PLANCK’S CONSTANT AND THE ENERGY OF RADIATION

By: Clarence L. Dulaney

2226 Fairgreen Drive

Missouri City, TX 77489

e-mail cldtx1@sbcglobal.net                                 

 

Abstract: To explain the radiation from "black bodies" M. Planck developed the theory that light was emitted discontinuously. From this work came the constant "h" which turned out to be a universal constant in that, for atomic or molecular scale processes involving a frequency, the energy therof is hf where f is the frequency. The photoelectric, and Compton effects are discussed, along with Debye’s theory of specific heats of monatomic solids.

THE "COMPLETE RADIATOR"

A cavity with a small hole in it that is heated (usually to incandescence) approximates a "complete radiator" 1, commonly called a "black body". A perfect black body absorbs all the radiation that falls on it.

Several physicists around the turn of the century had proposed formulas for calculating the properties of a black body. Wien in 1896 made the closest approximation to the spectral radiancy at various wavelengths, Rl. Wien assumed that the light of the black body could be compared to the particles considered by the statistics of Maxwell and Boltzmann. Here, the energies of the "light-particle" were to vary continuously over the range of possible energies. His formula compared Rl at various wavelengths with the Maxwell - Boltzmann statistics. The equation he obtained was:

Rl = c1l-5 (1/ex) where x = c2/lT and T is the Kelvin Temperature

This equation works quite well for all except for the longest wavelengths for which it diverges slightly.

Max Planck in 1900 made an empirical adjustment to the Wien formula in the exponential term, replacing "x" by ((c2 /lT) - 1). With this addition, the equation worked for all wavelengths.

Planck who was trained as a statistical thermodynamicist, attempted to find a theoretical explanation for the success of his simple alteration, but could only do so if the radiation took place in an intermittent fashion rather than in a continuous, random manner as had been assumed by Wien. The energy of each of the "light-particles" had to be independent of all the others.

The energy of the radiation was to be taken as the frequency thereof multiplied by the constant "h", or hfi where fi is the frequency of the ith "light-particle". The constant h came to be known as Planck’s constant. This constant is equal to c2k/c where k is Boltzmann’s constant, 1.38 x 10-16 erg/°K/molecule, and c is the speed of light. (Note that the x in Wien’s expression may be written as hf/kT, since f = c/l. Thus Planck did not really invent "h".) "h" is numerically equal to

624 x 10--277erg sec.

To set the stage for the ideas of radiation in the late 19th century, it is to be noted that the electron had only been discovered in 1897, and that the electron theory of matter was yet to be developed. It was not known exactly how radiation was produced, or what it even was , there being competing theories, first of corpuscular radiation due to Newton primarily and secondly the wave theory of Huygens et al. At this point the wave theory had pretty well prevailed. (Note that even today there is no real explanation of how radiation actually occurs, or whether it is corpuscular or wavelike, or both.) At any rate Planck’s explanation was even hard for him to accept, and harder for his colleagues. It was not until Einstein’s photoelectric explanation was made that much credence was given to Planck’s ideas.

In "The Stationary Hydrogen Atom"22, it is explained how the line spectrum of Hydrogen is generated with the energy of the lines being equal to the difference in energy of successive "orbits".  It is postulated that all spectra are similar, all being made up of individual lines.  The lines each have their own frequency.  This explains why the energy of the "light particles" are all different .(See also the paper "Why’C’?"3, for an explanation of how wavelets may be carried by neutrinos.  Thus, light really does perform like a wave, and in some instances like a particle.)

A UNIVERSAL CONSTANT

Subsequent research has shown that for any atomic scale process involving a frequency, the energy involved is "h" times the frequency, indicating that "h" is truly a universal constant. Applications include, but are not limited to, the photoelectric effect, the Compton effect, and the specific heat of solids.

THE PHOTOELECTRIC EFFECT

The photoelectric effect was discovered by Hertz in 18874,5 He found that an electric spark had more energy if the electrodes were irradiated with ultraviolet light. He determined that the spark was initiated at the negative electrode (indicating that the ultraviolet light enhanced the negative charge on the electrode).

A number of investigators then conducted experiments on the phenomenon. Based on a number of ingenious experiments they found:

1. The particles involved are electrons.

2. The number of electrons is proportional to the illumination per unit area.

3. For each frequency there is a maximum velocity.

4. Maximum electron velocity increases with increasing frequency.

5. The condition of the surface has a marked effect on the frequency response and on the          total number of electrons produced.

6. If the surface has at least a slight negative charge, production of electrons is virtually instantaneous.

In 1905 A. Einstein6 propounded his famous photoelectric theory and equation, based on Planck’s hypothesis: ½mv2 = hf -hf0 where m and v refer to the electron and the f’s to the light, with f0 being the "threshold" frequency, below which no action occurs.  (Note particularly that the electrons do not reach "relativistic" velocities.)  He also, to account for the "instantaneous" production of electrons, proposed that light travels as bundles of energy, particles of light, which he called photons. He reasoned that it would take too long for light waves, which  he thought to spread out over the whole surface, to impart enough energy to a single electron to cause its eviction form the metal surface. According to Einstein, it required a particle to knock the electron out immediately. (However, note that Hull7, points out that an electron cannot absorb all the energy of a photon if energy and momentum are conserved.)

(Note that experiments on electron emissions from surfaces irradiated by high intensity laser pulses by Panarella8 and others indicate "photo-emission" with light of substantially lower frequencies than f0. Although Panarella has a "photon enhancement" explanation, it is believed by this author that the effect is thermal emission.)

AN ALTERNATIVE EXPLANATION

Most of the definitive studies on photoelectricity were done on alkali metals which have very slight control of their valence electrons. Witness their extreme chemical reactivity. Consider Cs which has an ionization energy of only 3.394 ev. This is equalled with light of 3659 Å which is in the visible range. Light of this or higher frequency would tend to make the Cs surface negative (Hertz, above) and thus allow electrons to leave the surface readily. Furthermore, as long as light of this frequency was maintained, the negative potential would exist even as electrons are leaving the surface.

The maximum energy of the electrons would be obtained when the surface potential is most negative, ie when the frequency is highest. Increasing the intensity of the light would simply increase the number of electrons produced, not their energy. Thus, it would not require a "bullet" of light to cause the photoelectric effect.

 

THE COMPTON EFFECT

A. H. Compton9 and others studied the scattering of x-rays by matter. For x-rays of 1 Å or greater, or less than 0.4 Å wavelength, the scattering followed the classical theory. That is, electrons were knocked straight out of paraffin wax or the like, with no deflection or alteration of the x-rays.

With wavelengths between 0.4 and 1.0 Å, a different type of scattering occurs. Compton found that when x-rays of this wavelength range were scattered by paraffin, electrons were expelled at an angle to the direction of the x-ray, and the wavelength of the x-ray was increased by 0.024 x (1 - cosf) Å, where f is the angle of deflection of the ray9 from the original path10.( Reference 12 indicates the mimimum effective diameter of a "photon" is l/p.)

In this case, the x-ray acts like a particle, and since the electron has a diameter of about 1 Å (See reference 11), the two interact, with energy being transferred from the x-ray to the electron.  If the electron is scattered at a right angle, this amounts to 0.042 increase in wavelength.  This is the difference from the photoelectric case. 

Compton calculates12 the energies involved, based on STR . as:    h(f-f0) = m0c2((1/(1-b2)½) -1),  where f is the final frequency, and m0 is the "rest" mass of the electron.  He did not calculate the speed of the electron, just assumed it would be "relativistic".  Actually, the velocity comes out  2.49 x 109 cm/sec which is less than 0.1c  Even calculated by    h(f-f0) = ½mv2,  v comes out 2.62 x 109  which again is not relativistic.  Calculations will be given on request..

At any rate, the energy is dependent on the frequency of the x-ray..

SPECIFIC HEAT OF SOLIDS

P. Debye14 developed a theory describing specific heats of monatomic solids by assuming that solids act as isotropic, elastic media which vibrate at characteristic frequencies with energies hf. He was able to develop the concept of a "characteristic temperature" q = hfm/k, where fm is the maximum frequency for the particular solid, and k is Boltzmann’s constant.

Debye plotted the specific heat at constant volume, Cv vs T/q for a considerable number of monatomic solids, and found that all fell on the same straight line to a high degree of approximation. Considering the very simple model, the agreement was excellent, and explained anomalous results for carbon and other solids.

Born and von Karman15 extended Debye’s work to nonisotropic crystals.

SUMMARY

It appears Planck’s constant "h" is a universal constant for atomic and molecular scale processes involving a frequency, in that the energy concerned with the process is hf, where f is the frequency involved.

Because light theory was in its infancy when Planck proposed his theory, it seemed as if energy was produced in packets, and this concept was abhorrent to physicists at the time. Radiation from atoms is produced in wavelets from electronic transitions between various energy levels in atoms, and these "lines" are distinct for each atom.

. As is shown above, it is not necessary to invoke photons to explain the photoelectric effect, rather the light has enough energy to keep the metal surface potential negative.

In the Compton Effect, the x-ray neutrino does act like a particle, which loses energy in a collision with an electron which has a similar diameter.

Among other applications, that of the prediction of specific heats by assigning a frequency and vibrational energy hf to crystalline solids is a particularly striking application of Planck’s hypothesis.

Bibliography

[1] T. Osgood, et al.,"Atoms, Radiation and Nuclei", John Wiley & Sons, NY, (1955),p68ff.

[2] C. Dulaney ,"The Stationary Hydrogen Atom"

[3] C. Dulaney,”Why ‘C’/"

[4] T. Osgood, op. cit. P77ff.

[5] G. Hull, "Elementary Modern Physics", Macmillan Co., NY, (1949), p57ff.

[6] A. Einstein, Ann. der Physik, 17, 132, !!905)

[7] G. Hull, op cit.,p67 and p130

[8] E. Panarella in "Quantum Uncertanties" edited by W. Honig, D. Kraft and E. Panarella, Plenum, NY, (1989) p-237ff

[9] A. H. Compton, Phys. Rev. 21, 483, (1923)

[10] H. Semat, "Introduction to Atomic Physics" , Rinehart & Co., NY, (1946), p141ff.

[11] C. Dulaney, "What is an Atom?"

[12] G. Hunter and R. Wadlinger in "Quantum Uncertainties". edited by W. Honig, D. Kraft and E. Panarella, Plenum, NY, (1989) p-331ff

[13] H. Semat, op cit., p397ff.

[14] P. Debye, Ann. der Physik, 39, 789, (1912). .

[15] M. Born and T. von Karman, Phys. Zeits. 13, 297 (1912) and 14, 65, (1913).

 All my papers may be found at:  http://mywebpage.netscape.com/clarencedulaney

 © 8/15/2006   Clarence L. Dulaney