabilities of absorption and emission of atoms in a radiation field as equal to ${B}_{a}\ensuremath{\rho}$ and $A+{B}_{e}\ensuremath{\rho}$ (the well-known Einstein equations) Strum has in a recent paper assumed them to be respectively equal to ${B}_{a}\ensuremath{\rho}+{C}_{a}{\ensuremath{\rho}}^{2}+\ensuremath{\cdots}$ and $A+{B}_{e}\ensuremath{\rho}+{C}_{e}{\ensuremath{\rho}}^{2}+\ensuremath{\cdots}$. This leads to a modification of Planck's law. It is shown in this paper that this assumption effectively amounts to replacing the new statistical law of interaction of Nordheim e.g. $f\left(1\ifmmode\pm\else\textpm\fi{}\frac{{f}^{*}}{A}\right){{f}_{1}}^{*}\left(1\ifmmode\pm\else\textpm\fi{}\frac{{{f}_{1}}^{*}}{{A}_{1}}\right)$ by the more general $(f+a{f}^{2}+\ensuremath{\cdots})\left(1\ifmmode\pm\else\textpm\fi{}\frac{{f}^{*}}{A}\ifmmode\pm\else\textpm\fi{}\frac{a{f}^{*2}}{A}\ifmmode\pm\else\textpm\fi{}\ensuremath{\cdots}\right)({f}_{1}+a{{f}_{1}}^{2}+\ensuremath{\cdots})\left(1\ifmmode\pm\else\textpm\fi{}\frac{{{f}_{1}}^{*}}{{A}_{1}}\ifmmode\pm\else\textpm\fi{}\frac{a{{f}_{1}}^{*2}}{{A}_{1}}\ifmmode\pm\else\textpm\fi{}\ensuremath{\cdots}\right)$ where $f$, ${f}_{1}$, ${f}^{*}$, ${{f}_{1}}^{*}$ are the distribution functions for the projectiles corresponding to the numbers of each before and after collision; $A$, ${A}_{1}$ the number of cells respectively associated with them. On this basis a general $H$ function defined by $H=\ensuremath{\int}(f+a{f}^{2}+\ensuremath{\cdots}) log(f+a{f}^{2}+\ensuremath{\cdots})\ensuremath{\mp}(A\ifmmode\pm\else\textpm\fi{}f\ifmmode\pm\else\textpm\fi{}a{f}^{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\cdots})log(A\ifmmode\pm\else\textpm\fi{}f\ifmmode\pm\else\textpm\fi{}a{f}^{2}\ifmmode\pm\else\textpm\fi{}\ensuremath{\cdots})$ is considered. As a consequence of this we get a generalized distribution function given by the relation $f+a{f}^{2}+b{f}^{3}+\ensuremath{\cdots}=\frac{A}{C\mathrm{exp}[c{|v|}^{2}]\ensuremath{\mp}1}$ If we take only two terms we get approximately $J=\frac{A}{C{e}^{c|v|2}\ensuremath{\mp}1}\ensuremath{-}\frac{{A}^{2}a}{{(C{e}^{c|v|2}\ensuremath{\mp}1)}^{2}}=\frac{A}{C\mathrm{exp}[c{|v|}^{2}]\ensuremath{\mp}1}\ensuremath{-}\frac{{A}^{2}a}{{(C\mathrm{exp}[c{|v|}^{2}]\ensuremath{\mp}1)}^{2}}.$ By substituting the appropriate values for $A$ in the cases corresponding to the photons and the atoms, we get Strum's modified radiation formula, and a new distribution function for a gas. Thus as a result of a generalized analysis of the $H$-theorem we get a new statistics the successive approximations of which are a more accurate statistics which includes Strum's formula, the statistics of Bose-Einstein, Fermi-Dirac, the classical statistics of Maxwell-Boltzmann. It is suggested that this new statistics may be of importance in the study of dense matter at high temperature.

The melting point of palladium

The melting point of palladium is a convenient reference point for the measurement of high temperatures. In fixing a scale of temperature the aim is, of course, to approximate as closely as possible to the thermodynamic scale. From the absolute zero up to moderately high temperatures this ideal scale is realised most directly and accurately through the medium of the gas thermometer. However, with increase of temperature beyond a certain limit, the experimental difficulties of gas thermometry multiply rapidly, so that ultimately it becomes necessary to adopt another basis for obtaining the scale. This is conveniently found in the laws governing the radiation from a black body, which have a sound theoretical foundation and permit the use of measuring instruments of precision. The establishment of a practical scale of temperature on the lines above indicated has been the subject of considerable discussion between the national standardising laboratories of Germany, Great Britain and the United States of America. As a result, proposals for the definition of an "International Temperature Scale" were submitted to the 7th General Conference of Weights and Measures, and approved by them. In effect, the basis of the scale up to the melting point of gold is the gas thermometer, and beyond this temperature the Wien or Planck law of radiation with an agreed value for the constant c 2 . Owing to the difficulty of absolute measurements of radiation, no attempt has so far been made to place the radiation scale on an independent basis by fixing the other constant in the Wien or Planck equation. Consequently the scale is defined, for the present, relatively to a fixed point on the thermodynamic scale, as given by the gas thermometer, namely the melting point of gold (1063°C.).

The Fermi Statistical Postulate; Examination of the Evidence in Its Favor

In 1924 Bose,1 undertaking to deal statistically with light quanta, hit upon a method of using generalized co6rdinates which enabled him to derive the Planck law of black-body radiation without making use, as others had done, of conceptions taken from the classical mechanics. Einstein,2 adopting the Bose statistical method, applied it to the discussion of quantization of energy in an ideal monatomic gas, and the consequent degeneration [Entartung] of the gas-that is, the degree of its departure from the classical gas-laws. He reached the following conclusion: The mean energy of the gas molecules (as well as the pressure) at the temperature is therefore always less than the classical value. In 1926 Fermi3 attacked the same problem that Einstein had discussed. He used the same general statistical method that Bose had invented but introduced in its application a certain limiting assumption, with the following explanation: Now a short time ago the rule was proposed by Pauli4 . . . that when in an atom an electron exists the quantum numbers of which (the magnetic quantum number included) have definite values, there can be in the atom no other electron whose path is characterized by the same numbers. In other words, a quantum orbit (in an external magnetic field) is already completely occupied by a single electron. Since this rule of Pauli has proved extremely fruitful in the explanation of spectroscopic facts, we propose to inquire whether it is not also of some use for the problem of the quantization of ideal gases. We shall therefore in what follows assume that at most one molecule with given quantum numbers can exist in our gas; as quantum numbers not only those here come into consideration which belong to the internal motions of the molecule but also those which determine its motion of translation.

Maxwell's Law, and the Absorption and Emission of Radiation

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .

LXXXIV. On the derivation of planck's law from einstein's equation

LXXXIV. <i>On the derivation of planck's law from einstein's equation</i>

Influence of Radiation on lonisation Equilibrium

IN a recent letter (NATURE, March 14, p. 377) Saha and Swe develop the theory of the stationary state of a medium traversed by radiation of a temperature different from that belonging to the medium. Their argument seems to follow thermodynamic lines; supposing this to be the case, it may be of interest to point out how the same results may be obtained starting from Einstein's classical paper on Planck's radiation law (Phys. Zeit., 18, p. 121, 1917) and from Milne's extension to the photo-electric effect (Phil. Mag., 47, p. 209). The principle of this alternative method, already well known (Milne in this way recently has derived a formula equivalent to formula (3) of Saha and Swe, in his investigation of the equilibrium of a Ca+ chromosphere, Mon. Not. R.A.S. 85, p. 119), consists in formulating the conditions required for a stationary state by means of probability laws governing the elementary processes involved. These probability laws themselves have been derived from the consideration that in thermodynamic equilibrium the number of elementary processes in one direction ought to balance the number occurring in the opposite direction.

Colour Temperature and Brightness of Moonlight

OUR more complete knowledge of full or black-body radiation embodied in Planck's law makes it possible to speak of the temperature of radiation as well as the temperature of radiating bodies. Thus, the temperature of any visible radiation is the temperature to which a black body must be raised to emit light as nearly as possible of the same integral colour or quality as that of the radiation in question.

XXVIII. A derivation of Planck's law of radiation by means of the adiabatic hypothesis

(1923). XXVIII. A derivation of Planck's law of radiation by means of the adiabatic hypothesis. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science: Vol. 45, No. 266, pp. 300-303.

XXXVI. On the origin of spectra and Planck's law

XXXVI. <i>On the origin of spectra and Planck's law</i>

Wheatstone bridges and accessory apparatus for resistance thermometry

This chapter provides an overview of blackbody radiation. All objects that have a temperature at any value other than absolute zero continuously emit and absorb radiation. The radiation characteristics of certain surfaces are completely specified if the temperature is known. These surfaces radiate continuously through the optical spectrum and are known as ideal thermal radiators or blackbodies. Blackbody radiation is described by Planck's equation. Slide rules are available that provide rapid calculation of blackbody quantities with good accuracy. The Planck radiation formula shows that the spectrum of the radiation shifts toward shorter wavelengths as the temperature of the radiator is increased. The derivative of the Planck equation with respect to wavelength yields the Wien displacement law that gives the wavelength for which maximum radiation occurs for a given temperature. The total power radiated per unit area of a blackbody is obtained by integrating the Planck's radiation law over all wavelengths. Two well-known approximations to Planck's law are readily obtained: Rayleigh-Jeans' law and Wien's radiation law. The chapter also highlights emissivity, Kirchhoff's law, and Lambert's cosine law. Emissivity of a surface is a function of wavelength, temperature, and direction. Lambert's cosine law signifies that the amount of energy in a given solid angle varies in proportion to the cosine of the angle between the direction in question and the normal to the surface.

XXIV. A method of deriving Planck's law of radiation

XXIV. <i>A method of deriving Planck's law of radiation</i>

A new relation derived from Planck's law

A new relation derived from Planck's law

A New Relation Derived from Planck's Law

A New Relation Derived from Planck's Law

A new relation from Planck's law

A new relation from Planck's law

Societies and Academies

Societies and Academies